Dynamical Control of Many-Body Interactions in Driven Quantum Matter Citation Yang, Christopher Kai-Chen (2025) Dynamical Control of Many-Body Interactions in Driven Quantum Matter. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/fh52-tw61. https://resolver.caltech.edu/CaltechTHESIS:04112025-180844471 Abstract Strongly driven Floquet systems have emerged as promising platforms for exotic non-equilibrium physics, but their instability to heating motivates practical questions about how Floquet engineering can be useful. Although drive-induced heating is often attributed to interactions, this thesis adopts a different perspective, identifying regimes where dissipative many-body dynamics can stabilize Floquet physics and define remarkable new drive-tunable properties. This principle enables highly tunable many-body steady states with minimal heating, leading to a novel regime where drive control over single-particle Floquet states can extend to many-body interactions. Our theoretical and experimental results in Parts II and III center around two themes. The first theme focuses on discovering controllable and stable many-body Floquet states. The second explores further into what the future holds--envisioning the prospects for unconventional Floquet physics with nontraditional driving fields and three-dimensional materials. Part II of this thesis leverages kinematic constraints on low-dimensional many-body scattering as new principles for tuning and stabilizing Floquet phases. First, we predict that a circularly polarized laser can drive slow electrons of moiré systems into a subsonic regime where they decouple from the intrinsic 2D acoustic phonons of the system. This "slow-electron regime" enables optical control over the steady-state occupation of topological Floquet states and the resulting anomalous Hall conductivity. Second, we present experimental transport signatures of steady Floquet physics in graphene irradiated by a continuous-wave laser. Our experiment, performed at 3-4 K lattice temperatures with lasers off-resonant to optical phonons, creates electron-phonon scattering bottlenecks that stabilize persistent low-temperature phases with light-induced longitudinal transport characteristics. The long-lived many-body phase represents the first experimental signatures of steady Floquet physics in a metallic solid. Part III presents emerging opportunities for many-body Floquet engineering beyond traditional optically-driven, low-dimensional materials. We first explore beyond-optical driving fields, revealing the emergence of quantized charge transport in 1D systems driven by coherent phonons. Incoherent phonons relax electrons into a topological spatiotemporal Floquet state with quantized group velocity set by the coherent phonon, realizing topological charge pumping in a highly non-adiabatic setting. Finally, we address the topological effects of time-periodic drives beyond low-dimensional systems, revealing that THz-frequency, circularly polarized light can induce topological chiral plasmons in Weyl semimetals with band anisotropy, broken time-reversal symmetry, and broken inversion symmetry. The theoretical and experimental work in this thesis represent key progress towards realizing persistent Floquet physics for diverse applications in quantum device engineering. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Condensed matter theory, Floquet engineering, topological quantum matter, light-matter interactions Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Refael, Gil Thesis Committee: Hsieh, David (chair) Refael, Gil Alicea, Jason F. Motrunich, Olexei I. Defense Date: 14 May 2025 Non-Caltech Author Email: christopherkcyang (AT) gmail.com Record Number: CaltechTHESIS:04112025-180844471 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04112025-180844471 DOI: 10.7907/fh52-tw61 Related URLs: URL URL Type Description https://doi.org/10.1103/PhysRevLett.131.026901 DOI Article adapted for Chapter 5 https://doi.org/10.1038/s41467-025-57335-2 DOI Article adapted for Chapter 6 https://doi.org/10.1103/PhysRevLett.133.226301 DOI Article adapted for Chapter 7 ORCID: Author ORCID Yang, Christopher Kai-Chen 0000-0002-9462-9074 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17150 Collection: CaltechTHESIS Deposited By: Christopher Yang Deposited On: 30 May 2025 23:36 Last Modified: 06 Jun 2025 22:15 Thesis Files PDF - Final Version See Usage Policy. 20MB Repository Staff Only: item control page

Dynamical Control of Many-Body Interactions in Driven Quantum Matter Thesis by Christopher Kai-Chen Yang In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2025 Defended May 14th, 2025 ii © 2025 Christopher Kai-Chen Yang ORCID: 0000-0002-9462-9074 All rights reserved iii ACKNOWLEDGEMENTS I would like to thank my advisor, Gil Refael, for his guidance and encouragement on this journey into the quantum world of electrons. Gil has a remarkable ability to explain quantum physics from a bird’s eye view—without compromising on any precision or accuracy. From his unique perspective of the quantum world, Gil taught me to look further into the horizons of physics, identify the boundaries of our knowledge, and establish unlikely connections with seemingly distant research topics. Most importantly, Gil taught me that research was a responsibility in the greater context of things: a commitment to guide the broader scientific community— and the general public—with honest results rooted in a fundamental respect for scientific integrity. Caltech’s Bridge Hall of Physics is an inspiring place where numerous brilliant physicists have visited and worked over the years. I thank Jason Alicea, Lesik Motrunich, and David Hsieh for always encouraging me, sharing their wisdom during daily condensed matter lunch meetings, and providing me with concrete advice on my research. I’ve also had the great pleasure of working with former Caltech postdocs Cyprian Lewandowski, Iliya Esin, and Frederik Nathan. Cyprian was my first postdoctoral mentor at Caltech. He always encouraged me to take the lead in meetings, even when he understood the research topics better than I did. He taught me how to navigate the academic world and build a network in the community. During my nearly annual trips to Tallahassee, Florida, Cyprian and his wife, Marilena, would invite me to their home for board games and dinner. I thank him for teaching me much of what I know about solid-state systems and for making my PhD journey possible. He is a true friend and mentor. I feel deeply fortunate to have worked with Iliya over the past five years. I increasingly find that my approach to physics and research is influenced by Iliya’s example as a scientist. From my perspective, Iliya’s research abilities are analogous to conducting an orchestra—he has a remarkable intuition for what details of which components to tune and perfect, an effort that pays off when the components play together in a beautiful symphony. His commitment to scientific rigor complements his enthusiasm for quantum dynamics, which is always grounded in solid factual evidence. Iliya also taught me how to communicate science effectively, patiently leaving hundreds of suggestions on my early drafts and referee replies, even though iv I knew it would’ve been easier for him to edit the paper directly. I am particularly grateful for his belief in me throughout my PhD—for reminding me to celebrate small progress and focus on putting forth solid results, even if they may not seem to lead anywhere at first. I thank him for teaching me how to produce quality work as a scientist. Frederik has been an equally exceptional mentor. Despite living thousands of miles away and across multiple time zones, he involved me in his projects on Weyl semimetals, for which I am deeply grateful. Frederik has a remarkable ability to simplify complex results, and he’d often predict results that would typically require extensive calculations. His excitement for research is contagious, inspiring me to think visually about electronic physics. I would also like to thank the condensed matter theory group at Caltech, including the members of the institute I’ve had the pleasure of knowing: Loly Ekmekjian, Liam O’Brien, Andrew Ivanov, Swati Chaudhary, Nandu Manoj, Sara Vanovac, Yue Liu, Yinan Chen, Stephen Naus, Valerio Peri, Étienne Lantagne-Hurtubise, Gal Shavit, Federica Surace, and others. The greatest gift they gave me was their genuine deep interest in my research. Beyond Caltech, I feel grateful for the quantum experts with whom I’ve interacted. I remember, in particular, meeting Mikael Rechtsman during one of Caltech’s colloquium lunches at the Athenaeum. Mikael asked me why I hadn’t attempted to realize my theoretical work experimentally and offered some new angles to pursue. In retrospect, his encouragement played an important role at the start of my first theory-experiment collaboration a few years later. Speaking of the experimental collaboration, I would like to thank Yijing Liu and the Paola Barbara and Nikolai Kalugin groups for the privilege of working with them on experimental Floquet physics. I met Yijing by chance at the APS exhibition hall in Minnesota in 2024, having previously attended his talk. The resulting collaboration has been a true delight. I am grateful to the Georgetown University and New Mexico Tech groups for giving me a real-world view of the quantum world. Paola has also been incredibly supportive of my postdoc and career endeavors. Beyond quantum physics, I have many people to thank. I thank the DOE for its fellowship support during my PhD and for inspiring me early on through its internship program. I thank my early mentors—Robin Rehagen, Jay Salmonson, Christopher Young, and Aritoki Suzuki for teaching me how to become a researcher v and showing me the beauty of science on all length scales—from the galaxies in the skies to atoms at ultra-low temperatures. I also want to acknowledge my friends at Caltech, especially Jerry Zhang, who introduced me to photography. Without Jerry, I wouldn’t have discovered the beauty of California and explored its stunning landscapes and wildlife. I am thankful to Jerry for the encouragement and laughter that helped me overcome the challenging aspects of my PhD. His kindness and generosity define what it means to be a genuine friend. My parents have been the source of my strength and motivation. I thank my mom for packing my childhood days with every possible creative activity to spark my imagination. I thank her for sharing laughter and tears and ensuring I never weather the storms alone. She made great sacrifices for my career on the unconditional basis of love. I thank my dad for inspiring my interest in science. He shows his care for me in unique ways–taking me on bike trips, adding new accessories to my bike and car, and teaching me how to build and fix things. I thank my grandmother for taking care of every aspect of my life and for bravely coming to this country as a single mother. She could not speak much English, yet she navigated this country better than I ever could, working until age 77 and building a remarkable life and family for her children. I thank Tzu Chi and Dharma Master Cheng Yen for shaping how I see the world and teaching me to hold love and care for others close to my heart. I thank Mr. Joe Wang and Mrs. Shiu-Yun Tsai for taking care of me just like their own child and inspiring me to go to Tijuana for charity missions. During my near-weekly trips to Tijuana with Tzu Chi, I’ve gotten to know children living in houses made of cardboard and styrofoam. It makes me reflect on how lucky I am and how challenges make us stronger. In both Floquet physics and Buddhism, periodicity appears as a common theme. It may not be possible to comprehend the periodicity if you stand on just one trough of the wave. Nevertheless, the gift of persistence gives you a new perspective, revealing remarkable phenomena that only emerge on long timescales. vi ABSTRACT Strongly driven Floquet systems have emerged as promising platforms for exotic non-equilibrium physics, but their instability to heating motivates practical questions about how Floquet engineering can be useful. Although drive-induced heating is often attributed to interactions, this thesis adopts a different perspective, identifying regimes where dissipative many-body dynamics can stabilize Floquet physics and define remarkable new drive-tunable properties. This principle enables highly tunable many-body steady states with minimal heating, leading to a novel regime where drive control over single-particle Floquet states can extend to many-body interactions. Our theoretical and experimental results in Parts II and III center around two themes. The first theme focuses on discovering controllable and stable many-body Floquet states. The second explores further into what the future holds— envisioning the prospects for unconventional Floquet physics with nontraditional driving fields and three-dimensional materials. Part II of this thesis leverages kinematic constraints on low-dimensional manybody scattering as new principles for tuning and stabilizing Floquet phases. First, we predict that a circularly polarized laser can drive slow electrons of moiré systems into a subsonic regime where they decouple from the intrinsic 2D acoustic phonons of the system. This “slow-electron regime” enables optical control over the steady-state occupation of topological Floquet states and the resulting anomalous Hall conductivity. Second, we present experimental transport signatures of steady Floquet physics in graphene irradiated by a continuous-wave laser. Our experiment, performed at 3-4 K lattice temperatures with lasers off-resonant to optical phonons, creates electron-phonon scattering bottlenecks that stabilize persistent lowtemperature phases with light-induced longitudinal transport characteristics. The long-lived many-body phase represents the first experimental signatures of steady Floquet physics in a metallic solid. Part III presents emerging opportunities for many-body Floquet engineering beyond traditional optically-driven, low-dimensional materials. We first explore beyondoptical driving fields, revealing the emergence of quantized charge transport in 1D systems driven by coherent phonons. Incoherent phonons relax electrons into a topological spatiotemporal Floquet state with quantized group velocity set by the coherent phonon, realizing topological charge pumping in a highly non-adiabatic setting. Finally, we address the topological effects of time-periodic drives beyond vii low-dimensional systems, revealing that THz-frequency, circularly polarized light can induce topological chiral plasmons in Weyl semimetals with band anisotropy, broken time-reversal symmetry, and broken inversion symmetry. The theoretical and experimental work in this thesis represent key progress towards realizing persistent Floquet physics for diverse applications in quantum device engineering. viii PUBLISHED CONTENT AND CONTRIBUTIONS [1] Yijing Liu*, Christopher Yang*, Gabriel Gaertner, John Huckabee, Alexey V. Suslov, Gil Refael, Frederik Nathan, Cyprian Lewandowski, Luis E. F. Foa Torres, Iliya Esin, Paola Barbara, and Nikolai G. Kalugin. Signatures of floquet electronic steady states in graphene under continuous-wave mid-infrared irradiation. Nature Communications, 16(1), February 2025. ISSN 2041-1723. doi: 10.1038/s41467-025-57335-2. URL http://dx.doi.org/10.1038/ s41467-025-57335-2. *indicates equal contribution. C.K.Y. participated in the conception of the theory-experiment collaboration, carried out the steady-state Floquet simulations, and participated in the writing of the manuscript. [2] Christopher Yang, Gil Refael, and Frederik Nathan. Chiral plasmons and topological photocurrent in weyl semimetals. in preparation. This work is in preparation. C.K.Y. participated in the conception of the project, carried out the analytical and numerical calculations, and participated in the writing of the manuscript. [3] Christopher Yang, Iliya Esin, Cyprian Lewandowski, and Gil Refael. Optical control of slow topological electrons in moiré systems. Phys. Rev. Lett., 131: 026901, Jul 2023. doi: 10.1103/PhysRevLett.131.026901. URL https:// link.aps.org/doi/10.1103/PhysRevLett.131.026901. C.K.Y. participated in the conception of the project, carried out the analytical and numerical calculations, and participated in the writing of the manuscript. [4] Christopher Yang, Will Hunt, Gil Refael, and Iliya Esin. Quantized acoustoelectric floquet effect in quantum nanowires. Phys. Rev. Lett., 133:226301, Nov 2024. doi: 10.1103/PhysRevLett.133.226301. URL https://link.aps. org/doi/10.1103/PhysRevLett.133.226301. C.K.Y. participated in the conception of the project and the analytical and numerical calculations, and participated in the writing of the manuscript. ix TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . viii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxx I Introduction 1 Chapter I: General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2 Chapter II: Topology in Adiabatic Dynamics . . . . . . . . . . . . . . . . . . 9 2.1 Adiabatic Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Berry Curvature and Chern Number . . . . . . . . . . . . . . . . . . 10 2.3 Manifestations of Topology in Adiabatic Transport Dynamics . . . . 12 2.4 Edge States and Topology in Graphene . . . . . . . . . . . . . . . . 16 Chapter III: Coherent Dynamics and Topology of Floquet Systems . . . . . . 19 3.1 Floquet Theory: Separating Stroboscopic Dynamics from Micromotion 20 3.2 Analytic Approximations to the Floquet Hamiltonian . . . . . . . . . 24 3.3 Numerical Approach: Frequency-Space Expansion . . . . . . . . . . 27 3.4 Floquet Hamiltonian of Irradiated Graphene and Semiconducting Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.5 The Floquet Topological Insulator . . . . . . . . . . . . . . . . . . . 31 3.6 Topological Transport in Floquet Systems . . . . . . . . . . . . . . . 35 Chapter IV: Incoherent Dynamics of Floquet Systems . . . . . . . . . . . . . 37 4.1 Floquet-Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . 38 4.2 Solving for the Steady State . . . . . . . . . . . . . . . . . . . . . . 40 4.3 Survival of Floquet-Bloch States Under Continuous-Wave Driving . . 45 II Persistent Floquet Quantum Matter 48 Chapter V: Optical Control of Slow Topological Electrons in Moiré Systems . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Phenomenological Analysis . . . . . . . . . . . . . . . . . . . . . . 5.5 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter A: Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 49 50 52 53 55 57 58 59 x A.1 Details of the Models . . . . . . . . . . . . . . . . . . . . . . . . . 59 A.2 Choice of Physical Parameters . . . . . . . . . . . . . . . . . . . . . 62 A.3 Anomalous Hall Conductivity Calculations for the Continuum Model 63 A.4 Direct Variation of the Phonon Speed 𝑐 ph . . . . . . . . . . . . . . . 64 A.5 Symmetries and Berry Curvature Distributions of the Toy and Continuum Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 A.6 Full Phenomenological Model . . . . . . . . . . . . . . . . . . . . . 67 A.7 Formal Definition, Numerical Evaluation, and Phenomenological Model of Ain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 A.8 Predicting E ∗ for the Toy Model . . . . . . . . . . . . . . . . . . . . 74 A.9 Different Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . 77 A.10Gap Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 A.11Floquet Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . 78 A.12Form Factor Details . . . . . . . . . . . . . . . . . . . . . . . . . . 80 A.13Validity of the Diagonal Density Matrix Approximation . . . . . . . 81 A.14Dielectric Function of Strontium Titanate . . . . . . . . . . . . . . . 82 A.15Monkhorst-Pack Grid, Numerical Integration, and Convergence . . . 83 A.16Berry Curvature Calculations . . . . . . . . . . . . . . . . . . . . . 84 A.17The Drive Amplitude Perturbative Regime at Different Twist Angles 85 Chapter VI: Signatures of Floquet Electronic Steady States in Graphene Under Continuous-Wave Mid-Infrared Irradiation . . . . . . . . . . . . . . . . . 86 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.2 Devices and photoconductive measurements . . . . . . . . . . . . . 88 6.3 Theory of Floquet Steady States . . . . . . . . . . . . . . . . . . . . 89 6.4 Optically-controlled photoresponse . . . . . . . . . . . . . . . . . . 91 6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.6 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Chapter B: Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 B.1 Sample Irradiation with a Continous-Wave Mid-Infrared Laser Beam 99 B.2 Measurement Sequence . . . . . . . . . . . . . . . . . . . . . . . . 100 B.3 Transmission property of ITO top gate for Mid-IR irradiation . . . . 102 B.4 Cyclotron Resonance of the Top-Gated Epitaxial Graphene Samples . 103 B.5 Gate Efficiency Calibration . . . . . . . . . . . . . . . . . . . . . . 104 B.6 Device Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 B.7 Characterization of Single-Particle Dynamics via Floquet States . . . 107 B.8 Steady State Simulations of Laser-Illuminated Graphene . . . . . . . 108 B.9 Transport in the Floquet Steady State . . . . . . . . . . . . . . . . . 110 B.10Density of States and Berry Curvature Calculation . . . . . . . . . . 112 B.11Transverse Hall Voltage . . . . . . . . . . . . . . . . . . . . . . . . 112 B.12Reproducibility of the Results . . . . . . . . . . . . . . . . . . . . . 114 B.13Effect of Hall contact misalignment on the transverse signal measured under illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 B.14Linearity in the device response . . . . . . . . . . . . . . . . . . . . 116 B.15Chopper frequency dependence of the photoresponse . . . . . . . . . 116 xi B.16Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 IIIFloquet Physics Beyond Optical Drives and Beyond LowDimensional Systems 120 Chapter VII: Quantized Acoustoelectric Floquet Effect in Quantum Nanowires 121 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.2 Phonon-Driven Carbon Nanotube. . . . . . . . . . . . . . . . . . . . 123 7.3 Floquet-Boltzmann Equation. . . . . . . . . . . . . . . . . . . . . . 125 7.4 Phenomenological Rate Equation . . . . . . . . . . . . . . . . . . . 126 7.5 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Chapter C: Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 C.1 Microscopic Details of the Carbon Nanotube . . . . . . . . . . . . . 131 C.2 Details of the coherent phonon drive . . . . . . . . . . . . . . . . . 133 C.3 Details on the Steady State Calculation . . . . . . . . . . . . . . . . 137 C.4 Effective Description Using Lindblad Master Equation . . . . . . . . 138 Chapter VIII: Chiral Plasmons and Topological Photocurrent in Weyl Semimetals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8.2 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.3 Origin of Topological Response . . . . . . . . . . . . . . . . . . . . 147 8.4 Nonzero Filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 8.5 Photoresponse of a Single WN . . . . . . . . . . . . . . . . . . . . 149 8.6 Symmetry Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 151 8.7 Numerical Verification . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.8 Plasmon Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . 153 8.9 Chiral Plasmons in WSMs with broken TRS . . . . . . . . . . . . . 154 8.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 8.11 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Chapter D: Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 D.1 Steady State Electronic Distribution . . . . . . . . . . . . . . . . . . 157 D.2 Photoresponse of an Isolated, Untilted Weyl Node . . . . . . . . . . 157 D.3 Consequences of Finite tilt . . . . . . . . . . . . . . . . . . . . . . . 165 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 xii LIST OF ILLUSTRATIONS Number Page 3.1 Schematic illustration of slow (stroboscopic-timescale) dynamics and fast (micromotion) dynamics of a Floquet system with a time-periodic Hamiltonian 𝐻 (𝑡) = 𝐻 (𝑡 +𝑇). A system is initialized at time 𝑡0 in the state |Ψ(𝑡0 )⟩. We compare the time-dependence of the expectation value of an operator 𝑂ˆ under evolution by the full time-dependent Schrodinger equation |Ψ(𝑡)⟩ = 𝑈 (𝑡, 𝑡i0 )|Ψ(𝑡0 )⟩ (orange curve), where h ∫𝑡 𝑈 (𝑡, 𝑡0 ) = T exp −(𝑖/ℏ) 𝑡 𝐻 (𝑡 ′)𝑑𝑡 ′ , and under the effective time0 independent Floquet Hamiltonian | Ψ̃(𝑡)⟩ = 𝑒𝑖𝐻𝐹 (𝑡−𝑡0 ) |Ψ(𝑡0 )⟩ (blue curve). The Floquet Hamiltonian 𝐻 𝐹 captures the slow coursegrained dynamics of the system on timescales longer than the period of the drive 𝑇 (see blue curve), while the full solution to the timedependent Schrodinger equation captures the fast dynamics (micromotion) within each time period (see orange curve). The orange and blue curves agree at stroboscopic times given by 𝑡0 +𝑛𝑇, where 𝑛 ∈ Z, where |Ψ(𝑡0 + 𝑛𝑇)⟩ = | Ψ̃(𝑡0 + 𝑛𝑇)⟩. Figure based on Ref. [290]. . . . 21 3.2 Floquet electronic bands of graphene irradiated by a circularly-polarized laser. (a) Energy spectrum of graphene as a function of 𝑘 𝑥 for fixed 𝑘 𝑦 = 0. (b) Quasienergy spectrum of graphene under circularly polarized laser irradiation, exhibiting an off-resonant Haldane gap of size Δ𝐾 and a resonant gap of size Δ 𝑅 . . . . . . . . . . . . . . . . . . 30 3.3 Pseudospin winding under circularly polarized irradiation. (a) Here we box in orange the 𝛼 = − quasienergy band. (b) Plot of 𝛼 = − quasienergy band vs. 𝑘 𝑥 and 𝑘 𝑦 . Red arrows indicate the orientation of the pseudospin 𝑛ˆ 𝒌− (0). We focus on the pseudospin along the rings sketched in green. (c) Pseudospin in the region 𝑘 < 𝑘 𝑅 , which is roughly aligned with the 𝑧ˆ-axis, with orientation set by the sign of the mass gap at 𝑘 = 0. (d) Pseudospin along the resonance ring 𝑘 = 𝑘 𝑅 . (e) Pseudospin in the region 𝑘 > 𝑘 𝑅 , which approaches the ˆ . . . . . . . . . . . . . . . . . 32 equator, parameterized by 𝑛ˆ 𝒌− (0) = 𝒌. xiii Path (red curve) traced out by the pseudospin 𝑛ˆ 𝒌− (0) upon variation across the resonance ring [see Figure 3.3(d)]. (a) The case of sign(Δ𝜉 𝜉𝜂) > 0, in which case the pseudospin winds around the polar axis (sketched by a black line) 2 times. (b) The case of sign(Δ𝜉 𝜉𝜂) < 0, in which case the pseudospin does not wind around the polar axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The quasienergy of the driven system is defined modulo ℏ𝜔. In our discussion of the Floquet Boltzmann equation, it is convenient to consider two choices of the quasienergy bands. (a) The “folded” Floquet Brillouin Zone (FBZ) picture is analogous to the rotating frame in the rotating wave approximation, where states with band character corresponding to the valence band of the undriven system are shifted in energy by +ℏΩ. (b) The “unfolded” picture where the quasienergies are shifted such that quasienergy spectrum resembles the original bands, except with a gap opening at resonance. Dashed lines indicate ±ℏΩ/2. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Sketch of “regular” electron-phonon and electron-electron scattering processes that do not rely on photoemission and photoabsorption to conserve energy and momentum. In the folded picture [panels (a) and (c)], these scattering processes only connect the same Floquet copies [i.e. states sketched with the same color]. When the processes are sketched on the unfolded bands [see panels (b) and (d)], it becomes clear that no photoabsorption or emission is required to conserve energy-momentum in the scattering processes. . . . . . . . . . . . . 4.3 Sketch of electron-phonon Floquet-Umklapp scattering processes that absorb or emit a single photon. In the folded picture [panel (a)], these scattering processes connect different Floquet copies [i.e., states sketched with different color]. When the processes are sketched on the unfolded bands [see panels (b-d)], it is clear that the scattering relies on an absorption or emission of a single photon (squiggly blue lines) to conserve energy and momentum. . . . . . . . . . . . . . . . 3.4 34 41 42 43 xiv 4.4 Sketch of electron-electron Floquet-Umklapp scattering processes that absorb or emit a single photon, categorized as Floquet-Auger I and II which transfer a single or pair of electrons across distinct Floquet bands. We sketch the scattering processes in the folded picture in panels (a) and (c) and the unfolded picture in panels (b) and (d). In the unfolded picture, it is clear that the scattering relies on an absorption or emission of a single photon (squiggly blue lines) to conserve energy and momentum. . . . . . . . . . . . . . . . . . . 44 4.5 Sketch of an electron-electron collision process (green arrows) facilitating interband electronic scattering across the Floquet gap. . . . . . 46 5.1 (a) Schematic experimental design. Circularly polarized laser induces non-trivial Berry-curvature in the narrow bands (see Figure 5.2(bc)), resulting in an anomalous Hall conductivity 𝜎𝑥𝑦 . TBG lies on top of a dielectric and metallic gate that screen electron-electron interactions. (b) Anomalous Hall conductivity vs. drive amplitude E for 𝜁 ≈ 0.5 and various values of 𝜒 indicated on the scale (see below Eqs. 5.4 and 5.7 for definitions of 𝜁 and 𝜒). The 𝜎𝑥𝑦 features a rapid drop with E below the critical amplitude E ∗ (dashed line). Here, E0 = ℏ𝑣 𝐹 /(𝑒𝐿 2𝑀 ) ≈ 7.2 × 104 V/m. (c) Critical amplitude vs. 𝑐 ph /𝑣 0eff , where 𝑣 0eff = 𝑣 eff (0) is an effective electron velocity defined in the text. Enlarged red circle: E ∗ in (b). . . . . . . . . . . . . . . . 50 5.2 (a) Zoom-in on schematic narrow bands in a moiré system. Drive with angular frequency Ω resonantly couples states along resonance rings (green curves). (b) Undriven spectrum of TBG along a line in the Brillouin Zone indicated by the orange curve in (c). Dashed frame encloses optically-active, narrow central bands 𝜈 = ±1. (c) (𝜉) Berry curvature B 𝒌+ in the upper Floquet band, with blue color   (𝜉) (𝜉) intensity proportional to tanh 2B 𝒌+ /𝐿 2𝑀 (color bar) so B 𝒌+ peaks (𝜉) are more visible. Dashed lines indicate areas enclosing B 𝒌+ peaks at the Dirac points and resonance ring. (d) Periodic quasienergy Floquet spectrum of the driven system, having two central bands shown in (a). The Floquet spectrum exhibits the upper (UFB, 𝛼 = +) and lower (LFB, 𝛼 = −) Floquet bands, separated by off-resonant gaps Δ𝐾 at the Dirac 𝐾, 𝐾 ′ points and a Rabi-like gap Δ 𝑅 along the resonance ring [130, 236, 239]. . . . . . . . . . . . . . . . . . . . . . . . . . . 51 xv 5.3 5.4 A.1 A.2 A.3 A.4 (a) Schematics of the Floquet spectrum and one of the phonon lightcones originating from the area S𝐾 in the UFB. The intersection between the UFB (LFB) and all cones centered in S𝐾 form Sin (Sout ). As E → E ∗ , the area of Sin vanishes. (b-d) Numerical verification of the phenomenological model. (b) Area of Sin , Ain , vs. E for three values of 𝑐 ph /𝑣 0eff . (c) Average occupation in S𝐾 . (d) Anomalous Hall conductivity 𝜎𝑥𝑦 for same parameters as (b, c). At E ∗ (dashed (𝜉) lines), Ain , 𝐹𝐾+ , and 𝜎𝑥𝑦 sharply change. . . . . . . . . . . . . . . . (a) Left column: steady-state occupation of the UFB when 𝜒 = ∞ (calculated on a 163 × 163 momentum grid). Right column: steadystate occupation when 𝜒 = 0.24 (calculated on a 73 × 73 momentum grid). Bottom row: strong-drive case (E = 4.3E0 > E ∗ ). Top row: weak-drive case (E = 0.97E0 < E ∗ ). Zoom-in boxes: the 𝐾, 𝐾 ′ points have reduced occupation when E > E ∗ relative to when E < E ∗ . (b) Visibility V vs. 𝜒. (c) Value of 𝜒 for various 𝜖 and gate distances 𝑑. (d) Value of 𝜂 for various 𝜖 and deformation potentials 𝐷, with 𝑑 = 4 nm. Points in (b, d): parameters used in Figure 5.1(b). (a) The quasienergy band structure of the toy model with for the parameters used in the main text. (b) The quasienergy band structure of the continuum model at valley 𝜉 = +1. In both panels, the first Floquet Brillouin zone is shaded. See Section A.2 for details and justification for the parameters we have used. . . . . . . . . . . . . . (a) Left: the steady-state occupation of the lower Floquet band in valley 𝜉 = +1 of the continuum model [25, 149]. Right: the Berry curvature of the same band, which peaks near the Dirac points and the resonance ring. (b) The anomalous Hall conductivity 𝜎𝑥𝑦 as a function of drive strength E. . . . . . . . . . . . . . . . . . . . . . . Anomalous Hall conductivity of the toy model as a function of the ratio 𝑐 ph /𝑣 0eff for three different drive field strengths E/E0 . The same electron-phonon decoupling process is visible as 𝜎𝑥𝑦 plateaus. . . . . Comparing the Berry curvature distribution in the upper Floquet band in the (a) continuum model and (b) toy model at a drive amplitude of E/E0 ≈ 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 57 61 64 65 67 xvi A.5 A.6 A.7 A.8 A.9 A.10 A.11 A.12 (𝜉) The occupation 𝐹𝐾+ as predicted by the phenomenological model in ph Section A.6 for different ratios 𝜒 ≡ 𝜏𝐾el /𝜏𝐾 . Note that for large 𝜒, the occupation is lower than the non-interacting case at weak drive amplitudes, an effect of reduced Pauli blocking in the electron-phonon interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) The intersection S𝐾 (Figure 3(a)) as viewed on the Brillouin zone. The outer radius along the path 𝐾 𝑅 is ℎ 𝑏 (E). (b) Quasienergy (pink) along the path 𝐾 𝑅, with the phonon light cone (grey) that determines the outer radius of Ain . The intersections 𝑘 + and 𝑘 − between the cone and the upper Floquet band determines ℎ 𝑏 (E) = 𝑘 + − 𝑘 − . . . . . (a) Width of the intersection S𝐾 , 𝑤 𝑏 (E). (b) Circular coordinate system with arc length 𝑤 (increasing counterclockwise) that we use to determine 𝑤 𝑏 (E) = 𝑤 + − 𝑤 − . . . . . . . . . . . . . . . . . . . . . Comparing numerical evaluation of E ∗ (points) to an analytic fit to Eq. A.65. We use the same fitting parameters 𝑓2 = 0.778, 𝑓1 = 0, and 𝛿(𝑁) = 0.006 for both panels. . . . . . . . . . . . . . . . . . . . Comparing the dependence of 𝜎𝑥𝑦 on E for (a) the frequency considered in the main text and (b) a lower frequency where FloquetUmklapp processes are stronger. Note that the frequency in panel (b) is inaccessible without generating two-photon resonances in the continuum model due to the peaked shape of the 𝜈 = ±1 bands near the Γ point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the fitted Δ 𝑅 and predicted Δ𝐾 in Equations A.72 and A.76 (solid lines) to those obtained from numerics (points), using ℏΩ = 5 meV in the toy model. Here, we fit Δ 𝑅 with factors of 𝑓1𝑅 = 0.04 and 𝑓2𝑅 = 0.0184 (see Eq. A.72). . . . . . . . . . . . . . . 𝜉𝜈 ′ 𝜈 Comparing the average toy and continuum model form factors W𝒌,𝒒+𝑮 , with |𝒒+𝑮 | measured in units of 108 m−1 . (a-c) The intraband (𝜈 = 𝜈′) form factors at the 𝐾, 𝑀, and Γ points. (d-f) The interband (𝜈 ≠ 𝜈′) form factors at the 𝐾, 𝑀, and Γ points. . . . . . . . . . . . . . . . . The color represents the ratio ℏ/(𝜏𝒌tot Δ𝜀 𝒌 ) at different points in momentum space (a) for the upper Floquet band and (b) for the lower Floquet band for the case 𝜒 ≈ 2.8 and 𝜁 ≈ 0.5. . . . . . . . . . . . . 69 72 73 75 76 77 80 82 xvii Convergence of anomalous conductivity with grid size for (a) 𝑁 (mod 3) = 1 and (b) 𝑁 (mod 3) = 2. Due to the positioning of grid points near the 𝐾 point, the results at low grid resolutions show significant disagreement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 A.14 The requirement that the laser drive strength E is perturbative, i.e. a fraction of electron bandwidth 𝑒E 𝐿 𝑀 < 𝑊, narrows the range of E values that can be used. As a result, the range of 𝑐 ph whose E ∗ is visible is limited as well - we postulate that they are pushed to higher drive strengths E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.1 (a) Device layout with two contacts for longitudinal bias, 𝑉Bias , and two contacts for measurements of the transverse voltage, 𝑉𝑦 . The device has a square geometry with a 5-𝜇m side and an indium tin oxide (ITO) top gate with an Al2O3 dielectric sublayer, for applying a gate voltage 𝑉Gate . (b) Drive-modified Floquet bands of the graphene Dirac cone, which exhibits a Floquet gaps of size at the resonance energies 𝜀 = ±ℏΩ/2, where ℏ is the reduced Planck constant and Ω is the angular frequency of the laser. (c) Density of states of the Floquet bands calculated numerically for a circularly and linearly polarized laser of power density 𝑃 = 3 mW/𝜇m2 . (d) Photo-induced change of source-drain current (Δ𝐼) as a function of Fermi energy 𝐸 𝐹 for circularly and linearly polarized laser irradiation, as measured at a constant source-drain voltage of 6 mV. Δ𝐼 is measured with a lock-in amplifier using a chopper modulation as a reference. The circles represent data points and the solid lines are obtained by adjacent point averaging. The dotted lines mark the 𝐸 𝐹 values corresponding to ±ℏΩ/2, with the related uncertainty indicated by the gray stripes. The uncertainty is attributed to the gate efficiency calibration and the detail is described in Appendix B. The dips in Δ𝐼 at energies ±ℏΩ/2 are much broader than the Floquet gap Δ(see c) and arise from the non-equilibrium steady-state distribution of electrons. The measurements were performed on sample A at a cryostat temperature of ∼ 3.5 K, laser photon energy of 117 meV, and laser power density of 1.1 mW/𝜇m2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 A.13 xviii 6.2 (a) Key scattering processes between the graphene 𝐾 and 𝐾 ′ valleys facilitated by photons (arrow 1), surface acoustic phonons (arrows 2, 3), and graphene acoustic phonons (arrow 4) that contribute to the steady state. (b) Steady state distribution (black solid curve) for 𝑃 = 1.9 mW/𝜇m2 and equilibrium distribution (blue dashed curve) of electrons at doping 𝐸 𝐹 = 0, where the longitudinal photoconductivity is enhanced relative to equilibrium. Here, 𝜀 denotes the quasienergy and 𝐹𝒌𝛼 denotes the electronic occupation. (c) Steady state distribution (black solid curve) of electrons at doping 𝐸 𝐹 = 0.35ℏΩ and 𝑃 = 0.8 mW/𝜇m2 , where the photoconductivity is suppressed relative to an equilibrium distribution expectation (blue dashed curve). (d) Same as (c) but for a larger power density 𝑃 = 1.9 mW/𝜇m2 , where the steady state distribution exhibits additional electron density above the Floquet gap. e Steady state distribution (black solid curve) of electrons for 𝑃 = 1.9 mW/𝜇m2 at doping 𝐸 𝐹 = 0.75ℏΩ, where the photoconductivity is approximately identical to that of an equilibrium distribution (blue dashed curve), with a slightly raised effective temperature due to multi-photon heating processes. The horizontal dotted lines in (b–e) mark 𝜀 values corresponding to ±ℏΩ/2. 90 xix 6.3 (a) Photo-induced change of source-drain current (Δ𝐼) as a function of 𝐸 𝐹 under irradiation at various peak power densities, as measured on sample C at a constant source-drain voltage of 6 mV and a cryostat temperature of 3.2 K. The photon energy is ℏΩ = 117 meV. The laser beam is circularly polarized. The circles represent data points and the solid lines are the fitted curve by a Gaussian-like function described in the Methods. The dotted line marks the 𝐸 𝐹 value corresponding to −ℏΩ/2, with the related uncertainty indicated by the gray stripes. (b) Theoretically predicted Δ𝐼 as calculated from the Floquet Boltzmann equation. The dotted line marks the 𝐸 𝐹 value corresponding to −ℏΩ/2. (c) Depth 𝑎 0 of the photocurrent dip and associated error, as calculated from a Gaussian-like fit (see details in the Methods section). For both the theoretical and experimental data, 𝑎 0 decreases with the power density 𝑃 for large 𝑃 due to enhanced heating processes in the Floquet steady state. (d) Experimental and theoretical FWHM of the photocurrent dip and the Floquet gap size (red dotted line), Δ, as a function of the power density. The FWHM exceeds Δ, indicating the emergence of photoexcited electrons in the non-equilibrium Floquet steady state. The detailed error analysis of FWHM is described in Appendix B. The uncertainty in the laser power density due to power fluctuation in (c) and (d) are estimated by the change in the laser power before and after each transport measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 xx 6.4 B.1 B.2 (a) Longitudinal photocurrent as a function of at a fixed source-drain voltage of 6 mV, with laser spot in focus and defocused. For all three curves, the laser power is ∼ 3.6 W, corresponding to a peak power density of 0.6 mW/𝜇m2 at focus. Under defocused irradiation, with the laser spot slightly moved to the side, the power density drops by ∼ 3 times (gray curves). Circular polarization of 10.6 𝜇m wavelength laser radiation. The cryostat temperature stabilized at ∼ 3.4 K. b Theoretically calculated photocurrent for weak drives, exhibiting shallower dips near resonance as a function of decreasing driving power, in agreement with (a). (c) Temperature dependence of the longitudinal photocurrent as a function of 𝐸 𝐹 . The photocurrent decreases, and the dips at 𝐸 𝐹 = ±ℏΩ/2 disappear at high temperatures. The laser beam is circularly polarized. (d) Theoretically calculated photocurrent for laser power density 1.4 mW/𝜇m2 , the photocurrent dip becomes less visible at higher temperatures. The dotted lines in (a)–(d) mark the 𝐸 𝐹 value corresponding to ±ℏΩ/2, where the related uncertainty is indicated by the gray stripes in (a) and (c). . . . . . . . 93 Scheme of the sample illumination system. Schematic illustration of the optical setup used for delivering and focusing the mid-infrared irradiation. The beamsplitters are used only during the preliminary alignment of optical elements. The inset image shows the visible laser spot on the sample used for the preliminary alignment of the molybdenum mirrors and the ZnSe focusing lens. . . . . . . . . . . . 99 Measurement sequence in time. Experimentally measured cryostat temperature (top panel) and longitudinal current (bottom panel) with laser blocked and unblocked intervals. Note that the actual graphene lattice temperature, 𝑇 (𝑡), is expected to be further elevated relative to the cryostat temperature. When the laser is blocked, the sample cools down to 𝑇bg , giving rise to a strong bolometric effect in the longitudinal current. The laser illumination is blocked/unblocked in the optical path between the chopper and the sample (light coral stripes indicate when laser is unblocked), the average laser power density is 0.125 mW/𝜇m2 and the peak power density is 1.25 mW/𝜇m2 , ℏΩ = 117 meV (10.6 𝜇m wavelength). . . . . . . . . . . . . . . . . . 100 xxi B.3 B.4 B.5 B.6 B.7 Laser intensity and temperature dependence of the longitudinal conductance. Gate voltage dependence of the source-drain current at different temperatures with laser blocked (gray) and under irradiation at the base temperature (red), with 𝑃 = 0.6 mW/𝜇m2 and with source-drain voltage fixed at 50 mV. . . . . . . . . . . . . . . . . . . 101 Mid-Infrared transmission spectrum of ITO. Example of spectrum measured from our typical 110-nm thick ITO layer in the range around 10.6 𝜇m wavelength (corresponding to the wavenumber 943 cm−1 ). . 102 Longitudinal photoresponse vs. carrier density at different B fields. Longitudinal voltage photoresponse as a function of magnetic field and carrier density, under chopper-modulated irradiation with circular polarization, at a laser intensity of 20 𝜇W/𝜇m2 and photon energy ℏΩ = 117 meV. The sample is biased at a fixed current 𝐼𝑥 of 1 𝜇A, and the photoinduced change in source-drain voltage Δ𝑉𝑥𝑥 is measured with a lock-in amplifier. . . . . . . . . . . . . . . . . . . . . . . . . 103 Gate efficiency calibration. (a) (Black) Source-drain voltage and (blue) transverse voltage from the Hall effect measurements for 𝑉𝑔 = 7 V and a constant bias (source-drain) current 𝐼sd = 1 𝜇A. The green and red lines are examples of the set of “4 points” used for extracting the actual transverse voltage. (b) (Black) Conductance at different gate voltages. (Blue) Carrier density at different 𝑉𝑔 from Hall measurements. (Red) Linear fitting of carrier density at different 𝑉𝑔 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Scheme of key steps in top-gated graphene device fabrication. (a) Patterning of graphene Hall bars with a thin Pd/Au protection layer. (b) Optical image of the graphene devices prior to removing the metal protection layer and the top gate fabrication. (c) Illustration of top gate fabrication using Al2 O3 as the dielectric layer and ITO as the gate contact. (d) Optical image of the top-gated devices. The dashed line marks the area of devices shown in (b). . . . . . . . . . . . . . . 107 xxii B.8 Characteristics of the single-particle quasienergy spectrum. (a) Singleparticle quasienergy bandgap Δ𝐸 1 opening at 𝜀 𝑘𝛼 = ±ℏΩ/2, with 𝜀 𝑘𝛼 = 0 corresponding to the Dirac point. The axes indicate the laser power density and photon energy. (b) Predicted light-induced modifications of the time-averaged density of states of graphene (see section J for definition) as a function of Fermi energy shift from the Dirac point for different power densities, with circular polarization (top) and linear polarization (bottom). Notice that the laser-induced gap at the Dirac point, which is a second-order process, is not resolved in the experiment for the chosen laser intensities. (See the main text for details.) The vertical dotted lines mark the 𝐸 𝐹 , value at ±ℏΩ/2. . 109 B.9 Temperature dependence of the longitudinal conductivity at 𝐸 𝐹 = 0 without irradiation, which indicates charge puddle effects. . . . . . . 111 B.10 Transverse voltage response under different conditions. Transverse voltage as a function of 𝐸 𝐹 , under light irradiation (ℏΩ = 117 meV) with left-hand circular polarization. The dotted lines mark the values corresponding to 𝐸 𝐹 = ±ℏΩ/2, with the related uncertainties indicated by the gray stripes. (a) Transverse voltage measured at different bias voltage values. The cryostat temperature is 3.1 K, and the laser beam is circularly polarized. (b) Transverse voltage measured under different irradiation power. The curves are vertically shifted for clarity. The cryostat temperature ranges from 3.0-3.3 K depending on laser intensity. The laser beam is circularly polarized. The circles represent data points, and the solid lines are obtained by adjacent point averaging. (c) Transverse voltage measured at different temperatures. For both (a) and (c), the laser intensity is 0.8 mW/𝜇m2 and for (b) and (c) the bias voltage is 50 mV. The measurements are from sample C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 xxiii B.11 Transverse voltage response under different conditions. (a) Transverse voltage as a function of 𝐸 𝐹 , under light irradiation (ℏΩ = 117 meV) with different polarizations of the laser. (b) helicitydependent Hall voltage in Sample A. (c) Hall conductivity in Sample A. Here, the laser intensity is 1.4 mW/𝜇m2 , bias source-drain voltage is 6 mV, and cryostat temperature is 3.4 K. (d) Theoretically calculated Floquet Hall conductivity with nontrivial 𝐸 𝐹 -dependence arising from the tunable electronic steady state. The dotted lines in (a)-(d) mark the values corresponding to 𝐸 𝐹 = ±ℏΩ/2, with the related uncertainties indicated by the gray stripes in (a)-(c). In (a)-(c), the circles represent data points, and the solid lines are obtained by adjacent point averaging. . . . . . . . . . . . . . . . . . . . . . . . . 114 B.12 Floquet signatures in the longitudinal photocurrent in additional devices. The dotted lines mark the 𝐸 𝐹 values corresponding to ±ℏΩ/2, with the related uncertainty described by the gray stripes. (a) Longitudinal photocurrent as a function of 𝐸 𝐹 under different polarization illumination in sample B. Parameters: Laser power density 1.4 mW/𝜇m2 and bias voltage 6 mV. (b) Longitudinal photocurrent as a function of 𝐸 𝐹 , under different polarization illumination in sample C. Parameters: Laser power density 1.6 mW/𝜇m2 , bias voltage 6 mV, and cryostat temperature 3.4 K. . . . . . . . . . . . . . . . . . 115 B.13 Transverse voltage response in additional devices. Transverse voltage as a function of 𝐸 𝐹 , under irradiation with different polarizations in sample C [panel (a)] and the difference between the transverse voltage response from circular polarizations of the laser beam with opposite chirality extracted from the curves in panel (a) [see panel (b)], and upon reversing the source-drain bias 𝑉sd . Laser power density 0.5 mW/𝜇m2 , cryostat temperature 3.1 K. The dotted lines mark the 𝐸 𝐹 values corresponding to ±ℏΩ/2, with the related uncertainty described by the gray stripes. . . . . . . . . . . . . . . . . . . . . . . 116 xxiv B.14 Effect of Hall contacts misalignment. (a) (Red) 𝑉Misalignment and (black) 𝑉sd measured at the same source-drain current. Room temperature. (b) Misalignment contribution to the measured transverse voltage for (top) circular and (bottom) linear irradiation. Solid lines represent the transverse voltages measured by the lock-in amplifier; dotted lines represent the contribution resulting from misalignment in the Hall contacts. The experimental data from Figure B.11 are shown for reference. . . . . . . . . . . . . . . . . . . . . . . . . . . 117 B.15 Bias dependence of sample response. (a) I-V curve from sample C at different values of 𝑉𝑔 . (b) Photo-induced current change Δ𝐼 as a function of bias voltage at different values of 𝑉𝑔 . (c) Normalized longitudinal photoresponse Δ𝐼 as a function of 𝐸 𝐹 at different bias voltages, under circularly polarized irradiation, with a power density of 1.5 mW/𝜇m2 . The curves are vertically shifted for clarity. The cryostat temperature is 3.4 K. The dotted lines mark the 𝐸 𝐹 values corresponding to ±ℏΩ/2, with the related uncertainty described by the gray stripes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 B.16 Longitudinal photoresponse as a function of chopper frequency. Photo-induced current change measured in sample C under 1.5 mW/𝜇m2 of circularly polarized irradiation. The sample is biased at a fixed 6 mV. The cryostat temperature is 3.4 K. Error bars indicate the maximum deviation in the lock-in amplifier reading during the measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 xxv 7.1 (a) Schematic experimental setup. A THz-frequency coherent phonon wave (with atomic displacements sketched by orange arrows) of momentum 𝑞 and angular frequency 𝜔 propagates from the phonon source on the right to the absorbing material on the left. A screening material of dielectric constant 𝜖 is placed below the CNT. Black sinusoid represents the coherent phonon potential. Electrons (black dots) generate quantized current 𝐽 = 𝑒𝜔/𝜋 when confined to the troughs of the potential, achieved when incoherent phonon relaxation (ep) is much faster than electron-electron heating (ee). (b) Current 𝐽 vs. Floquet gap Δ/ℏ𝜔 for different 𝜖 (see inset). Vertical line denotes Δ = ℏ𝜔. Inset: Δ∗ vs. 𝜖, where Δ∗ is the minimal Floquet gap at which 𝐽 is nearly quantized, set as 0.96𝑒𝜔/𝜋. (c) Band structure of a (10, 0) armchair CNT. Inset: Fermi energy lies near the band bottom of the lowest positive-energy band, and the electronic density 𝑛𝑒 is chosen to be commensurate to 𝑞. (d) Quasienergy spectrum of the driven system. Blue shading on the 𝛼 = 0 band indicates the optimal filling resulting in quantized current. Incoherent phonon scattering transitions (black arrows) relax electrons into the 𝛼 = 0 band. . . . . 122 7.2 (a) Floquet bands upon driving by a coherent phonon wave. Dashed light purple arrows (1-3): dominant electron-phonon intraband and cooling processes relaxing electrons into the 𝛼 = 0 band. Solid red arrows (4-5): electron-phonon heating processes exciting electrons into the 𝛼 = 1 band, which are kinematically suppressed when Δ ≫ 𝛿𝜀. (b) Pairs of zigzag and squiggly red arrows: electron-electron scattering exciting electrons into the 𝛼 = 1 band. Dashed light purple arrow: electron-phonon process relaxing excited electrons into S+ . (c)-(e) Occupation 𝐹𝑖 of the patch S𝑖 for 𝑖 = +, −, 0 [see panel (a)] vs Δ for various dielectric constants 𝜖. Vertical lines: Δ = Δ∗ (𝜖). Note 𝐹+ → 0, 𝐹− → 1 as Δ → Δ∗ , resulting in quantized current. . . . . . 126 xxvi 7.3 (a) Steady state occupation of the phonon-driven CNT for a weak phonon drive (Δ < Δ∗ ), dielectric constant 𝜖 = 80, and optimal doping 𝑛𝑒 = 𝑞. Inset shows excitations in the 𝛼 = 1 band near the Floquet gap. (b) Same as (a) but for a strong drive amplitude (Δ > Δ∗ ) where the occupation of the 𝛼 = 1 band is negligible, and the 𝛼 = 0 band is fully occupied. (c)-(d) Same as (b), but with two different electronic densities 𝑛𝑒 away from optimal doping. (e) Steady state current vs 𝑛𝑒 evaluated at Δ = 0.8ℏ𝜔 > Δ∗ for 𝜖 = 80. . . . . . . . . 128 C.1 Phonon transfer mechanism when a phonon source (grey block) is placed under the right end of the carbon nanotube [see Figure 1(a) in the main text for the complete, proposed experimental setup]. A carbon atom of mass 𝑀𝑐 (grey circle in magnified picture) experiences a Van der Waals force across the interface, with effective spring constant 𝑘. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 C.2 The ratio 𝜁 of the maximum interband scattering rate to the Floquet band energy separation vs. Floquet gap Δ for various dielectric constants 𝜖. Note that 𝜁 ≪ 1, so steady state coherences are suppressed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 C.3 Simplified model of the quantized acoustoelectric Floquet effect. The black curve represents the moving coherent phonon potential 𝑉 (𝑥, 𝑡), and the grey dashed curve sketches the potential at a slightly later time. Electrons (green dots) generate quantized current when confined to the troughs of the potential, which is achieved when the incoherent phonon relaxation time 𝜏ep is much faster than the heating time 𝜏h . . . 140 C.4 Steady state current 𝐽 as calculated from the Lindblad master equation [Eq. (C.33)]. (a) Steady state current in the regime ℏ/( 𝜏Δ) ¯ ≪ 1 analyzed in the main text using the Floquet Boltzmann equation. The current is quantized when the system is driven adiabatically,

i.e., log ℏ𝜔/Δ < 0. In the non-adiabatic regime log ℏ𝜔/Δ > 0, the current attains a quantized value in the regime where incoherent phonon relaxation dominates 𝜏ep ≪ 𝜏h . (b) Steady state current for a weak driving potential amplitude Δ satisfying ( 𝜏Δ)/ℏ ¯ = 0.3. In contrast to panel (a), the current deviates from quantization in the
adiabatic regime log ℏ𝜔/Δ < 0, and instead only attains a quantized value when 𝜏ep ≪ 𝜏h . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 xxvii 8.1 Light-induced charge pumping and chiral plasmons. (a) We consider the optical response of a noncentrosymmetric WSMs, where opposite-chirality Weyl nodes (WNs) can have distinct effective chemical potentials (𝜇± ) and tilts (𝜒± ). We show that circularly polarized driving in the near-adiabiatic regime can induce a topological charge pumping effect leading to a strong photocurrent 𝑗¯𝜂𝑧 along the beam axis, where 𝜂 =⟲, ⟳ denotes the helicity of the laser. (b) Numeri𝑧 cally computed photocurrent, 𝑗¯⟲ , for 𝜇− = 15 m𝑒𝑉, 𝜇+ = 30 m𝑒𝑉, 𝑧 and ℎ 𝑓 = 5 meV, with 𝜒± = ±𝜒. (c) Maximal 𝑗¯𝜂𝑧 , denoted 𝑗max , attained numerically (scatter points) and predicted semiclassically (dashed line) [Eq. (8.7)]. Discrepancies arise as ℎ 𝑓 is raised beyond the adiabatic limit. (d) Plasmon enhancement factor 𝐺 𝜂 ≡ 𝐸/𝐸 ext vs ℎ 𝑓 and 𝐸, where 𝐸 ext (𝐸) denotes the drive-field amplitude outside (inside) the WSM. Dashed curve indicates plasmon resonance predicted semiclassically. (e) Same as (d), but for a WSM with broken TRS and a pair of WNs with opposite chirality and tilt. Here, the chiral plasmon resonance differs for left (left panel) and right (right panel) circular polarizations. . . . . . . . . . . . . . . . . . . . . . . 146 8.2 Topological photocurrent as quantized charge pumping. (a) Brillouin zone region (purple) surrounding a single WN (bright yellow) at 𝜇 = 0. Under illumination by circularly polarized light, all electronic states with a given 𝑘 𝑥 and 𝑘 𝑦 (yellow vertical line) traces out a cylinder (pink) in momentum space during a period of the drive. This generates a quantized charge transfer per cycle along the 𝑧 direction, given by −𝑒 times the total flux of Berry curvature through the surface of the cylinder (differential area element indicated by dark purple). When the cylinder encloses the WN, this integral is nonzero and given by ±1. (b) Total time-averaged current I⟳𝑧 ( 𝒌 ⊥ ) in the system generated by electrons with a fixed in-plane momentum 𝒌 ⊥ = (𝑘 𝑥 , 𝑘 𝑦 ). Left and right panels depict 𝑣 𝐹 𝑒 𝐴/ℏ ≈ 5ℎ 𝑓 and 𝑣 𝐹 𝑒 𝐴/ℏ ≈ 0.2ℎ 𝑓 , respectively, corresponding to adiabatic and nonadiabatic driving. The dashed curve indicates the trajectory of the WN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 xxviii 8.3 Photoresponse from doped and tilted WNs, with parameters 𝐸 = 1.5 MV/m, ℎ 𝑓 = 5 meV and 𝜏 = 5 ps, corresponding to 𝜏 𝑓 ≈ 6. (a) Photocurrent 𝑗¯𝜂𝑧 vs. 𝜇 for an isolated, untilted WN. The orange curve indicates semiclassical prediction [see discussion below Eq. (8.6)]. (b)-(c) Photocurrent 𝑗¯𝜂𝑧 from a pair of opposite-chirality WNs with opposite tilts, 𝜒 ≡ 𝜒+ = −𝜒− under left (b) and right (c) laser polarization. In the limit 𝜒 → 0, the photoresponse is dominated by the chirality-dependent component generated by the anomalous 𝑧 𝑧 velocity. (d) Chirality-independent component 𝑗¯𝐺𝑧 = ( 𝑗¯⟲ + 𝑗¯⟳ )/2 of 𝑗¯𝜂𝑧 . Dashed line indicates prediction from semiclassical analysis [Eq. (8.6)]. (e) Steady state occupation function 𝑔1 ( 𝒌, 0) at time 𝑡 = 0 in the 𝜈 = 1 band, for the same parameters as panel (a). Black dot indicates the position of the WN, and vectors indicate the oscillating current components generated parallel 𝑗 ∥ and perpendicular 𝑗⊥ to the internal electric field 𝑬 (𝑡) and the resulting polarization 𝑷𝜂 (𝑡), along with induced field 𝑬 ind (𝑡). . . . . . . . . . . . . . . . . . . . . . . . 151 D.1 (a) Topological current produced by conduction band (𝜈 = 1) states in an isolated Weyl node, calculated by numerically integrating the semiclassical equations of motion [see Eq. (D.14)]. Here, 𝑗top = −𝑒 𝑓 (𝑒 𝐴/ℏ) 2 /(4𝜋) denotes the topological current magnitude produced by a fully-occupied valence band [see Eq. (1) in the main text], and 𝑥 𝐹 = ℏ𝑘 𝐹 /(𝑒 𝐴). (b) Log-scale plot of panel (a), with fits (solid lines) to the numerical data. . . . . . . . . . . . . . . . . . . . 160 D.2 Illustration of the coordinate system used to calculate the saturation 𝑧 value of 𝑗1,𝜂 for 𝑘 𝐹 ≫ 2𝑒 𝐴/ℏ. Here, 𝒌 ′ = 𝒌 + 𝑒 𝑨(𝑡)/ℏ is the shifted coordinate system, which fixes the WN at the origin. Purple sphere indicates the Fermi volume in the conduction band, which is centered at the WN (center white dot). Under illumination by circularly polarized light, populated electronic states with a given 𝒌 ⊥ = (𝑘 𝑥 , 𝑘 𝑦 ) traces out a cylinder S𝑡 ( 𝒌 ⊥ ) (dark purple) in momentum space during a period of the drive. Due to the finite Fermi momentum 𝑘 𝐹 , finite Berry curvature 𝛀1 penetrates √︃ the “caps” S𝑡 ( 𝒌 ⊥ ) (light green) of the cylinder located at 𝑘 𝑧 = ± 𝑘 2𝐹 − | 𝒌 ⊥ | 2 . The Berry flux through the 𝑧 caps S𝑡 ( 𝒌 ⊥ ) produces a saturation value of 𝑗1,𝜂 in the limit 𝑘 𝐹 ≫ 2𝑒 𝐴/ℏ.160 xxix D.3 (a) Illustration of the characteristic in-plane momentum dynamics relevant for analyzing the photocurrent. Solid black circle indicates the edge of the instantaneous equilibrium Fermi volume of 𝐻 ( 𝒌, 𝑡) as a function of in-plane momentum, 𝒌 ⊥ = (𝑘 𝑥 , 𝑘 𝑦 ), for an arbitrary, fixed value of 𝑘 𝑧 . Black dot indicates the instantaneous location of the Weyl node, which is located at an azimuthal angle of 2𝜋 𝑓 𝑡 from the 𝑘 𝑥 axis (red arrow). We approximate the instantaneous equilibrium eq occupation 𝑓1 ( 𝒌, 𝑡) [Eq. (D.30)] to be given by 1 if 𝒌 ⊥ is located within the annulus segment (shaded blue) and between the two angles indicated by green lines, and 0 otherwise. (For the choice of 𝒌 ⊥ eq sketched in the figure, 𝒌 ⊥ is positioned such that 𝑓1 ( 𝒌, 𝑡) = 0.) Such an approximation is accurate in the limit where the Fermi momentum 𝑘 𝐹 is much less than the vector potential 𝑒 𝐴/ℏ. (b) Occupation function √︃ 𝑔1 ( 𝒌, 𝑡) as a function of the in-plane momentum magnitude 𝑘 ⊥ = 𝑘 𝑥2 + 𝑘 2𝑦 in the slow relaxation time limit 𝜏 𝑓 ≫ 1, plotted for several values of the 𝑧-momentum 𝑘 𝑧 . . . . . . . . . . . . . . . . . . 164 D.4 Origin of the laser helicity-independent photocurrent in a tilted WN. In the limit 𝜏 𝑓 ≫ 1, the steady state occupation of bands at any given time 𝑡 is the time average of the instantaneous equilibrium occupation function 𝜃 (𝜀 𝒌1 (𝑡 ′)−𝜇) at all other times 𝑡 ′. The black solid and dashed gray curves represent the electronic bands at representative times 𝑡 and 𝑡 ′, respectively. The instantaneous equilibrium distribution at time 𝑡 ′ (orange) preferentially fills states with 𝜕𝜀 𝒌1 (𝑡)/𝜕𝑘 𝑧 < 0, producing a net helicity-independent current. . . . . . . . . . . . . . . . . . . . 169 xxx LIST OF TABLES Number Page B.1 Fitting results of Hall measurements at different gate voltages . . . . 106 B.2 𝑉𝑔 corresponds to 𝐸 𝐹 = ℏΩ/2 . . . . . . . . . . . . . . . . . . . . . 106 Part I Introduction 1 2 Chapter 1 GENERAL INTRODUCTION The length scales and timescales in the quantum and macroscopic worlds differ by many orders of magnitude, inspiring questions about how quantum degrees of freedom can be manipulated in a laboratory. Efforts to control quantum phenomena have set the stage for highly tunable platforms to study quantum information and matter. Time-periodic fields, in particular, offer versatile methods for controlling quantum systems, with wide-ranging applications across condensed matter physics. For example, ultra-low-frequency driving fields can unveil the adiabatic topological dynamics of quantum matter, such as the Thouless charge pumping effect [7, 29, 32, 47, 71, 82, 111, 126, 133, 154, 159, 173, 201, 201, 221, 234, 278, 279, 302]. Low-amplitude time-periodic fields can also serve as a probe of quantum phases by inducing perturbative responses through optical spectroscopy [53, 195, 303]. While these examples already illustrate some practical applications of time-periodic fields, this thesis primarily focuses on the strong and non-adiabatic driving regime far beyond linear response theory, where Floquet engineering can fundamentally design nonequilibrium quantum phases with exotic properties [15, 36, 45, 64, 74– 76, 80, 81, 83, 99, 131, 140, 152, 165, 167, 169, 189, 214, 235, 239, 258, 260, 281, 291–294, 304]. Floquet-engineered systems exhibit dynamics that roughly separate into “micromotion” and “stroboscopic dynamics,” with frequencies faster and slower than the driving frequency, respectively [216, 239, 241, 290]. Micromotion can underlie exotic magnetic properties [204], reveal anomalous Floquet topological phases [240], and be detected with high-resolution time-resolved probes [65]. On the other hand, the coarse-grained stroboscopic dynamics can be modeled by a time-independent Floquet Hamiltonian that captures the time-averaged dynamics of the system on timescales much longer than the driving period. The Floquet Hamiltonian could generally differ significantly from the Hamiltonian of the equilibrium system and even exhibit drive-induced topology [167, 215, 216]. Additionally, the Floquet Hamiltonian governs the low-frequency stroboscopic dynamics, allowing its properties to be probed with conventional DC or low-frequency measurements as long as the drive pulse duration is sufficiently long. These properties highlight a key motivation for pursuing Floquet engineering: the Floquet Hamiltonian behaves like 3 a tunable static Hamiltonian, with potentially exotic phases that can be manipulated in situ. Despite the promising opportunities for Floquet engineering to control and manipulate quantum matter, the statistical mechanics of Floquet systems is a challenging problem that poses several issues [195, 196]. First, experimental observables depend not just on the single-particle electronic dynamics governed by the Floquet Hamiltonian but also on how the Floquet states are populated. In particular, the chemical potential of Floquet systems is ill-defined because Floquet states are coherent superpositions of distinct energy levels, potentially above and below the chemical potential of the equilibrium system [74, 76, 196, 239, 257, 258]. The resulting highly-nonequilibrium distribution of electrons are also susceptible to absorption of energy from the driving field, facilitated by electronic interactions, that could heat the distribution to a high-temperature state [54, 138, 158]. Lastly, in any heated system, incoherent scattering processes facilitated by electron-phonon and electronelectron scattering are also enhanced, drawing questions about whether Floquet states can remain coherent in the presence of rapid incoherent scattering processes [6, 257, 258]. These problems have become such a concern that the uncontrolled heating of Floquet systems is sometimes referred to as the “heat death” of Floquet physics [238]. The field of Floquet engineering has introduced several potential solutions to stabilize low-temperature driven phases. The umbrella term for these solutions is sometimes called the “non-thermal pathway” to Floquet engineering, which manipulates quantum properties not through heating but by accessing and controlling low-temperature degrees of freedom through coherent coupling to the driving field [59]. One promising opportunity for non-thermal Floquet engineering is to drive systems with ultra high-frequencies, which can produce a long-lived, low-temperature Floquet phase known as the Floquet pre-thermal state [1, 3, 4, 155, 223]. On the other hand, ultrafast laser pulses with frequencies resonant with the level spacing of the system can also produce transient Floquet effects at high driving intensities while minimizing heating due to its short pulse duration [143, 174, 185, 187, 260, 263, 301, 329]. In this thesis, we seek to use the intrinsic coupling of low-dimensional Floquet solid-state systems to their surrounding environment—the phonons in the lattice and the screening from surrounding dielectrics—as a primary mechanism for stabilizing low-temperature Floquet physics under continuous-wave (CW) driving. The 4 dissipation of energy to a low-temperature bath of phonons, combined with the CW driving, leads electrons to relax into a steady-state electronic distribution with stable transport effects. Furthermore, we show that Pauli blocking in the steady state can suppress scattering rates between Floquet states, enabling Floquet states to remain coherent even with driving amplitudes much weaker than those used in ultrashort-pulsed laser experiments. Our theoretical and experimental results provide key principles to realize Floquet physics in CW-driven solid-state systems and enhance their controllability with many-body interactions. This thesis is organized as follows. Part I reviews Floquet engineering and the incoherent dynamics of Floquet systems. In Chapter 2, we provide an overview of the connection between topology in solid-state quantum systems and its response to adiabatic driving fields. The topological adiabatic dynamics inspire questions about whether similar universal topological signatures of quantum matter can also emerge when the driving source is non-adiabatic. In Chapter 3, we relax the assumptions of adiabaticity by introducing Floquet theory, which describes the quantum dynamics of systems driven by time-periodic fields. We discuss the emergence of fast micromotion and slow stroboscopic dynamics in Floquet systems and introduce the time-independent Floquet Hamiltonian to model the stroboscopic dynamics. We derive the effective Floquet Hamiltonian for gapless systems such as graphene and semiconducting systems such as TMDs, characterize the topology of the Floquet Hamiltonian, and discuss how its band geometry can be probed through transport measurements. In Chapter 4, we discuss the effects of weak many-body interactions in Floquet systems and introduce the Floquet-Boltzmann equation. The remainder of the thesis, Parts II and III, are centered around two key themes. Part II: Tunability and Stability of Many-Body Floquet Topological Phases Part II of this thesis presents theoretical and experimental results centered around two primary goals. The first objective is to optically control many-body interactions in Floquet systems, and the second is to stabilize Floquet physics on long timescales by leveraging dissipative dynamics. Optical Control of Many-Body Interactions To understand how periodic drives can manipulate many-body interactions, it is helpful to review how optical drives can control single-particle electronic physics. A notable example of this is drive-induced topology in Floquet systems. In 2009, Takashi Oka and Hideo Aoki predicted that monolayer graphene systems irradiated 5 by a circularly polarized laser drive could be described by an effective Floquet Hamiltonian with a topological Haldane gap and a drive-induced DC anomalous Hall conductivity [215]. This drive-induced Haldane gap was a first demonstration of an off-resonant topological Floquet gap—where the gap opening (centered at the zero-energy Dirac point between the conduction and valence bands) was energetically far-separated from the single-photon resonance in the bands. In 2011, Netanel Lindner, Gil Refael, and Victor Galitski noticed that nontrivial band topology could also be induced in trivial bands by resonantly-driving the bands with a laser field [167]. The drive-induced topological system was named the “Floquet topological insulator” (FTIs). Further work formalized the topological characterization of Floquet states and elucidated its transport properties [139, 152]. Importantly, these works presented Floquet engineering as a novel method to optically control single-particle electronic physics on demand. While the single-particle dynamics of Floquet systems already present remarkable opportunities to engineer new quantum matter, an accurate modeling of Floquet systems must also consider many-body interactions. In driven quantum matter, many-body collisions only conserve energy up to an integer multiple of the driving frequency, enabling a dramatically different population distribution from equilibrium. In 2015, Seetharam et al. inroduced a microscopic modeling of many-body interactions in optically-driven semiconducting systems through the Floquet Boltzmann equation (FBE) approach [257, 258], which built on previous theoretical works studying interacting driven open quantum systems [90, 144]. The FBE leveraged Fermi’s golden rule for driven states to microscopically model perturbative many-body collisions. Crucially, it demonstrated that low-temperature steady states under continuous-wave irradiation could be induced by coupling the Floquet system to a low-temperature bosonic bath and suppressing heating processes via external leads. Further work by Esin et al. [74–76, 78] elucidated how many-body interactions in Floquet systems could be engineered to produce finite, and even quantized, topological transport characteristics and symmetry broken phases. Overall, these groundbreaking examples extended optical control over single-particle dynamics in continuously-driven Floquet systems to many-body interactions. This thesis seeks to leverage many-body interactions in low-dimensional systems as new principles for controlling Floquet topological insulators. Previous works that analyzed Floquet systems with the FBE (mentioned above) considered lowdimensional systems coupled to 3D phononic baths, where energy-momentum 6 constraints on electron-phonon scattering were relaxed. Chapter 5 of this thesis considers a new regime where phonons and electrons are both low-dimensional. Here, kinematic bottlenecks on electron-phonon scattering are highly constrained in phase space and can become highly sensitive to driving parameters. We identify a tunable kinematic constraint uniquely possible in moiré materials, where electronic and phononic speeds are comparable. In particular, we show that variations in the laser amplitude can drive Floquet states into a slow regime where their group velocity is subsonic to acoustic phonons, weakening electron-phonon coupling. In this regime, the drive-induced topological transport signatures are sharply modified, providing a novel example of optically-controllable topological electronic physics. Stabilizing Persistent Floquet Physics Having discussed theoretical proposals for Floquet topological insulators, we now discuss the experimental feasibility of Floquet engineering. Most experimental realizations of Floquet physics in solid-state systems thus far have utilized ultrafast lasers with pulse times in the range of femto to picoseconds to limit the effect of heating and leverage the intense laser amplitudes achievable by ultrafast laser sources. Notably, Floquet modifications to the single-particle electronic bandstructure have been confirmed by ARPES experiments [174, 187, 301, 329]. ARPES experiments have also offered some insights into the stability of Floquet states under many-body scattering in solid-state systems, establishing that large Floquet gaps, combined with low lattice temperatures, could be a necessity for Floquet-Bloch states to survive in the presence of the scattering of photoexcited carriers [6]. While ARPES experiments have provided direct probes of the hallmark Floquet bandstructure of periodically-driven systems, the signatures of Floquet bands have also been detected through optical signatures such as second-harmonic generation [260] and transport experiments such as the light-induced anomalous Hall effect in graphene [185]. Ultrafast laser experiments have been remarkably successful in producing transient Floquet effects, but stabilizing long-living signatures of Floquet physics in solids remains a long-sought goal. A truly continuous-wave (CW) irradiated Floquet system could increase the accessibility of Floquet effects and extend its tunability and utility. However, the heating and stability of continuously-driven Floquet systems seem to pose serious problems to its experimental realization. In this thesis, we seek to understand what experimental considerations are necessary to leverage dissipation dynamics to stabilize CW irradiated Floquet systems. Chapter 6 presents the first experimental signatures of Floquet steady states in a CW irradiated solid-state 7 metallic system. The frequency of the drive and temperature of the lattice were deliberately chosen to maintain low-temperature phonons, providing principles for cooling the system to a low-temperature steady state. We show that the longitudinal transport signatures are consistent with the predictions of the Floquet-Boltzmann equation and that Pauli blocking in the steady state can preserve the coherence of Floquet-Bloch states in the presence of scattering. Part III: Floquet Topological Physics Beyond Optical Driving Fields and LowDimensional Systems While Part II of this thesis primarily addresses the stability and topological characteristics of low-dimensional driven systems under continuous optical driving, Part III of this thesis seeks to extend persistent topological Floquet physics to (i) beyond-optical drives and (ii) 3D systems. The intense theoretical and experimental research on Floquet engineering has motivated new efforts to realize Floquet physics with novel driving sources, such as angular-momentum carrying light [8, 137], spatially-patterned light fields [255], multi-tone laser drives [180, 205], and spatiotemporal drives such as coherent phonon lasers [77, 224]. The remarkable topological signatures of these effects extend far beyond the predictions of Floquet topological physics driven by single-tone optical lasers. For example, a simultaneous application of incommensurate driving frequencies can produce the topological frequency conversion effect [180, 205]. Coherent phonon lasers, which transfer both energy and momentum, can also produce exotic topological bands with spatiotemporal symmetries [40, 91, 114, 224]. A notable new category of driving sources consists of high-frequency coherent phonon sources in the THz range. Historically, coherent phonon sources have been limited to ultra-low frequencies around the MHz or GHz range. While MHz and GHz frequency phonon sources are suitable for producing adiabatic dynamics, the more recent emergence of THz-frequency phonon sources offers exciting potential for Floquet engineering. These THz-frequency coherent phonon waves can be generated optically, by directly driving an IR-active phonon mode [68], by excitation of electrons that produce coherent phonons through electron-phonon coupling [67, 274, 317], or by biasing metals to produce the acoustic Cerenkov [10, 151, 213, 327], Klein-Zehner [5, 104], or phaser effect [77]. Chapter 7 of this thesis establishes new Floquet engineering applications for these high-frequency coherent phonon laser sources. In particular, we introduce a new type of Floquet phase with quantized 8 transport driven by THz-frequency coherent phonons. We use Floquet-Boltzmann theory to show that incoherent phonon dissipation in the system can relax electrons into a topological spatiotemporal Floquet band with a quantized group velocity. The resulting topological charge pumping effect remains remarkably robust in a highly non-adiabatic regime, producing currents orders of magnitude stronger than those in an adiabatic Thouless pump setup [7, 29, 32, 47, 71, 82, 111, 159, 201, 201, 278, 279]. Finally, Chapter 8 discusses the emergence of topological plasmons and photocurrents in Weyl semimetals irradiated by THz-frequency circularly-polarized lasers. In contrast with the works in other chapters of this thesis, the phenomenon is adiabatic in nature, but importantly highlights the interplay between time-periodic driving fields and transport phenomena beyond low-dimensional systems. In summary, this thesis presents key progress towards realizing persistent Floquet physics with optical and beyond-optical driving fields, addressing critical questions about the role of many-body interactions in the stability and tunability of Floquet quantum matter. 9 Chapter 2 TOPOLOGY IN ADIABATIC DYNAMICS We begin by reviewing some of the earliest demonstrations of topology in solidstate systems. These examples establish the intricate connection between topology and the adiabatic evolution of quantum systems. Section 2.1 defines when the dynamics of a time-dependent Hamiltonian can be classified as adiabatic. Section 2.2 reveals the connection between adiabatic electronic dynamics, Berry curvature, and topology. Section 2.3 derives explicit relations between the Berry curvature and electronic transport dynamics. Finally, Section 2.4 closes with discussions of the topology of 2-level systems and edge states, all of which will be extended to Floquet topological systems in Chapter 3. 2.1 Adiabatic Evolution The following section is based on Refs. [97, 287, 307, 330]. In this section, we model the adiabatic evolution of quantum systems and define the adiabatic condition. To study the dynamics produced by a time-dependent Hamiltonian, we consider a Hamiltonian 𝐻𝝀(𝑡) parameterized by a time-dependent vector of parameters, 𝝀(𝑡). By diagonalizing the instantaneous Hamiltonian at each time 𝑡, we obtain the in𝑛 , which satisfy the eigenvalue stantaneous eigenstates |𝑛𝝀(𝑡) ⟩ and eigenenergies 𝐸 𝝀(𝑡) relation 𝑛 𝐻𝝀(𝑡) |𝑛𝝀(𝑡) ⟩ = 𝐸 𝝀(𝑡) |𝑛𝝀(𝑡) ⟩, (2.1) where 𝑛 indexes the eigenstates. In our analysis, we assume that the instantaneous eigenenergies are non-degenerate at all times. To find the solutions to the timedependent Schrodinger equation, given by 𝑖ℏ𝜕𝑡 |Ψ(𝑡)⟩ = 𝐻𝝀(𝑡) |Ψ(𝑡)⟩, we expand the solution |Ψ(𝑡)⟩ in terms of the instantaneous eigenstates: ∫ ∑︁ 𝑖 𝑡 𝑚 ′ ℏ 0 𝐸 𝝀 (𝑡 ′ ) 𝑑𝑡 |𝑛𝝀(𝑡) ⟩, |Ψ(𝑡)⟩ = 𝑐 𝑚 (𝑡)𝑒 (2.2) (2.3) 𝑛 where the form of the coefficients 𝑐 𝑚 (𝑡) is chosen for convenience. The Schrodinger equation, Eq. (2.2), can now be written as a matrix equation in terms of the 10 coefficients 𝑐 𝑛 (𝑡), described by the rate equations ∫ ∑︁ 𝑖 𝑡 𝑚 𝑛 ′ ℏ 0 (𝐸 𝝀 (𝑡 ′ ) −𝐸 𝝀 (𝑡 ′ ) )𝑑𝑡 𝜕𝑡 𝑐 𝑚 (𝑡) = − ⟨𝑚 𝝀(𝑡) |𝜕𝑡 𝑛𝝀(𝑡) ⟩𝑐 𝑛 (𝑡)𝑒 . (2.4) 𝑛 Eq. (2.4) indicates that the time-dependence of the Hamiltonian generates transitions between distinct pairs of eigenstates |𝑛𝝀(𝑡) ⟩ and |𝑚 𝝀(𝑡) ⟩, with a transition rate of magnitude |⟨𝑚 𝝀(𝑡) |𝜕𝑡 𝑛𝝀(𝑡) ⟩𝑐 𝑛 (𝑡)|. We will now prove that the probability of tunneling transitions is suppressed if the parameters 𝝀(𝑡) are varied slowly. To simplify the analysis, we consider the time evolution of a system initialized at time 𝑡 = 0 in the 𝑛-th eigenstate of the instantaneous Hamiltonian, i.e., |Ψ(0)⟩ = |𝑛𝝀(0) ⟩. We focus on a regime where 𝝀(𝑡) is varied slowly such that ⟨𝑚 𝝀(𝑡) |𝜕𝑡 𝑛𝝀(𝑡) ⟩ and 𝑐 𝑚 (𝑡) vary with frequencies 𝑚 − 𝐸 𝑛 )/ℏ between eigenstates of the much slower than the level spacing (𝐸 𝝀(𝑡) 𝝀(𝑡) instantaneous Hamiltonian. In this limit, Eq. (2.4) can be integrated by parts to obtain ⟨𝑚 𝝀(𝑡) |𝜕𝑡 𝑛𝝀(𝑡) ⟩ ∫ 𝑡 (𝐸 𝑚 ′ −𝐸 𝑛 ′ )𝑑𝑡 ′ 𝑒 0 𝝀 (𝑡 ) 𝝀 (𝑡 ) 𝑐 𝑛 (𝑡) (2.5) 𝑐 𝑚 (𝑡) ≈ −𝑖ℏ 𝑚 𝑛 𝐸 𝝀(𝑡 ′ ) − 𝐸 𝝀(𝑡 ′) for 𝑛 ≠ 𝑚. The solution to the time-dependent Schrodinger equation can therefore be written as ∑︁ ⟨𝑚 𝝀(𝑡) |𝜕𝑡 𝑛𝝀(𝑡) ⟩ |Ψ(𝑡)⟩ ∝ |𝑛𝝀(𝑡) ⟩ − 𝑖ℏ |𝑚 𝝀(𝑡) ⟩, (2.6) 𝑚 𝑛 𝐸 − 𝐸 ′ ′ 𝝀(𝑡 ) 𝝀(𝑡 ) 𝑚≠𝑛 where we have dropped the normalization factor for clarity. The adiabatic limit is realized when 𝝀(𝑡) varies sufficiently slowly that the second term in Eq. (2.6) is negligible, and |Ψ(𝑡)⟩ ∝ |𝑛𝝀(𝑡) ⟩ at all times. To quantify the adiabatic limit, we note that differentiation of Eq. (2.1) yields ⟨𝑚 𝝀(𝑡) |𝜕𝑡 𝑛𝝀(𝑡) ⟩ = ⟨𝑚 𝝀(𝑡) |𝜕𝑡 𝐻𝝀(𝑡) |𝑛𝝀(𝑡) ⟩ . 𝑛 − 𝐸𝑚 𝐸 𝝀(𝑡) 𝝀(𝑡) (2.7) Upon substitution into Eq. (2.6), we find that the adiabatic condition is given by 𝑛 𝑚 2 |⟨𝑚 𝝀(𝑡) |ℏ𝜕𝑡 𝐻𝝀(𝑡) |𝑛𝝀(𝑡) ⟩| ≪ (𝐸 𝝀(𝑡) − 𝐸 𝝀(𝑡) ) . 2.2 (2.8) Berry Curvature and Chern Number The following section is based on Refs. [307]. As we showed in Section 2.1, an adiabatically-evolved system initialized in an instantaneous eigenstate of the Hamiltonian remains in the eigenstate at all times, satisfying |Ψ(𝑡)⟩ ∝ |𝑛𝝀(𝑡) ⟩. However, the state vector |Ψ(𝑡)⟩ can acquire a nontrivial 11 modulus-1 phase factor during the evolution. To restore the phase factor, we note that the matrix equation given in Eq. (2.4) contains only one nonzero element in the adiabatic limit, 𝜕𝑡 𝑐 𝑛 (𝑡) = −⟨𝑛𝝀(𝑡) |𝜕𝑡 𝑛𝝀(𝑡) ⟩𝑐 𝑛 (𝑡). The solution to this equation is given by |Ψ(𝑡)⟩ = 𝑒𝑖𝛾𝑛 (𝑡) |𝑛𝝀(𝑡) ⟩, (2.9) ∫𝑡 where 𝛾𝑛 (𝑡) = 𝑖 0 ⟨𝑛𝝀(𝑡 ′ ) |𝜕𝑡 𝑛𝝀(𝑡 ′ ) ⟩𝑑𝑡 ′. This phase 𝛾𝑛 (𝑡) can be expressed in terms of the Berry connection, 𝑨𝑛 (𝝀) = ⟨𝑛𝝀(𝑡) |∇𝝀 |𝑛𝝀(𝑡) ⟩, (2.10) using the relation ∫ 𝑨𝑛 (𝝀) · 𝑑𝝀, 𝛾𝑛 = 𝑖 (2.11) C where C defines the path traced out by 𝝀(𝑡) as a function of time. Furthermore, if C is a closed loop in 3D parameter space, then 𝛾𝑛 (𝑡) is called the Berry phase. For such a closed path, C encloses a closed surface, denoted S, and Stokes’ theorem implies that ∫ 𝛀𝝀𝑛 · 𝑑𝑺, 𝛾𝑛 = (2.12) S where 𝑑𝑺 is a differential area element on the surface S, and 𝛀𝝀𝑛 = ∇𝝀 × 𝑨𝑛 (𝝀) = 𝑖⟨∇𝝀 𝑛𝝀(𝑡) | × |∇𝝀 𝑛𝝀(𝑡) ⟩ (2.13) is the Berry curvature. A special case worth noting is that of a closed surface S, where 𝑒𝑖𝛾𝑛 (𝑡) is singled value, so 𝛾𝑛 = 2𝜋𝐶, and 𝐶 ∈ Z is known as the Chern number. Example: two-level system. Throughout this thesis, we will primarily focus on the dynamics and topological properties of low-temperature quantum matter, whose electronic physics can often be captured by a two-level Hamiltonian. In its most generic form, the Hamiltonian for a 2-level system is given by 𝐻 = 𝒅 · 𝝈, where 𝝈 = (𝜎𝑥 , 𝜎𝑦 , 𝜎𝑧 ) is the vector of Pauli matrices and 𝒅 is parameterized by three variables, 𝜃, 𝜙, and 𝑟 as follows: 𝒅 = 𝑟 (cos 𝜃 cos 𝜙, cos 𝜃 sin 𝜙, sin 𝜃). The eigenstates of the Hamiltonian are given by
!
! −𝑒 −𝑖𝜙 sin 𝜃/2 𝑒 −𝑖𝜙 cos 𝜃/2 |𝜓0 ⟩ = and |𝜓1 ⟩ = ,

cos 𝜃/2 sin 𝜃/2 (2.14) (2.15) 12 with eigenenergies 𝐸 0 = −𝑟 and 𝐸 1 = 𝑟, respectively. In the cartesian coordinates defined by 𝑥 = 𝑟 cos 𝜃 cos 𝜙, 𝑦 = 𝑟 cos 𝜃 sin 𝜙, and 𝑧 = 𝑟 sin 𝜃, explicit calculation yields the Berry curvature given by 𝛀 𝒅𝑛 = 𝑖⟨∇ 𝒅 𝜓𝑛 | × |∇ 𝒅 𝜓𝑛 ⟩ = (−1) 𝑛 𝒅 . | 𝒅| 3 (2.16) where 𝑛 = 0, 1 indexes the eigenstates |𝜓𝑛 ⟩. The Chern number of the system emerges when considering the Berry curvature along a closed surface S in parameter space, which is quantized with an integer value given by ∮ 1 𝐶𝑛 = 𝛀 𝒅𝑛 · 𝑑𝑺, (2.17) 2𝜋 S where 𝑑𝑺 is a differential element on the surface S. To prove the quantization of 𝐶𝑛 , note that ∇ 𝒅 𝛀 𝒅𝑛 = (−1) 𝑛 𝛿( 𝒅), where 𝛿( 𝒅) is the Dirac-Delta function. Therefore, applying the divergence theorem yields 𝐶𝑛 = (−1) 𝑛 if S encloses the origin 𝒅 = 0, and 𝐶𝑛 = 0 otherwise. 2.3 Manifestations of Topology in Adiabatic Transport Dynamics The Berry curvature can be detected in systems evolving adiabatically in time, producing robust topological transport signatures in electronic systems. Below, we provide two examples. Thouless Pumping in a Slow-Moving Turnstile The following section is based on Ref. [232]. The Thouless pump provides a simple example of topological transport in a 1D system. The system consists of electrons in a slowly-propagating wave potential, described by the Hamiltonian
ℏ2 𝑘ˆ 2 𝐻ˆ = + 2𝑉 cos 𝑞𝑥 − 𝜔𝑡 , 2𝑚 (2.18) where 𝑘ˆ = −𝑖𝜕𝑥 is the momentum operator, 𝑚 is the mass of the electron, 𝑉 is the energy scale of the potential wave, 1/𝑞 is the wavelength of the wave, and 𝜔 is its frequency. Because the moving wave potential is periodic in space, the electronic band structure can be folded into a Brillouin zone covering the crystal momenta 𝑘 ∈ [0, 𝑞]. We will focus on the effective Hamiltonian of the system surrounding 𝑞/2, given by  2 𝑞 1 + 𝑘 𝑉 𝑒 −𝑖𝜔𝑡 ª © 2𝑚 2 ­ 𝐻eff = ­ (2.19)   2 ®® . 𝑞 1 𝑖𝜔𝑡 𝑉𝑒 − + 𝑘 2𝑚 2 « ¬ 13 Up to a constant energy offset, 𝐻eff can be expressed in the form 𝐻eff = 𝒅 · 𝝈, with 𝒅 √︁ √︁ given in Eq. (2.14) and 𝑟 = (𝑘𝑞/2𝑚) 2 + 𝑉 2 , cos 𝜃 = (𝑘𝑞/2𝑚)/ (𝑘𝑞/2𝑚) 2 + 𝑉 2 and 𝜙 = 𝜔𝑡. The Hamiltonian has two instantaneous eigenstates, given by
!
! 𝑒 −𝑖𝜙 cos 𝜃/2 −𝑒 −𝑖𝜙 sin 𝜃/2 and |𝜓+ ⟩ = (2.20) |𝜓− ⟩ =

cos 𝜃/2 sin 𝜃/2 with energy 𝐸 − = −𝑟 and 𝐸 + = 𝑟, respectively. We assume that the system is initialized in the state |𝜓− ⟩ at 𝑡 = 0 and consider the current produced by the adiabatic evolution of the system as a function of 𝑡. To calculate the current, we consider the solution |Ψ(𝑡)⟩ to the time-dependent Schrodinger equation 𝑖ℏ𝜕𝑡 |Ψ(𝑡)⟩ = 𝐻eff (𝑡)|Ψ(𝑡)⟩ to the first order non-adiabatic correction, given by |Ψ(𝑡)⟩ ≈ |𝜓− ⟩ − 𝑖ℏ ⟨𝜓+ |𝜕𝑡 𝜓− ⟩ |𝜓+ ⟩ 𝐸+ − 𝐸− (2.21) up to an overall phase factor [see Eq. (2.6)]. The time-averaged total current therefore is given by ∫ ∫ 𝑞/2 𝑒𝜔 2𝜋/𝜔 1 𝜕𝐻eff 𝐽= |Ψ(𝑡)⟩, (2.22) 𝑑𝑡 𝑑𝑘 ⟨Ψ(𝑡)| 2𝜋 0 ℏ 𝜕𝑘 −𝑞/2 where 𝑒 is the electron charge. Using Eq. (2.21), we find that 1 𝜕𝐻eff 1 𝜕𝐻eff |Ψ(𝑡)⟩ = ⟨𝜓− | |𝜓− ⟩ − Ω(𝑘, 𝑡), (2.23) ℏ 𝜕𝑘 ℏ 𝜕𝑘   where Ω(𝑘, 𝑡) = 𝑖 ⟨𝜕𝑘 𝜓− |𝜕𝑡 𝜓− ⟩ − ⟨𝜕𝑡 𝜓− |𝜕𝑘 𝜓− ⟩ is the Berry curvature in (𝑘, 𝑡)space. The first term on the right side of Eq. (2.23) is the group velocity contribution to the current, and the second term is the quantum geometric contribution—the anomalous velocity. ⟨Ψ(𝑡)| To calculate 𝐽, we note that the group velocity [first term in Eq. (2.23)], is zero after integration, because 𝐸 − | 𝑘=𝑞/2 − 𝐸 − | 𝑘=−𝑞/2 = 0. Therefore, only the anomalous velocity remains: ∫ ∫ 𝑞/2 𝑒𝜔 2𝜋/𝜔 𝐽=− 𝑑𝑡 𝑑𝑘 Ω(𝑘, 𝑡). (2.24) 2𝜋 0 −𝑞/2 Already, we can see that the current should be quantized in integer multiples of 𝑒𝜔/2𝜋, because the integral over 𝑡 and 𝑘 traces out a curve in 𝒅-space fully enclosing the origin. We can arrive at the same result by performing the calculation explicitly, first through computation of the Berry connection A(𝑘, 𝑡) = ⟨𝜓− (𝜃, 𝜙)|𝜕𝑘 |𝜓− (𝜃, 𝜙)⟩ 𝑘ˆ + ⟨𝜓− (𝜃, 𝜙)|𝜕𝑡 |𝜓− (𝜃, 𝜙)⟩ 𝑡ˆ, (2.25) 14 where 𝑘ˆ and 𝑡ˆ are unit vectors in the (𝑘, 𝑡) phase space. Using Eq. (2.20), we find that     1 − cos 𝜃 𝜕𝜃 ˆ 𝜕𝜙 𝑑 1 − cos 𝜃 A(𝑘, 𝑡) = 𝑡ˆ and Ω(𝑘, 𝑡) = . (2.26) 𝑘+ 2 𝜕𝑘 𝜕𝑡 𝑑𝑘 2 Upon substitution into Eq. (2.24), we find that   𝜔 1 − cos 𝜃 1 − cos 𝜃 𝑒𝜔 𝐽= | 𝑘=𝑞/2 − | 𝑘=−𝑞/2 = , 2𝜋 2 2 2𝜋 (2.27) implying that a single electron is pumped across the system per driving period. Weak Electric Field and Quantized Hall Conductivity The following section is based on Refs. [97, 232, 307]. The Berry curvature of electronic states in a 2D lattice can be detected from the current response to a weak, DC electric field 𝑬. We consider a Bloch Hamiltonian 𝐻 ( 𝒌) parameterized by the crystal momentum 𝒌. The weak electric field is equivalent to a slowly-varying vector potential 𝑨(𝑡) = 𝑬 (𝑡)𝑡, which produces adiabatic electronic dynamics via minimal coupling 𝒌 → 𝒌 ′ = 𝒌 + 𝑒 𝑨(𝑡)/ℏ, yielding the time-dependent Hamiltonian 𝐻 ( 𝒌 ′) with instantaneous eigenstates |𝑢 𝒌 ′ 𝑛 ⟩ and eigenenergies 𝐸 𝒌 ′ 𝑛 indexed by the integer 𝑛, which satisfy the eigenvalue relation 𝐻 ( 𝒌 ′)|𝑢 𝒌 ′ 𝑛 ⟩ = 𝐸 𝒌 ′ 𝑛 |𝑢 𝒌 ′ 𝑛 ⟩. (2.28) Here, we have absorbed the time dependence into 𝒌 ′ for simplicity of the notation. We consider the solution |𝜓 𝒌𝑛 (𝑡)⟩ to the time-dependent Schrodinger equation 𝑖ℏ𝜕𝑡 |𝜓 𝒌𝑛 (𝑡)⟩ = 𝐻 ( 𝒌 ′)|𝜓 𝒌𝑛 (𝑡)⟩ initialized in the 𝑛-th instantaneous eigenstate at time 𝑡 = 0. To the first order non-adiabatic correction, ∑︁ ⟨𝑢 ′ |𝜕𝑡 𝑢 ′ ⟩ 𝒌𝑚 𝒌𝑛 |𝑢 𝒌 ′ 𝑚 ⟩ (2.29) |𝜓 𝒌𝑛 (𝑡)⟩ ≈ |𝑢 𝒌 ′ 𝑛 ⟩ − 𝑖ℏ ′ 𝐸 − 𝐸 𝒌𝑚 𝒌′𝑛 𝑚≠𝑛 up to an overall phase factor. The group velocity, given by 𝒗 𝒌𝑛 = ⟨𝜓 𝒌𝑛 (𝑡)| 1 𝜕𝐻 ( 𝒌) |𝜓 𝒌𝑛 (𝑡)⟩, ℏ 𝜕𝒌 (2.30) can be evaluated by using the relation 𝜕/𝜕𝑡 = −(𝑒/ℏ)𝑬 · ∇ 𝒌 and noting that ⟨𝜓 𝒌𝑚 (𝑡)|(𝜕𝐻 ( 𝒌)/𝜕 𝒌)|𝜓 𝒌𝑛 (𝑡)⟩ = (𝐸 𝒌𝑚 − 𝐸 𝒌𝑛 )⟨𝜓 𝒌𝑚 (𝑡)|∇ 𝒌 |𝜓 𝒌𝑛 (𝑡)⟩ [as determined by differentiating both sides of Eq. (2.28) with respect to 𝒌]. Combining these results, we find that 1 𝜕𝐸 𝒌𝑛 𝑒 𝒗 𝒌𝑛 = − 𝑬 × 𝛀 𝒌𝑛 , (2.31) ℏ 𝜕𝒌 ℏ 15 where 𝛀 𝒌𝑛 = 𝑖⟨∇ 𝒌 𝑢 𝒌𝑛 | × |∇ 𝒌 𝑢 𝒌𝑛 ⟩ (2.32) is the Berry curvature. Eq. (2.31) is the semiclassical equations of motion for electrons under weak electric fields, and the second term in Eq. (2.31) is known as the anomalous velocity. The semiclassical equations of motion enable us to estimate the current flowing through the system, given by ∑︁ ∫ 𝑑2 𝒌 𝑱=𝑒 𝒗 𝑓 , (2.33) 2 𝒌𝑛 𝒌𝑛 BZ (2𝜋) 𝑛 where 𝑛 sums over the bands of the system, the integral over 𝒌 covers the full Brillouin zone (BZ), and 𝑓 𝒌𝑛 denotes the occupation function of the state |𝑢 𝒌𝑛 ⟩. Without loss of generality, we can align 𝑬 = 𝐸 𝑥 𝑥ˆ and compute Hall conductivity 𝜎𝑥𝑦 = 𝐽𝑦 /𝐸 𝑥 , given by ∑︁ 𝑒 2 ∫ 𝑑 2 𝒌 𝛀 𝒌𝑛 · 𝑧ˆ 𝑓 𝒌𝑛 . (2.34) 𝜎𝑥𝑦 = ℏ BZ 𝑛 If the chemical potential of the system is chosen such that an integer number of bands are fully-filled, then the Hall conductivity is proportional to the sum over the Chern numbers 𝐶𝑛 of the fully-filled bands, where ∫ ∑︁ 𝑒 2 1 𝐶𝑛 and 𝐶𝑛 = 𝑑 2 𝒌 𝛀 𝒌𝑛 · 𝑧ˆ. (2.35) 𝜎𝑥𝑦 = 2𝜋ℏ 2𝜋 BZ 𝑛∈filled In the Brillouin zone, it can be shown that 𝐶𝑛 ∈ Z by noting that ∫ ∮ 1 𝑖 2 𝐶𝑛 = 𝑑 𝒌 𝛀 𝒌𝑛 · 𝑧ˆ = 𝑑 𝒌 · ⟨𝑢 𝒌𝑛 |∇ 𝒌 |𝑢 𝒌𝑛 ⟩ 2𝜋 BZ 2𝜋 (2.36) where the integral on the left hand side is performed along the boundary of the first Brillouin zone (1BZ). To prove the quantization of 𝐶𝑛 , we leverage the periodicity of the Brillouin zone, noting that the Bloch function |𝑢 𝒌𝑛 ⟩ should return to itself upon the transformation 𝒌 → 𝒌 + 𝑮, where 𝑮 is a reciprocal lattice vector, but only up to a phase factor. We can account for the phase factor by defining |𝑢 𝒌𝑛 ⟩ = 𝑒 −𝑖𝜃 𝑛 ( 𝒌) | 𝑢˜ 𝒌𝑛 ⟩, where 𝜃 𝑛 ( 𝒌) is a real-valued phase factor and | 𝑢˜ 𝒌𝑛 ⟩ is invariant upon 𝒌 → 𝒌 + 𝑮. ∮ Noting that 𝑑 𝒌 · ⟨𝑢˜ 𝒌𝑛 |∇ 𝒌 | 𝑢˜ 𝒌𝑛 ⟩ = 0 due to the periodicity of | 𝑢˜ 𝒌𝑛 ⟩ upon the transformation 𝒌 → 𝒌 + 𝑮, we find that ∮ 1 𝐶𝑛 = 𝑑 𝒌 · ∇ 𝒌 𝜃 𝑛 ( 𝒌). (2.37) 2𝜋 16 Furthermore, because 𝑒 −𝑖𝜃 𝑛 ( 𝒌) is single-valued on a closed loop around the 1BZ, 𝐶𝑛 must be quantized. Two-level system. In a two-level system [see definition above Eq. (2.14)], we can derive the Chern number from a simple geometric picture. It can be shown [see Ref. [232]] that the Chern number can be expressed as ∫ 𝑛 1 ˆ 𝐶𝑛 = (−1) 𝑑 2 𝒌 𝒅ˆ · (𝜕𝑘 𝑥 𝒅ˆ × 𝜕𝑘 𝑦 𝒅), (2.38) 4𝜋 BZ where 𝑛 = 0, 1 which counts the oriented solid angle swept by 𝒅ˆ ≡ 𝒅/| 𝒅| upon variation across the Brillouin zone (BZ), divided by 4𝜋. Because the BZ is a closed manifold, 𝐶 is quantized. Thus, we can interpret 𝐶 as the number of times 𝒅ˆ fullycovers the Bloch sphere (or, equivalently, the number of times the surface traced out by 𝒅 fully encloses the origin 𝒅 = 0) upon variation across the BZ. 2.4 Edge States and Topology in Graphene The following section is based on Ref. [97]. To illustrate the emergence of topology in a solid-state system, we consider the Hamiltonian of graphene modified by the addition of a mass gap. Near the 𝐾 and 𝐾 ′ valleys, indexed by 𝜉 = +1, −1, respectively, the system can be described by the Dirac Hamiltonian 𝐻𝜉 ( 𝒌) = ℏ𝑣 𝐹 (𝑘 𝑥 𝜎𝑥 + 𝜉 𝑘 𝑦 𝜎𝑦 ) + 𝑚 𝜉 𝜎𝑧 , (2.39) where 𝑣 𝐹 is the Fermi velocity and 𝑚 𝜉 is the mass gap. The eigenergies of the √︃ √︃ Hamiltonian are given by 𝐸 𝑛 ( 𝒌) = (−1) 𝑛 (ℏ𝑣 𝐹 𝑘) 2 + 𝑚 2𝜉 , where 𝑘 = 𝑘 𝑥2 + 𝑘 2𝑦 and 𝑛 = 0, 1 indexes the quasienergy. Topology To determine the topology of the system, we consider the winding of the pseudospin √︃ ˆ 𝒅(𝑘, 𝜙) = (ℏ𝑣 𝐹 𝑘 cos 𝜙 𝒌 , ℏ𝑣 𝐹 𝑘 sin 𝜙 𝒌 , 𝑚 𝜉 )/ (ℏ𝑣 𝐹 𝑘) 2 + 𝑚 2𝜉 (2.40) √︃ within each valley, where we have used the polar coordinates 𝑘 = 𝑘 𝑥2 + 𝑘 2𝑦 and
𝜙 𝒌 = arctan 𝑘 𝑦 /𝑘 𝑥 . To count the winding of the pseudospin, we track the evolution ˆ ˆ of 𝒅(𝑘, 𝜙) as a function of of 𝑘, starting at 𝑘 = 0. At 𝑘 = 0, 𝒅(𝑘, 𝜙) = sign(𝑚 𝜉 ) 𝑧ˆ is aligned with the 𝑧-axis, with an orientation determined by the sign of the mass term 𝑚 𝜉 . As 𝑘 is increased, the pseudospin begins to approach the 𝑘 𝑥 -𝑘 𝑦 plane, winding 17 clockwise as a function of increasing 𝜙 𝒌 if 𝜉 = +1, or counterclockwise if 𝜉 = −1. Thus, the pseudospin covers half of the Bloch sphere, tracing out a solid angle of 𝜉 2𝜋 sign(𝑚 𝜉 )𝜉 and contributing a Chern number of 𝐶𝑛 = (−1) 𝑛 sign(𝑚 𝜉 )𝜉/2 to the band indexed by 𝑛 = 0, 1. In order to calculate the total Chern number 𝐶𝑛 of the electronic bands in the Brillouin zone, we add the contributions from the 𝐾 and 𝐾 ′ valleys, using the relation 𝐶𝑛 = 𝐶𝑛+1 + 𝐶𝑛−1 . For a regular mass term, given by 𝑚 𝜉 = 𝑚, the bands are trivial (𝐶𝑛 = 0) because 𝐶𝑛+1 = −𝐶𝑛−1 . For a Haldane (topological) mass term, given 𝑚 𝜉 = 𝜉𝑚 with opposite signs in the 𝐾 and 𝐾 ′ valleys, we find that 𝐶𝑛 = (−1) 𝑛 sign(𝑚), giving rise to topologically-nontrivial Chern bands. Edge States An important consequence of topologically-nontrivial bands is the emergence of edge states localized to the spatial interfaces between topologically-distinct phases. Let us consider a slab of graphene with a Haldane mass gap, extending infinitely along the 𝑦-direction but with a finite length along the 𝑥-direction, terminating at the right and left edges positioned at 𝑥 = 𝑥 𝑅 and 𝑥 = 𝑥 𝐿 , respectively, where 𝑥 𝐿 < 𝑥 𝑅 . The edges in the 𝑥-direction are interfaced with graphene with a regular mass gap. To understand the emergence of edge states, we analyze the Schrodinger equation near the right edge for different values of 𝑘 𝑦 , first for a single valley indexed by 𝜉. Starting at 𝑘 𝑦 = 0, we find that the Schrodinger equation is given by ! ! ! 𝑚 𝜉 (𝑥) −𝑖ℏ𝑣 𝐹 𝜕𝑥 𝜓1 𝜓1 =𝜀 (2.41) −𝑖ℏ𝑣 𝐹 𝜕𝑥 −𝑚 𝜉 (𝑥) 𝜓2 𝜓2 where 𝑚 𝜉 = 𝜉𝑚 (corresponding to a Haldane gap) in the region 𝑥 𝐿 < 𝑥 < 𝑥 𝑅 , and 𝑚 𝜉 = 𝑚 elsewhere. To determine the energy 𝜀 of the edge mode, we focus on the Hamiltonian 𝐻𝜉 evaluated at 𝑘 𝑦 = 0, which anticommutes with 𝜎𝑦 , i.e., {𝐻𝜉 , 𝜎𝑦 } = 0. This symmetry relation implies that an eigenstate |𝜓⟩ with energy 𝜀 must come in a pair with an eigenstate 𝜎𝑦 |𝜓⟩ with energy −𝜀, except if 𝜀 = 0, implying that the edge mode must have zero energy, 𝜀 = 0. Solving Eq. (2.45) for the zero-energy state, we find that −𝑖ℏ𝑣 𝐹 𝜕𝑥 𝜓2 = −𝑚 𝜉 𝜓1 and − 𝑖ℏ𝑣 𝐹 𝜕𝑥 𝜓1 = 𝑚 𝜉 𝜓2 , whose solutions are given by ! ! 𝜓1 1 −𝑖 sign(𝑚 𝜉 ) −|𝑚||𝑥−𝑥 𝑅 |/(ℏ𝑣 𝐹 ) , =√ 𝑒 1 𝜓2 2 (2.42) (2.43) 18 and ! ! 𝜓1 1 𝑖 sign(𝑚 𝜉 ) −|𝑚||𝑥−𝑥 𝐿 |/(ℏ𝑣 𝐹 ) =√ 𝑒 𝜓2 1 2 (2.44) near the right and left edges, respectively. Notice that the edge states are localized to the edges of the system, located at 𝑥 = 𝑥 𝐿 , 𝑥 𝑅 . We can now generalize the analysis to nonzero values of 𝑘 𝑦 , where the Schrodinger equation now reads ! ! !    𝑚 𝜉 (𝑥) −𝑖ℏ𝑣 𝐹 𝜕𝑥  𝜓1 𝜓 1    −𝑖ℏ𝑣 𝜕 −𝑚 (𝑥) + 𝜉ℏ𝑣 𝐹 𝑘 𝑦 𝜎𝑦  𝜓 = 𝜀 𝜓 . 𝐹 𝑥 𝜉 2   2   (2.45) The wave vectors given in Eqs. (2.43)-(2.44) remain solutions to the Eq. (2.45), except with an energy shifted to 𝐸 𝑅 (𝑘 𝑦 ) = 𝜉 sign(𝑚 𝜉 )ℏ𝑣 𝐹 𝑘 𝑦 at 𝑥 = 𝑥 𝑅 and 𝐸 𝐿 (𝑘 𝑦 ) = −𝜉 sign(𝑚 𝜉 )ℏ𝑣 𝐹 𝑘 𝑦 at 𝑥 = 𝑥 𝐿 . Importantly, when combining the results for the 𝐾 and 𝐾 ′ valleys, the edge states are chiral, because the edge states exhibit opposite group velocity in the 𝑦-direction on the right and left edges. 19 Chapter 3 COHERENT DYNAMICS AND TOPOLOGY OF FLOQUET SYSTEMS The topological transport properties of adiabatically driven systems, reviewed in Chapter 2, motivate questions about how topology could play a role in the universal properties of non-adiabatically driven systems. Floquet theory leverages a common feature of many driving fields—time-periodicity—to model its effects on quantum systems without assumptions of adiabaticity. A remarkable prediction of Floquet theory is the emergence of drive-induced band topology, which leads to transport signatures such as an anomalous Hall conductivity. We provide an overview of these predictions in this chapter. Before discussing the topological properties of Floquet systems, we begin with a broad overview of the key results of Floquet theory. Floquet theory provides a framework to solve the time-dependent Schrodinger equation, 𝑖ℏ𝜕𝑡 |Ψ(𝑡)⟩ = 𝐻 (𝑡)|Ψ(𝑡)⟩, (3.1) when the Hamiltonian is time-periodic: 𝐻 (𝑡) = 𝐻 (𝑡 + 𝑇). In particular, the Schrodinger equation is solved by computing the unitary evolution operator −𝑖 ∫𝑡 𝑈 (𝑡, 𝑡0 ) = T 𝑒 ℏ 𝑡0 𝐻 (𝑡 ′ )𝑑𝑡 ′ (3.2) relating the state |Ψ(𝑡)⟩ = 𝑈 (𝑡, 𝑡0 )|Ψ(𝑡0 )⟩ at arbitrary times 𝑡 to the initial state |Ψ(𝑡0 )⟩ at time 𝑡0 , where T denotes time-ordering. The key result of Floquet theory is that a time-independent effective Floquet Hamiltonian 𝐻 𝐹 [𝑡0 ] can fully capture the dynamics of the system at stroboscopic times 𝑡 = 𝑡0 + 𝑚𝑇, where 𝑚 ∈ Z [196, 241, 290]. In particular, we can write 𝑈 (𝑡0 + 𝑚𝑇, 𝑡0 ) = 𝑒 −𝑖𝐻𝐹 [𝑡0 ]×𝑚𝑇/ℏ . (3.3) Furthermore, the set of eigenvectors |𝑢 𝑛 ⟩ and eigenvalues 𝜀 𝑛 (“quasienergy”) of the Floquet Hamiltonian 𝐻 𝐹 [𝑡0 ], indexed by the integer 𝑛, provides a simple expression for |Ψ(𝑡)⟩ at stroboscopic times, ∑︁ |Ψ(𝑡 0 + 𝑚𝑇)⟩ = 𝑎 𝑛 𝑒 −𝑖𝜀 𝑛 𝑚𝑇 |𝑢 𝑛 ⟩, (3.4) 𝑛 20 where 𝑎 𝑛 are time-independent coefficients dependent only on the initial state |Ψ(𝑡0 )⟩. The Floquet Hamiltonian 𝐻 𝐹 [𝑡0 ] therefore provides a simple description of the coarse-grained, slow dynamics of the system on stroboscopic timescales longer than the period of the drive. These slow dynamics are often sufficient to describe the salient properties of Floquet systems. For instance, many-body scattering processes and low-frequency transport measurements of optically-driven systems, which are the primary focus of this thesis, occur on timescales much longer than the period of most laser drives [74, 258]. Importantly, under certain conditions, 𝐻 𝐹 [𝑡0 ] can also possess topological properties [167, 215]. The rest of the chapter is structured as follows. In Section 3.1, we introduce the Floquet Hamiltonian 𝐻 𝐹 and the micromotion operator 𝐾 𝐹 , which respectively describe the dynamics of Floquet systems on timescales longer and shorter than the period of the drive. Section 3.2 seeks to find approximate analytic forms for 𝐻 𝐹 using the high-frequency expansion and the rotating wave approximation, respectively applicable when the driving frequency is off-resonant and resonant to the energy level spacing of the equilibrium system. Section 3.3 describes a numerical scheme to solve for |Ψ(𝑡)⟩ using a frequency-space expansion when the approximation methods in Section 3.2 may not apply. Section 3.4 provides an example of the Floquet Hamiltonian for graphene-based systems. Section 3.5 describes the emergence of topology in Floquet systems. And, finally, Section 3.6 describes how drive-induced topology can emerge in the transport properties of Floquet systems. 3.1 Floquet Theory: Separating Stroboscopic Dynamics from Micromotion The following section is based on Refs. [196, 241, 290]. In this section, we seek to separate the evolution operator 𝑈 (𝑡, 𝑡0 ) = T 𝑒 − ℏ𝑖 ∫𝑡 𝑡0 𝐻 (𝑡 ′ )𝑑𝑡 ′ (3.5) into components that dictate the dynamics of Floquet systems on timescales longer and shorter than the period of the drive. As we will see in Section 3.3 of this chapter, the slow dynamics can often be approximated in analytic form and provide key insights into the behavior of Floquet systems on long timescales. Slow and Fast Dynamics The evolution of a Floquet system from time 𝑡0 to time 𝑡 can be separated into two steps: a stroboscopic evolution from time 𝑡0 to time 𝑡 0 + 𝑛𝑇, where 𝑛 ∈ Z, and a 21 Figure 3.1: Schematic illustration of slow (stroboscopic-timescale) dynamics and fast (micromotion) dynamics of a Floquet system with a time-periodic Hamiltonian 𝐻 (𝑡) = 𝐻 (𝑡 + 𝑇). A system is initialized at time 𝑡0 in the state |Ψ(𝑡0 )⟩. We compare the time-dependence of the expectation value of an operator 𝑂ˆ under evolution by the full time-dependent Schrodinger equation |Ψ(𝑡)⟩ = 𝑈 (𝑡, 𝑡0 )|Ψ(𝑡0 )⟩ (orange h i ∫𝑡 ′ ′ curve), where 𝑈 (𝑡, 𝑡0 ) = T exp −(𝑖/ℏ) 𝑡 𝐻 (𝑡 )𝑑𝑡 , and under the effective time0 independent Floquet Hamiltonian | Ψ̃(𝑡)⟩ = 𝑒𝑖𝐻𝐹 (𝑡−𝑡0 ) |Ψ(𝑡0 )⟩ (blue curve). The Floquet Hamiltonian 𝐻 𝐹 captures the slow course-grained dynamics of the system on timescales longer than the period of the drive 𝑇 (see blue curve), while the full solution to the time-dependent Schrodinger equation captures the fast dynamics (micromotion) within each time period (see orange curve). The orange and blue curves agree at stroboscopic times given by 𝑡0 + 𝑛𝑇, where 𝑛 ∈ Z, where |Ψ(𝑡 0 + 𝑛𝑇)⟩ = | Ψ̃(𝑡0 + 𝑛𝑇)⟩. Figure based on Ref. [290]. sub-cycle evolution from the stroboscopic time 𝑡0 + 𝑛𝑇 to the final time 𝑡 = 𝑠 + 𝑛𝑇, where 𝑛 is chosen such that 0 ≤ 𝑠 − 𝑡0 < 𝑇. The evolution operator 𝑈 (𝑡, 𝑡0 ) can therefore be written as 𝑈 (𝑡, 𝑡0 ) = 𝑈 (𝑠 + 𝑛𝑇, 𝑡0 + 𝑛𝑇)𝑈 (𝑡 0 + 𝑛𝑇, 𝑡0 ). (3.6) Noting that 𝑈 (𝑠 + 𝑛𝑇, 𝑡0 + 𝑛𝑇) = 𝑈 (𝑡, 𝑡0 ), we find that 𝑈 (𝑡, 𝑡0 ) = 𝑈 (𝑠, 𝑡0 )𝑈 (𝑡0 + 𝑛𝑇, 𝑡0 ). (3.7) The operators 𝑈 (𝑠, 𝑡0 ) and 𝑈 (𝑡0 + 𝑛𝑇, 𝑡0 ) capture the micromotion and stroboscopic time evolution, respectively, which we now analyze separately: • Stroboscopic dynamics and the Floquet Hamiltonian. The stroboscopic evolution operator 𝑈 (𝑡0 + 𝑛𝑇, 𝑡0 ) can be separated into 𝑛 single-period evolutions: 22 𝑈 (𝑡0 + 𝑛𝑇, 𝑡0 ) = [𝑈 (𝑡0 + 𝑇, 𝑡0 )] 𝑛 . By definition, the evolution operator across each period is given by 𝑈 (𝑡0 + 𝑇, 𝑡0 ) = 𝑒 −𝑖𝐻𝐹 [𝑡0 ]𝑇/ℏ (3.8) where the time-dependent operator 𝐻 𝐹 [𝑡0 ] is called the Floquet Hamiltonian. • Micromotion and micromotion operator. If we wish to determine the state |Ψ(𝑡)⟩ at an arbitrary time 𝑡 = 𝑠 + 𝑛𝑇, it would be incorrect to evolve the state with the Floquet Hamiltonian 𝐻 𝐹 [𝑡0 ], i.e., |Ψ(𝑡)⟩ ≠ 𝑒 −𝑖𝐻𝐹 [𝑡0 ]×(𝑡−𝑡0 )/ℏ |Ψ(𝑡0 )⟩. Instead, another operator 𝐾 𝐹 [𝑡0 ] (𝑠) is needed to restore information about the sub-cycle micromotion of the system from time 𝑡0 to 𝑠, such that |Ψ(𝑡)⟩ = 𝑒 −𝑖𝐾𝐹 [𝑡0 ] (𝑠)/ℏ 𝑒 −𝑖𝐻𝐹 [𝑡0 ]×(𝑡−𝑡0 )/ℏ |Ψ(𝑡 0 )⟩. (3.9) The operator 𝐾 𝐹 [𝑡0 ] (𝑠) is called the micromotion (or kick) operator. The definitions of 𝐻 𝐹 [𝑡0 ] and 𝐾 𝐹 [𝑡 0 ] can be used to rewrite the evolution operator in the form 𝑈 (𝑡, 𝑡0 ) = 𝑒 −𝑖𝐾𝐹 [𝑡0 ] (𝑠)/ℏ 𝑒 −𝑖𝐻𝐹 [𝑡0 ]×(𝑡−𝑡0 )/ℏ . (3.10) State Vector With knowledge of the Floquet Hamiltonian 𝐻 𝐹 [𝑡0 ] and the micromotion operator 𝐾 𝐹 [𝑡0 ], we can now find an explicit form for |Ψ(𝑡)⟩ using the eigenstates |𝑢 𝑛 ⟩ and quasienergies 𝜀 𝑛 of the Floquet Hamiltonian 𝐻 𝐹 , which satisfy the eigenvalue relation 𝐻 𝐹 [𝑡0 ]|𝑢 𝑛 ⟩ = 𝜀 𝑛 |𝑢 𝑛 ⟩. (3.11) We assume that the system is initialized in the state |Ψ(𝑡0 )⟩ at time 𝑡 = 𝑡0 , which can be expanded in terms of |𝑢 𝑛 ⟩ as follows: ∑︁ |Ψ(𝑡0 )⟩ = 𝑎 𝑛 |𝑢 𝑛 ⟩. (3.12) 𝑛 Using Eq. (3.10) to evaluate |Ψ(𝑡)⟩ = 𝑈 (𝑡, 𝑡0 )|Ψ(𝑡 0 )⟩ yields ∑︁ |Ψ(𝑡)⟩ = 𝑎 𝑛 𝑒 −𝑖𝜀 𝑛 (𝑡−𝑡0 )/ℏ |Φ𝑛 (𝑡)⟩, (3.13) 𝑛 where |Φ𝑛 (𝑡)⟩ = 𝑒 −𝑖𝐾𝐹 [𝑡0 ] (𝑡)/ℏ |𝑢 𝑛 ⟩ (3.14) 23 is periodic in time, i.e., |Φ𝑛 (𝑡)⟩ = |Φ𝑛 (𝑡 + 𝑇)⟩. The primary result here is that solutions to the time-dependent Schrodinger equation, |Ψ(𝑡)⟩, [see Eq. (3.13)] can be expanded in the complete basis given by the Floquet states |𝜓𝑛 (𝑡)⟩ = 𝑒 −𝑖𝜀 𝑛 𝑡/ℏ |Φ𝑛 (𝑡)⟩, (3.15) where 𝜀 𝑛 is the quasienergy of the Floquet state. Coarse-Grained Dynamics When only the coarse-grained dynamics of the system on timescales much longer than the driving period are relevant, we can approximate the expectation values of observables by using the state ∑︁ | Ψ̃(𝑡)⟩ = 𝑎 𝑛 𝑒 −𝑖𝜀 𝑛 (𝑡−𝑡0 ) |𝑢 𝑛 ⟩, (3.16) 𝑛 which is exact at stroboscopic times, i.e., | Ψ̃(𝑡 0 + 𝑚𝑇)⟩ = |Ψ(𝑡0 + 𝑚𝑇)⟩, where 𝑚 ∈ Z. This can be deduced from Eq. (3.14) by noticing that |Φ𝑛 (𝑡0 + 𝑚𝑇)⟩ = |𝑢 𝑛 ⟩, because 𝐾 𝐹 [𝑡0 ] (𝑡0 ) = 0 by definition [see Eq. (3.10)]. In Figure 3.1, we provide a schematic illustration of how the expectation values of an operator 𝑂ˆ may differ when evaluated with | Ψ̃(𝑡)⟩ and |Ψ(𝑡)⟩. Notice that the two results agree at stroboscopic times and | Ψ̃(𝑡)⟩ can capture the coarsegrained dynamics on slow timescales. Only knowledge of the Floquet Hamiltonian 𝐻 𝐹 [𝑡0 ] is necessary to solve for | Ψ̃(𝑡)⟩. In Figure 3.1, we have schematically illustrated micromotion oscillations with amplitudes weaker than the stroboscopic dynamics. This can be justified more rigorously with the frequency space expansion, introduced in Section 3.3, which predicts a Wannier-Stark like localization of the Fourier harmonics of Floquet states that suppresses the amplitude of micromotion with frequencies much higher than the driving frequency [241]. Gauge Choice In the discussion above, we implicitly chose a gauge given by 𝐾 𝐹 [𝑡0 ] = 0. The gauge choice is not unique. For instance, a choice of a different initial time 𝑡0′ gives rise to a different Floquet Hamiltonian and micromotion operator given by 𝐻 𝐹 [𝑡0′ ] = 𝑈 (𝑡0′ , 𝑡0 )𝐻 𝐹 [𝑡0 ]𝑈 † (𝑡 0′ , 𝑡0 ) and ′ 𝑒𝑖𝐾𝐹 [𝑡0 ] (𝑡) = 𝑈 (𝑡0′ , 𝑡0 )𝑒𝑖𝐾𝐹 [𝑡0 ] (𝑡) , (3.17) respectively. Alternatively, the gauge defined by 𝐻 𝐹 = 𝑒𝑖𝐾𝐹 (𝑡0 ) 𝐻 𝐹 [𝑡0 ]𝑒 −𝑖𝐾𝐹 (𝑡0 ) and 𝑒𝑖𝐾𝐹 (𝑡) = 𝑒𝑖𝐾𝐹 (𝑡0 ) 𝑒𝑖𝐾𝐹 [𝑡0 ] (𝑡) (3.18) 24 can be chosen to obtain a Floquet Hamiltonian 𝐻 𝐹 and micromotion operator 𝐾 𝐹 (𝑡) with no initial time dependence. In the latter gauge choice, the evolution operator also takes a form 𝑈 (𝑡 𝑓 , 𝑡𝑖 ) = 𝑒 −𝑖𝐾𝐹 (𝑡 𝑓 ) 𝑒 −𝑖𝐻𝐹 (𝑡 𝑓 −𝑡𝑖 ) 𝑒𝑖𝐾𝐹 (𝑡𝑖 ) independent of 𝑡 0 . Note that substitution of the evolution operator into the Schrodinger equation provides another equivalent definition of 𝐻 𝐹 and 𝐾 𝐹 . In particular, the micromotion operator 𝐾 𝐹 is the operator that transforms the operator 𝑄(𝑡) = 𝐻 (𝑡) − 𝑖ℏ𝜕𝑡 , sometimes known as the “quasienergy operator,” into a time-independent operator, which is precisely the Floquet Hamiltonian: 𝐻 𝐹 = 𝑒𝑖𝐾𝐹 (𝑡) [𝐻 (𝑡) − 𝑖ℏ𝜕𝑡 ]𝑒 −𝑖𝐾𝐹 (𝑡) . (3.19) This definition can sometimes be useful when the form of 𝑄(𝑡) is simple and the form of 𝑒𝑖𝐾𝐹 (𝑡) can be guessed by observation, such as in the rotating wave approximation that will be presented in Section 3.2. Note that the form of 𝐾 𝐹 (𝑡) and 𝐻 𝐹 are still not unique, and Eq. (3.19) can still be satisfied by a different set of operators ′ 𝐻 𝐹′ = 𝑀 𝐻 𝐹 𝑀 † and 𝑒𝑖𝐾𝐹 (𝑡) = 𝑀𝑒𝑖𝐾𝐹 (𝑡) related by the unitary operator 𝑀. For instance, a choice of 𝐾 𝐹 (𝑡0 ) = 0 restores the gauge choice used at the beginning of this section. 3.2 Analytic Approximations to the Floquet Hamiltonian It is sometimes impossible to obtain closed analytic forms for the operators 𝐾 𝐹 and 𝐻 𝐹 . In this section, we discuss two limits in which approximate forms for 𝐾 𝐹 , 𝐻 𝐹 , or both can be obtained. Throughout this section, we will consider a generic time-periodic system described by the Hamiltonian 𝐻 (𝑡) = 𝐻0 + 𝐻1 (𝑡), (3.20) where 𝐻0 is the static Hamiltonian, 𝐻1 (𝑡) = 𝐻1 (𝑡 + 2𝜋/𝜔), and 𝜔 is the angular frequency of the driving field. High-Frequency Expansion The following section is based on Refs. [196, 236]. The high-frequency expansion approximates the Floquet Hamiltonian 𝐻 𝐹 when the spectral level spacing between relevant states of the static Hamiltonian 𝐻0 is much smaller than the frequency ℏ𝜔 of the drive. To perform the expansion, we first rewrite ∑︁ 𝐻1 (𝑡) = 𝐻1(𝑚) 𝑒𝑖𝑚𝜔𝑡 (3.21) 𝑚 25 and seek a Floquet Hamiltonian and micromotion operator of the form 𝐻 𝐹 = 𝐻0 + ∞ ∑︁ Ω𝑚 𝐾 𝐹 (𝑡) = and 𝜔𝑚 𝑚=1 ∞ ∑︁ Λ𝑚 (𝑡) , 𝜔𝑚 (3.22) 𝑚=1 satisfying Eq. (3.19). Depending on the gauge choice, two types of expansions can be obtained. Floquet-Magnus expansion. This expansion is obtained by imposing the gauge choice 𝐾 𝐹 (𝑡 0 ) = 0. In this case, we obtain the leading order terms Ω1 =  ∑︁   [𝐻 (−𝑚) , 𝐻 (𝑚) ]  1 1 2𝑚  𝑚≠0   and Λ1 (𝑡) = 𝑖 ∑︁ 𝐻 (𝑚) 1    [𝐻1(𝑚) , 𝐻0 ] −𝑖𝑚𝜔𝑡  0 + 𝑒 ,  𝑚   (3.23) (𝑒 −𝑖𝑚𝜔𝑡 − 𝑒 −𝑖𝑚𝜔𝑡0 ) (3.24) 𝑚 𝑚≠0 by substitution of Eq. (3.22) into Eq. (3.19). Van-Vleck expansion. This expansion removes the 𝑡 0 dependence by imposing the ∫𝑇 gauge choice 0 𝐾 𝐹 (𝑡)𝑑𝑡 = 0. In this case, Ω1 = ∑︁ [𝐻 (−𝑚) , 𝐻 (𝑚) ] 1 𝑚≠0 1 2𝑚 and Λ1 (𝑡) = 𝑖 ∑︁ 𝐻 (𝑚) 1 𝑒 −𝑖𝑚𝜔𝑡 . (3.25) 𝑚 𝑚≠0 Rotating Wave Approximation The rotating wave approximation (RWA) [86, 251] approximates the Floquet Hamiltonian in regions of phase space where the level spacing between states of 𝐻0 is nearly resonant to the frequency ℏ𝜔 of the drive. To illustrate the RWA, we consider a generic two-level Bloch Hamiltonian described by the static Hamiltonian 𝐻0 = 𝒅 · 𝝈, where 𝒅 = 𝐸 𝒌 (sin 𝜃 𝒌 cos 𝜙 𝒌 , sin 𝜃 𝒌 sin 𝜙 𝒌 , cos 𝜃 𝒌 ) (3.26) is parameterized by the crystal momentum 𝒌, and 𝝈 = (𝜎𝑥 , 𝜎𝑦 , 𝜎𝑧 ) is the vector of Pauli matrices. The Hamiltonian 𝐻0 has two eigenstates denoted |𝑣⟩ and |𝑐⟩ with energy −𝐸 𝒌 and 𝐸 𝒌 , respectively. We also assume that the time-dependent Hamiltonian 𝐻1 (𝑡) of the system only oscillates with a single frequency 𝜔, i.e., 𝐻1 (𝑡) = 𝑽 · 𝝈𝑒𝑖𝜔𝑡 + 𝑽 ∗ · 𝝈𝑒 −𝑖𝜔𝑡 , (3.27) 26 where 𝑽 is a 3-component vector of complex numbers. The RWA provides an approximate Floquet Hamiltonian in the limit where ℏ𝜔 ≈ |2𝐸 𝒌 |. To perform the RWA, we transform to the diagonal basis of 𝐻0 , where ! −𝐸 0 𝒌 𝐻˜ 0 = 𝑈 (𝜃 𝒌 , 𝜙 𝒌 )𝐻0𝑈 † (𝜃 𝒌 , 𝜙 𝒌 ) = , 0 𝐸𝒌 (3.28) and 𝑈 (𝜃 𝒌 , 𝜙 𝒌 ) = 𝑒 −𝑖(𝜙𝒌 /2)𝜎𝑧 𝑒 −𝑖(𝜃 𝒌 /2)𝜎𝑦 (3.29) relates the two bases (i.e., rotates 𝒅 into a new basis where it is aligned with the 𝑧-axis). Similarly, the driven Hamiltonian in this basis is given by ∗ 𝐻˜ 1 (𝑡) = 𝑽˜ · 𝝈𝑒𝑖𝜔𝑡 + 𝑽˜ · 𝝈𝑒𝑖𝜔𝑡 , (3.30) ∗ where 𝑽˜ is related to 𝑽 by a rotation [see Eq. (3.29)]. We use 𝐻˜ (𝑡) = 𝐻˜ 0 + 𝐻˜ 1 (𝑡) to denote the full Hamiltonian in the diagonal basis of 𝐻0 . Now, we perform the rotating wave approximation (RWA). In the interacting pic˜ ˜ ture [115], where the Hamiltonian is given by 𝐻˜ 𝐼 (𝑡) = 𝑒𝑖 𝐻0 𝑡/ℏ 𝐻˜ (𝑡)𝑒 −𝑖 𝐻0 𝑡/ℏ , we discard terms oscillating with frequency 𝜔 and 𝜔 + 2𝐸 𝒌 /ℏ (i.e., “counter-rotating” terms), keeping only low frequency terms oscillating with frequency 𝜔 − 2𝐸 𝒌 /ℏ. Transforming back to the Schrodinger picture yields 𝐻˜ (𝑡) ≈ 𝐻RWA (𝑡), where the RWA-approximated Hamiltonian is given by ! 0 (𝑉˜𝑥 − 𝑖𝑉˜ 𝑦 )𝑒 −𝑖𝜔𝑡 𝐻RWA (𝑡) ≡ . (3.31) (𝑉˜𝑥 + 𝑖𝑉˜ 𝑦 )𝑒𝑖𝜔𝑡 0 The utility of the RWA is that it allows us to easily find an effective Floquet Hamiltonian for the system. To see this, we note that the approximate quasienergy operator ˜ 𝑄(𝑡) ≈ 𝐻RWA (𝑡) − 𝑖ℏ𝜕𝑡 (3.32) [see discussion above Eq. (3.19) for the definition] can be made time-independent ˜ by a transformation to a rotating frame, where 𝐻 𝐹 = 𝑈 (𝑡) † 𝑄(𝑡)𝑈 (𝑡) is the effective Floquet Hamiltonian and ! 𝑒𝑖𝜔𝑡 0 𝑈 (𝑡) = (3.33) 0 1 implements the transformation. Explicitly, the Floquet Hamiltonian is given by 𝐻 𝐹 = (−𝐸 𝒌 + ℏ𝜔/2)𝜎𝑧 + 𝑉˜𝑥 𝜎𝑥 + 𝑉˜ 𝑦 𝜎𝑦 , (3.34) 27 and the eigenspectrum of 𝐻 𝐹 contains two quasienergies, given by √︃ 𝜀 𝒌𝛼 = (−1) 𝛼 (−𝐸 𝒌 + ℏ𝜔/2) 2 + 𝑉˜𝑥2 + 𝑉˜ 𝑦2 , (3.35) where 𝛼 = 0, 1 enumerates the quasienergy bands. From Eq. (3.35), it is clear that the spectrum exhibits a resonant Floquet gap of size √︃ Δ𝐾 = 2 𝑉˜𝑥2 + 𝑉˜ 𝑦2 , (3.36) which opens at momenta 𝒌 𝑅 satisfying the resonance condition given by 2𝐸 𝒌 𝑅 = ℏ𝜔. 3.3 Numerical Approach: Frequency-Space Expansion So far, we have seen that the high-frequency expansion and RWA can provide approximate expressions for 𝐻 𝐹 in certain limits. Alternatively, we can calculate the full (stroboscopic and micromotion) time-dependence of the Floquet state |Φ𝑛 (𝑡)⟩ by solving the time-dependent Schrodinger equation numerically in Fourier space. Here, we expand the Floquet state in a Fourier series: ∑︁ |Φ𝑛 (𝑡)⟩ = 𝑒 −𝑖𝑚𝜔𝑡 |𝜙𝑛(𝑚) ⟩, (3.37) 𝑚 where |𝜙𝑛(𝑚) ⟩ are referred to as Floquet harmonics and 𝜔 = 2𝜋/𝑇 is the angular frequency of the drive. Expanding the Hamiltonian ∑︁ 𝐻 (𝑡) = 𝑒 −𝑖𝑚𝜔𝑡 𝐻 (𝑚) , (3.38) 𝑚 in terms of its harmonic components 𝐻 (𝑚) , we use the time-dependent Schrodinger equation to obtain the matrix equation ∑︁ ′ ′ (𝜀 𝑛 + 𝑚ℏ𝜔)|𝜙𝑛(𝑚) ⟩ = 𝐻 (𝑚−𝑚 ) |𝜙𝑛(𝑚 ) ⟩, (3.39) 𝑚′ which can be solved by the eigenvalue relation H 𝝓𝑛 = 𝜀 𝑛 𝝓𝑛 , where .. .. .. .. © ª . . . . ­ ® (−1) (−2) (−3) ...® ­... 𝐻0 − (𝑚 − 1)ℏ𝜔 𝐻 𝐻 𝐻 ­ ® ­ ® H = ­... 𝐻 (1) 𝐻0 − 𝑚ℏ𝜔 𝐻 (−1) 𝐻 (−2) ...® ­ ® (2) (1) (−1) ...® ­... 𝐻 𝐻 𝐻 − (𝑚 + 1)ℏ𝜔 𝐻 0 ­ ® ­ ® .. .. .. .. . . . . « ¬ (3.40) 28 is an infinite-dimensional matrix and .. © ª . ­ (𝑚−1) ® ­ |𝜙𝑛 ⟩ ®® ­ ­ ® 𝝓𝑛 = ­ |𝜙𝑛(𝑚) ⟩ ® (3.41) ­ (𝑚+1) ® ­ |𝜙𝑛 ⟩ ®® ­ ­ ® .. . « ¬ is the vector of Floquet harmonics [241]. The eigenvalue relation can be solved numerically by restricting |𝑚| < 𝑚 𝑐 in the frequency expansion of the Floquet state |Φ(𝑡)⟩, see Eq. (3.37), where 𝑚 𝑐 ≫ 1 is a large cutoff. Such a cutoff captures the micromotion of the system roughly up to a cutoff angular frequency of 𝑚 𝑐 𝜔. 3.4 Floquet Hamiltonian of Irradiated Graphene and Semiconducting Systems This section is based on, and extends upon, the analysis in Refs. [167, 239, 284]. Having discussed numerical and analytical methods to study the dynamics of Floquet systems, we now use Floquet theory to reveal the properties of optically-driven graphene and semiconducting systems. In the absence of driving, the low-energy physics of such a system near a single valley of its Brillouin zone is described by the Hamiltonian 𝐻0 = 𝒅 · 𝝈 𝒅 = (ℏ𝑣 𝐹 𝑘 𝑥 , 𝜉ℏ𝑣 𝐹 𝑘 𝑦 , Δ𝜉 /2), (3.42) √︁ and its eigenenergies are given by ±𝐸 𝒌 , where 𝐸 𝒌 = (ℏ𝑣 𝐹 𝑘) 2 + (Δ𝜉 /2) 2 and √︃ 𝑘 = 𝑘 𝑥2 + 𝑘 2𝑦 . Here, Δ𝜉 defines the semiconducting gap size, and 𝜉 = +1, −1 indexes the 𝐾 and 𝐾 ′ valleys, respectively. This Hamiltonian can describe a wide class of electronic systems. For example, the gapless Δ𝜉 = 0 limit of the Hamiltonian can be used to model a sheet of monolayer graphene, and a finite gap Δ𝜉 ≠ 0 can be used to model the low-energy Hamiltonian for a single spin and valley sector of a direct-bandgap transition metal dichalcogenide. The coupling of the system to an optical drive is described by minimal coupling, 𝒌 = 𝒌 + 𝑒 𝑨(𝑡)/ℏ, to the magnetic vector potential 𝑨(𝑡) of the laser field. Due to the linearity of 𝐻0 in 𝒌, the driven Hamiltonian can be written in the form 𝐻 (𝑡) = 𝐻0 + 𝐻1 (𝑡), with where 𝐻1 (𝑡) = 𝑽 · 𝝈𝑒𝑖𝜔𝑡 + 𝑽 ∗ · 𝝈𝑒 −𝑖𝜔𝑡 , (3.43) where 𝑽 is a 3-component vector of complex numbers that depends on the details of 𝑨(𝑡). Our goal is to determine the the Floquet Hamiltonian of the system for different types of laser fields. 29 Circularly Polarized Laser A circularly polarized laser field is described by the magnetic vector potential 𝑨(𝑡) = 𝐴(cos 𝜔𝑡, 𝜂 sin 𝜔𝑡, 0), where 𝜂 = −1 and 𝜂 = +1 for left and right circularly polarized lasers, respectively. By minimal coupling to the static Hamiltonian 𝐻0 , we find that the driven Hamiltonian 𝐻1 (𝑡) is parameterized by 𝑽 = 21 𝑣 𝐹 𝑒 𝐴(−𝑖, 𝜉𝜂, 0). Off-Resonant Floquet Hamiltonian. We can use the Van Vleck high-frequency expansion [196, 215, 236] to derive the Floquet Hamiltonian for crystal momenta satisfying 2𝐸 𝒌 ≪ ℏ𝜔. Substitution into Eqs. (3.25) and (3.22) yields 𝐻 𝐹 = 𝐻0 − Δ𝐾 𝜎𝑧 , where 𝑒 2 𝑣 2𝐹 𝐴2 Δ𝐾 = 𝜂𝜉 , 𝜔 (3.44) and the quasienergy 𝜀 𝒌𝛼 = (−1) 𝛼 √︃ 𝐸 𝒌2 + [(Δ𝜉 − Δ𝐾 )/2] 2 . (3.45) is obtained by diagonalizing 𝐻 𝐹 . In a gapped systems, the second term in Eq. (3.44) indicates that the drive renormalizes the bandgap in a valley and laser helicity dependent manner. In graphene, notice that the drive-induced mass term, see Eq. (3.44), has opposite signs in opposite valleys, indicating that the drive dynamically produces a Haldane gap, a result first predicted by Takashi Oka and Hideo Aoki in 2009 [215]. Resonant Floquet Hamiltonian. Next, we use the RWA to estimate the Floquet Hamiltonian in the region of momentum space characterized by 2𝐸 𝒌 ≈ ℏ𝜔. Following the RWA presented in Section 3.3, we find that the effective Floquet Hamiltonian in the rotating frame is given by 𝐻 𝐹 = (−𝐸 𝒌 + ℏ𝜔/2)𝜎𝑧 + 𝑉˜𝑥 𝜎𝑥 + 𝑉˜ 𝑦 𝜎𝑦 , (3.46) 𝑉˜𝑥 = 𝑣 𝐹 𝑒 𝐴 sin 𝜙 𝒌 [𝛿𝜉𝜂,−1 cos2 (𝜃 𝒌 /2) − 𝛿𝜉𝜂,1 sin2 (𝜃 𝒌 /2)], (3.47) 𝑉˜ 𝑦 = 𝑣 𝐹 𝑒 𝐴 cos 𝜙 𝒌 [𝛿𝜉𝜂,−1 cos2 (𝜃 𝒌 /2) + 𝛿𝜉𝜂,1 sin2 (𝜃 𝒌 /2)]. (3.48) where and The quasienergy spectrum is therefore given by √︁ 𝜀 𝒌𝛼 = (−1) 𝛼 (−𝐸 𝒌 + ℏ𝜔/2) 2 + (Δ 𝑅 /2) 2 , (3.49) 30 where 𝛼 = 0, 1 enumerates the Floquet bands. Note that the quasienergy spectrum exhibits a Floquet gap along the resonance ring paramaterized by 2𝐸 𝒌 ≈ ℏ𝜔, of size     (𝑣 𝐹 𝑒 𝐴) cos2 (𝜃 𝒌 /2)  Δ𝑅 = 2    (𝑣 𝐹 𝑒 𝐴) sin (𝜃 𝒌 /2)  if 𝜉𝜂 = 1, (3.50) if 𝜉𝜂 = −1. Here, we can immediately infer the valley selection rules for transition metal dichalcogenides. When the helicity of the laser is of opposite sign than the chirality 𝜉 sign(Δ𝜉 ) of the valley, then the valley is more optically transparent because Δ 𝑅 is suppressed relative to the opposite valley [263]. Note that the same selection rule does not apply to graphene, where Δ = 0 and the Floquet gap size Δ 𝑅 = 𝑣 𝐹 𝑒 𝐴 is independent of the laser helicity [284]. (a) (b) Figure 3.2: Floquet electronic bands of graphene irradiated by a circularly-polarized laser. (a) Energy spectrum of graphene as a function of 𝑘 𝑥 for fixed 𝑘 𝑦 = 0. (b) Quasienergy spectrum of graphene under circularly polarized laser irradiation, exhibiting an off-resonant Haldane gap of size Δ𝐾 and a resonant gap of size Δ 𝑅 . Visualization of the Floquet Bands in Graphene. The quasienergy near the 𝐾 point and near the resonance ring, presented in Eqs. (3.45) and (3.49), respectively, can be stitched together to obtain the quasienergy spectrum for arbitrary 𝒌. To visualize the resulting Floquet bands, we show the undriven bands of graphene (Δ = 0) in Figure 3.2(a) and the driven Floquet bands in Figure 3.2(b). We indicate the opening of the resonance and Haldane gaps denoted Δ 𝑅 and Δ𝐾 , respectively. 31 Linearly Polarized Laser We now consider the effect of a linearly polarized laser field, described by the magnetic vector potential 𝑨(𝑡) = ( 𝐴𝑥 cos(𝜔𝑡), 𝐴 𝑦 cos(𝜔𝑡), 0). In this case, the driven Hamiltonian is parameterized by 𝑽 = 21 𝑣 𝐹 𝑒( 𝐴𝑥 , 𝐴 𝑦 , 0). Off-Resonant Floquet Hamiltonian. In the region of the Brillouin zone satisfying 2𝐸 𝒌 ≪ ℏ𝜔, the Van Vleck high frequency expansion, see Eq. (3.25), yields the Floquet Hamiltonian 𝐻 𝐹 ≈ 𝐻0 to first order in 𝜔−1 . Thus, the drive-induced Haldane gap predicted by Oka and Aoki [215] does not emerge for linearly-polarized irradiation. Resonant Floquet Hamiltonian. Using the RWA, we derive an approximate Floquet Hamiltonian for crystal momenta near the resonance ring parameterized by 2𝐸 𝒌 ≈ ℏ𝜔. For a linearly polarized laser field, we find that 1 𝑉˜𝑥 = 𝑒𝑣 𝐹 ( 𝐴𝑥 cos 𝜙 𝒌 + 𝐴 𝑦 sin 𝜙 𝒌 ) cos 𝜃 𝒌 , 2 (3.51) and 1 (3.52) 𝑉˜ 𝑦 = 𝑒𝑣 𝐹 ( 𝐴 𝑦 cos 𝜙 𝒌 − 𝐴𝑥 sin 𝜙 𝒌 ). 2 A notable case is graphene, where 𝜃 𝒌 = 𝜋/2, 𝑉˜ 𝑦 = 𝑣 𝐹 𝑒( 𝐴𝑥 cos 𝜙 𝒌 + 𝐴 𝑦 sin 𝜙 𝒌 ), 𝑉˜𝑥 = 0, and the Floquet gap along the resonance ring 2𝐸 𝒌 = ℏ𝜔 is momentumdependent, given by Δ𝐹 = 𝑣 𝐹 𝑒| 𝐴𝑥 cos 𝜙 𝒌 + 𝐴 𝑦 cos 𝜙 𝒌 |. (3.53) The Floquet gap vanishes when the polarization of the laser field ( 𝐴𝑥 , 𝐴 𝑦 ) is perpendicular to the crystal momentum 𝒌. 3.5 The Floquet Topological Insulator This section is based on, and extends upon, the analysis in Refs. [167, 239, 284]. In Sections 3.2 and 3.4, we discussed analytical methods to estimate the Floquet Hamiltonian and its quasienergy spectrum. Already, we have seen some potential signatures of topology in the Floquet Hamiltonian, such as the drive-induced Haldane gap in graphene, see Eq. (3.44). In this section, we provide a more complete discussion of the topological properties of Floquet systems. As an example, we focus on a gapped graphene-like system driven by a circularly polarized laser, described by the time-periodic Hamiltonian 𝐻 (𝑡) = 𝐻0 + 𝐻1 (𝑡), 32 (a) (b) (c) (d) (e) Figure 3.3: Pseudospin winding under circularly polarized irradiation. (a) Here we box in orange the 𝛼 = − quasienergy band. (b) Plot of 𝛼 = − quasienergy band vs. 𝑘 𝑥 and 𝑘 𝑦 . Red arrows indicate the orientation of the pseudospin 𝑛ˆ 𝒌− (0). We focus on the pseudospin along the rings sketched in green. (c) Pseudospin in the region 𝑘 < 𝑘 𝑅 , which is roughly aligned with the 𝑧ˆ-axis, with orientation set by the sign of the mass gap at 𝑘 = 0. (d) Pseudospin along the resonance ring 𝑘 = 𝑘 𝑅 . (e) Pseudospin in the region 𝑘 > 𝑘 𝑅 , which approaches the equator, parameterized by ˆ 𝑛ˆ 𝒌− (0) = 𝒌. defined in Section 3.4. In order to characterize the topology of the Floquet bands, we consider the pseudospin of electronic states in the 𝛼-th Floquet band, defined by 𝑛ˆ 𝒌𝛼 (𝑡) = ⟨𝜓 𝒌𝛼 (𝑡)|𝝈|𝜓 𝒌𝛼 (𝑡)⟩, (3.54) where |𝜓 𝒌𝛼 (𝑡)⟩ denotes the Floquet state [see Eq. (3.15) for definition] with quasienergy 𝜀 𝒌𝛼 . The Chern number counts the number of times 𝑛ˆ 𝒌𝛼 covers the Bloch sphere upon variation across a the Brillouin zone (BZ) [167, 239]: ∫ 𝑑2 𝒌 𝑛ˆ (𝑡) · [𝜕𝑘 𝑥 𝑛ˆ 𝒌𝛼 (𝑡) × 𝜕𝑘 𝑦 𝑛ˆ 𝒌𝛼 (𝑡)], (3.55) 𝐶𝛼 = 2 𝒌𝛼 BZ (2𝜋) which is a generalization of the definition for a two-level static Hamiltonian, see Eq. (2.38). For a gapped graphene system, variations of the pseudospin are confined to regions around the 𝐾 and 𝐾 ′ valley, so it is easiest to compute contributions to 𝛼 and 𝐶 𝛼 to 𝐶𝛼 from the 𝐾 and 𝐾 ′ valley separately. In what follows, we use 𝐶+1 −1 respectively denote the contributions to the Chern number from the 𝐾 and 𝐾 ′ valleys, 𝛼 + 𝐶𝛼 . the sum of which is the total Chern number of the band 𝛼, i.e., 𝐶𝛼 = 𝐶+1 −1 For concreteness, we begin by focusing on the 𝛼 = − Floquet band [see band boxed in orange in Figure 3.3(a)] near the 𝐾 valley of the gapped graphene system. To 33 count the winding of the pseudospin in the region near the 𝐾 valley, our approach is to consider the pseudospin along “rings” in momentum space with radius 𝑘 and centered around the 𝐾 point [see green curves in Figure 3.3(b) for an illustration]. We begin near the 𝐾 point and track the changes in the pseudospin configuration as the ring radius 𝑘 is increased. It is useful to first start by considering momenta far from the resonance ring (defined by | 𝒌 | = 𝑘 𝑅 and 2𝐸 𝒌 = ℏ𝜔) where the pseudospin configuration is particularly simple: • Region 𝑘 ≪ 𝑘 𝑅 : near 𝑘 = 0, the pseudospin is roughly aligned with the polar axis (𝑧ˆ-axis), with a direction set by the sign of mass gap Δ, i.e., 𝑛ˆ 𝒌− ≈ sign(Δ𝜉 ) 𝑧ˆ [see Figure 3.3(c)]. • Region 𝑘 ≫ 𝑘 𝑅 : in this regime, the pseudospin is roughly identical to the ˆ where 𝒌ˆ = 𝒌/𝑘 [see pseudospin of the undriven valence band, so 𝑛ˆ 𝒌− ≈ 𝒌, Figure 3.3(e)]. Thus, as the “rings” in momentum space [green curves in Figure 3.3(b)] increase in radius, the pseudospin is initially aligned with the polar axis of the Bloch sphere ˆ at large 𝑘, covering a (𝑛ˆ 𝒌− = sign(Δ𝜉 ) 𝑧ˆ) and eventually winds to its equator (𝑛ˆ 𝒌− = 𝒌) solid angle of 2𝜋 sign(Δ𝜉 ). However, it would be incorrect to immediately conclude that the Chern number is given by 𝐶+1 = 12 sign(Δ𝜉 ). In fact, there are multiple paths that the pseudospin can take to wind from the polar axis to the equator, which can in general cover the Bloch sphere an additional integer number of times. Therefore, we expect that 𝐶+1 = 12 sign(Δ𝜉 ) + 𝑛, where 𝑛 ∈ Z. Our next task is to determine 𝑛 based on the RWA Hamiltonian. The value of 𝑛 can be determined by counting the number of times the pseudospin winds around the polar axis along the resonance ring [see Figure 3.3(d)] [130, 167, 239]. To proceed, we use the RWA Hamiltonian, ! 0 (𝑉˜𝑥 − 𝑖𝑉˜ 𝑦 )𝑒 −𝑖𝜔𝑡 𝐻RWA (𝑡) ≡ , (3.56) (𝑉˜𝑥 + 𝑖𝑉˜ 𝑦 )𝑒𝑖𝜔𝑡 0 to calculate the pseudospin configuration on the resonance ring. We focus on the pseudospin texture at time 𝑡 = 0, but the results can be shown to be equivalent at all other times. We first transform 𝐻RWA (0) to the original sublattice basis of the Hamiltonian: 𝑈 † (𝜃 𝒌 , 𝜙 𝒌 )𝐻RWA (0)𝑈 (𝜃 𝒌 , 𝜙 𝒌 ) = 𝒅 RWA · 𝝈, (3.57) 34 where 𝑉˜𝑥 cos 𝜙 𝒌 cos 𝜃 𝒌 − 𝑉˜ 𝑦 sin 𝜙 𝒌 ª 1 ©­ ® (3.58) 𝒅 RWA = ­𝑉˜𝑥 sin 𝜙 𝒌 cos 𝜃 𝒌 + 𝑉˜ 𝑦 cos 𝜙 𝒌 ® . ® ­ 2 −2𝑉˜𝑥 sin 𝜃 𝒌 ¬ « The expression for 𝒅 RWA allows us to compute the pseudospin along the resonance ring, given by 𝑛ˆ 𝒌− (0) = − 𝒅ˆ RWA , where 𝒅ˆ RWA = 𝒅 RWA /| 𝒅 RWA |. In Figure 3.4, we plot the trajectory traced out by 𝑛ˆ 𝒌− (0) (red curve) as a function of 𝜙 𝒌 , with 𝜃 𝒌 fixed by the resonance condition:   Δ𝜉 /2 𝜃 𝒌 = arctan . (3.59) ℏ𝑣 𝐹 𝑘 𝑅 Figure 3.4(a) considers the case sign(Δ𝜉 𝜉𝜂) > 0 and shows that 𝒅ˆ RWA winds twice around the polar axis. Whether the pseudospin winds counterclockwise or clockwise around the polar axis is determined by 𝜂, so 𝑛 = 2𝜂. Figure 3.4(b) considers the case sign(Δ𝜉 𝜉𝜂) < 0 and shows that 𝒅ˆ RWA does not wind around the polar axis, in which case 𝑛 = 0. (a) z (b) z Figure 3.4: Path (red curve) traced out by the pseudospin 𝑛ˆ 𝒌− (0) upon variation across the resonance ring [see Figure 3.3(d)]. (a) The case of sign(Δ𝜉 𝜉𝜂) > 0, in which case the pseudospin winds around the polar axis (sketched by a black line) 2 times. (b) The case of sign(Δ𝜉 𝜉𝜂) < 0, in which case the pseudospin does not wind around the polar axis. The arguments can be generalized to the 𝐾 ′ valley. In general, the valley indexed by 𝜉 contributes a Chern number given by [239],     2𝜂  1 − 𝐶𝜉 = sign(Δ𝜉 𝜉) +  2  0  if sign(Δ𝜉 𝜉𝜂) > 0 if sign(Δ𝜉 𝜉𝜂) < 0. (3.60) 35 Case of graphene. In graphene, although there is no mass gap in the equilibrium system, the 𝐾 and 𝐾 ′ points are not gapless due to the opening of the drive-induced Haldane gap, see Eq. (3.44). In the analysis above, we therefore replace Δ𝜉 with Δ𝐾 , defined in Eq. (3.44). Because sign(Δ𝐾 ) = −𝜉𝜂, substitution into Eq. (3.60) yields 1 𝐶𝜉− = − 𝜂 + 2𝜂. (3.61) 2 In a gapless graphene system, the circularly polarized driving field therefore induces a Floquet band with a net Chern number of 𝐶𝛼 = −3𝜂(−1) 𝛼 . 3.6 Topological Transport in Floquet Systems The following section is based on Ref. [74]. Having understood the origin of drive-induced topology in Floquet systems, we now discuss how drive-induced topology can be detected from the transport responses of Floquet systems to weak electric fields. First, we introduce the anomalous velocity in the Floquet basis by defining the Berry curvature of a Floquet state, given by 𝛀 𝒌𝛼 (𝑡) = ∇ 𝒌 × 𝑨 𝒌𝑛 (𝑡), (3.62) where 𝑨 𝒌𝑛 (𝑡) = 𝑖⟨𝜓 𝒌𝑛 (𝑡)|∇ 𝒌 |𝜓 𝒌𝑛 (𝑡)⟩ denotes the Berry connection and |𝜓 𝒌𝑛 (𝑡)⟩ is the Floquet state of a time-periodic Hamiltonian 𝐻 (𝑡). Note that the Chern number of the Floquet band, introduced in Eq. (3.55), can equivalently be calculated via integration of 𝛀 𝒌𝛼 (𝑡) across the Brillouin zone: ∫ 1 𝑑 2 𝒌 𝛀 𝒌𝛼 (𝑡) · 𝑧ˆ. (3.63) 𝐶𝛼 = 2𝜋 For our purposes, we are interested in transport responses or many-body collision processes on timescales much longer than the frequency of the drive [257, 258]. In this case, we focus on the time-averaged anomalous velocity of electrons, which can be calculated using the time-averaged Berry curvature given by 𝛀 𝒌𝛼 = ∇ 𝒌 × 𝑨 𝒌𝑛 . (3.64) Í Here, 𝑨 𝒌𝑛 (𝑡) = 𝑖 𝑛 ⟨𝜙𝑛𝒌𝑛 |∇ 𝒌 |𝜙𝑛𝒌𝑛 ⟩ denotes the time-averaged Berry connection and |𝜙𝑛𝒌𝑛 ⟩ are the Floquet harmonics, obtained by expanding the Floquet state in the harmonic series: ∑︁ |𝜓 𝒌𝑛 (𝑡)⟩ = 𝑒 −𝑖𝜀𝒌𝑛 𝑒 −𝑖𝑚𝜔𝑡 |𝜙𝑛𝒌𝑛 ⟩. (3.65) 𝑛 36 In the presence of a weak electric field 𝑬, the average group velocity of electrons is therefore given by 1 𝑒 𝒗 𝒌𝛼 = ∇ 𝒌 𝜀 𝒌𝛼 + 𝑬 × 𝛀 𝒌𝛼 , (3.66) ℏ ℏ which is derived in a similar manner as Section 2.3, by assuming that the weak electric field produces an adiabatic evolution of the Floquet states. To determine the current produced by the electric field, we estimate the occupation 𝐹˜𝒌𝛼 of the Floquet state |𝜓 𝒌𝑛 (𝑡)⟩ in the presence of the electric field. When the occupation is spatially homogenous across the system, its time evolution can be estimated from the Boltzmann transport equation given by 𝐹˜𝒌𝛼 − 𝐹𝒌𝛼 𝑒 𝜕𝑡 𝐹˜𝒌𝛼 − 𝑬 · ∇ 𝒌 𝐹˜𝒌𝛼 = − . ℏ 𝜏𝒌𝛼 (3.67) Here, we have used the relaxation time approximation, which assumes that the occupation relaxes to its value 𝐹𝒌𝛼 in the absence of the electric field within a relaxation time 𝜏𝒌𝛼 . We are interested in the population distribution of electrons in the steady state, defined by 𝜕𝑡 𝐹˜𝒌𝛼 = 0, which is given by 𝑒 𝐹˜𝒌𝛼 = 𝐹𝒌𝛼 − 𝜏𝒌𝛼 𝐸 𝜕𝑘 𝑥 𝐹𝒌𝛼 . ℏ The Hall conductivity in the steady state is therefore given by ∫ 𝑑2 𝒌 𝑒 2 ∑︁ 𝛀 𝒌𝛼 · 𝑧ˆ 𝐹𝒌𝛼 , 𝜎𝑥𝑦 = ℏ 𝛼 (2𝜋) 2 and the longitudinal conductivity is given by ∫ 𝑒 2 ∑︁ 𝑑2 𝒌 𝜎𝑥𝑥 = 𝜏𝒌𝛼 (𝜕𝑘 𝑥 𝜀 𝒌𝛼 )(𝜕𝑘 𝑥 𝐹𝒌𝛼 ). ℏ 𝛼 (2𝜋) 2 (3.68) (3.69) (3.70) The discussion of transport signatures in Floquet systems thus far inspires questions about what sets the steady-state occupations of Floquet states 𝐹𝒌𝛼 before the application of the weak electric field. The steady state is governed by the incoherent dynamics of electrons generated by many-body collisions, which will be the topic of Chapter 4. 37 Chapter 4 INCOHERENT DYNAMICS OF FLOQUET SYSTEMS In an interacting electronic Floquet system, the laws of equilibrium statistical mechanics that normally guarantee Fermi-Dirac statistics no longer apply [257, 258]. Photo-assisted many-body collisions sharply modify the occupation of electronic states, producing a dramatically different population distribution from equilibrium. In this chapter, we show that electron-phonon coupling to a low-temperature phonon bath, in particular, can serve as a mechanism to stabilize a low-temperature “ideal Floquet topological insulator” population distribution supporting topological transport signatures. We focus on a model of a many-body interacting system coupled to an external phonon bath, described by the Hamiltonian 𝐻ˆ = 𝐻ˆ 0 + 𝐻ˆ el-ph + 𝐻ˆ el-el . Here, the single-particle Hamiltonian is given by ∑︁ 𝐸 𝒌𝜈 𝑐ˆ†𝒌𝜈 𝑐ˆ 𝒌𝜈 , 𝐻ˆ 0 = (4.1) (4.2) 𝒌,𝜈 where 𝑐ˆ†𝒌𝜈 creates a Bloch state | 𝒌𝜈⟩ with energy 𝐸 𝒌𝜈 . The electron-phonon interactions are described by the Hamiltonian ∑︁ 𝜈𝜈 ′ † 𝐻ˆ el-ph = 𝑀 𝒌,𝒒 𝑐ˆ 𝒌+𝒒,𝜈′ 𝑐 𝒌𝜈 ( 𝑏ˆ †𝒒, 𝑗 + 𝑏ˆ −𝒒, 𝑗 ) + H.c. (4.3) 𝒌,𝒒, 𝑗,𝜈,𝜈 ′ and the electron-electron interactions are modeled by ∑︁ 𝑖} 𝐻ˆ el-el = 𝑉𝒌{𝜈1 ,𝒌 𝑐ˆ† 𝑐ˆ† 𝑐ˆ 𝒌 2 𝜈3 𝑐ˆ 𝒌 1 𝜈4 , 2 ,𝒒 𝒌 1 +𝒒,𝜈1 𝒌 2 −𝒒,𝜈2 (4.4) 𝒌 1 ,𝒌 2 ,𝒒,{𝜈𝑖 } where 𝑏ˆ †𝒒, 𝑗 creates a phonon of frequency 𝜔 𝑗 (𝒒) with momentum 𝒒 and mode in′ 𝜈𝜈 and 𝑉 {𝜈𝑖 } dexed by 𝑗, and 𝑀 𝒌,𝒒 are the matrix elements for electron-phonon and 𝒌 1 ,𝒌 2 ,𝒒 electron-electron interactions, respectively [258]. Upon driving the system with a laser field of magnetic vector potential 𝑨(𝑡) and angular frequency Ω, the Hamiltonian becomes time-dependent 𝐻ˆ 0 (𝑡) via minimal coupling 𝒌 → 𝒌 + 𝑒 𝑨(𝑡)/ℏ, and the time-dependent Schrodinger equation is solved by the Floquet states |𝜓 𝒌𝛼 (𝑡)⟩ 38 with quasienergies 𝜀 𝒌𝛼 , as discussed in Chapter 3. The occupations of the Floquet † † states, given by 𝐹𝒌𝛼 (𝑡) = ⟨ 𝑓ˆ𝒌𝛼 (𝑡) 𝑓ˆ𝒌𝛼 (𝑡)⟩, where 𝑓ˆ𝒌𝛼 (𝑡) denotes the creation operator for |𝜓 𝒌𝛼 (𝑡)⟩, are determined by the incoherent dynamics of the electrons generated by 𝐻ˆ el-ph and 𝐻ˆ el-el . Our goal is to evaluate the occupations in the long time limit, when the system approaches a steady state with occupation given by 𝐹𝒌𝛼 , which is time-independent, 𝐹¤𝒌𝛼 = 0, up to micromotion. In Section 4.1, we introduce the Floquet-Boltzmann equation, which models the population dynamics of Floquet systems when the interactions 𝐻ˆ el-el and 𝐻ˆ el-ph are weak perturbations to the singleparticle Hamiltonian 𝐻ˆ 0 (𝑡). The remainder of the chapter is concerned with solving for the steady state and finding conditions that stabilize low-temperature phases, see Section 4.2, and understanding the stability of Floquet-Bloch states in a steady state, see Section 4.3. 4.1 Floquet-Boltzmann Equation The following section is based on Refs. [257–259]. The Floquet-Boltzmann equation provides a framework to characterize the manybody steady state of Floquet systems under the following assumptions: 1. Born-Markov approximation. We assume that the incoherent phonons thermalize much faster than the rate of electron-phonon scattering, such that the incoherent phonons remain in thermal equilibrium at all times and can be described by a Bose-Einstein distribution. 2. Diagonal ensemble and perturbative interactions. We assume that coherences between the Floquet states are negligible, i.e., ⟨ 𝑓ˆ𝒌†′ 𝛼′ (𝑡) 𝑓ˆ𝒌𝛼 (𝑡)⟩ ≈ 0 for 𝒌 ′ ≠ 𝒌 or 𝛼 ≠ 𝛼′. This is an appropriate approximation when the steady state is translation invariant and the interactions are perturbative, such that the scattering time 𝜏scat between Floquet states is much larger than the inverse spectral gap Δ between the Floquet states, i.e., 𝜏scat ≫ ℏ/Δ. 3. Slow scattering dynamics. We assume that the characteristic timescale for scattering 𝜏scat is much longer than the period of the drive, i.e., 𝜏scat ≫ 2𝜋/Ω, such that only the slow (stroboscopic-timescale) evolution of the electronic occupations affect the incoherent scattering processes. In this limit, we can simplify the analysis by discarding the micromotion of the population dynamics. 39 Explicitly, the Floquet-Boltzmann equation is given by el-ph el-el 𝐹¤𝒌𝛼 = 𝐼 𝒌𝛼 [{𝐹𝒌𝛼 }] + 𝐼 𝒌𝛼 [{𝐹𝒌𝛼 }], (4.5) el-ph el-el [{𝐹 }] are respectively the collision integrals for where 𝐼 𝒌𝛼 [{𝐹𝒌𝛼 }] and 𝐼 𝒌𝛼 𝒌𝛼 electron-phonon and electron-electron scattering. In the following subsections, we el-ph el-el [{𝐹 }] and provide an intuitive provide the explicit forms for 𝐼 𝒌𝛼 [{𝐹𝒌𝛼 }] and 𝐼 𝒌𝛼 𝒌𝛼 explanation for the population dynamics generated by the many-body collisions. Electron-Phonon Scattering The electron-phonon collision integral is given by el-ph 𝐼 𝒌𝛼 [{𝐹𝒌𝛼 }] = 2𝜋 1 ∑︁ ∑︁ ∑︁ ∑︁ 𝒌 ′ 𝛼′ |G𝒌𝛼 (𝑛, 𝑗)| 2 ℏ 𝑁 ′ 𝑛 𝛼′ 𝑗 𝒌 ∈BZ × h 𝐹𝒌 ′ 𝛼′ (1 − 𝐹𝒌𝛼 )N (ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌)) − 𝐹𝒌𝛼 (1 − 𝐹𝒌 ′ 𝛼′ ) [1 + N (ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌))] × 𝛿(𝜀 𝒌 ′ 𝛼′ + 𝑛ℏΩ − 𝜀 𝒌𝛼 + ℏ𝜔 𝑗 (𝒒))  + 𝐹𝒌 ′ 𝛼′ (1 − 𝐹𝒌𝛼 ) [1 + N (ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌))] − 𝐹𝒌𝛼 (1 − 𝐹𝒌 ′ 𝛼′ )N (ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌)) i × 𝛿(𝜀 𝒌 ′ 𝛼′ + 𝑛ℏΩ − 𝜀 𝒌𝛼 − ℏ𝜔 𝑗 (𝒒)) , (4.6) where ′ ′ 𝒌𝛼 G𝒌𝛼 (𝑛, 𝑗) = ∑︁ ∑︁ 𝑚 ′ 𝜈𝜈 𝑀 𝒌,𝒒 ⟨𝜙𝑛+𝑚 |𝜈′ 𝒌 ′⟩⟨𝜈𝒌 |𝜙𝑚𝒌𝛼 ⟩ 𝒌 ′ 𝛼′ (4.7) 𝜈,𝜈 ′ is the photon dressed electron-phonon matrix element in the Floquet basis, N (𝜀) = (𝑒 𝜀/𝑘 𝐵𝑇 − 1) −1 is the Bose-Einstein distribution for the incoherent phonons maintained at temperature 𝑇, and 𝑘 𝐵 is the Boltzmann constant. To provide intuition for the collision integral, it is useful to consider the low-temperature limit 𝑇 → 0, in which case N (𝜀) → 0. In this limit, the collision integral, Eq. (4.6), reduces to: el-ph lim 𝐼 𝒌𝛼 [{𝐹𝒌𝛼 }] = 𝑇→0 2𝜋 1 ∑︁ ∑︁ ∑︁ ∑︁ 𝒌 ′ 𝛼′ |G𝒌𝛼 (𝑛, 𝑗)| 2 ℏ 𝑁 ′ 𝑛 𝛼′ 𝑗 𝒌 ∈BZ × [−𝐹𝒌𝛼 (1 − 𝐹𝒌 ′ 𝛼′ )𝛿(𝜀 𝒌 ′ 𝛼′ + 𝑛ℏΩ − 𝜀 𝒌𝛼 + ℏ𝜔 𝑗 (𝒒)) (4.8) i + 𝐹𝒌 ′ 𝛼′ (1 − 𝐹𝒌𝛼 )𝛿(𝜀 𝒌 ′ 𝛼′ + 𝑛ℏΩ − 𝜀 𝒌𝛼 − ℏ𝜔 𝑗 (𝒒)) . The first term in brackets represents a sink term, scattering an electron out of the state with quasienergy 𝜀 𝒌𝛼 into a state with quasienergy 𝜀 𝒌 ′ 𝛼′ + 𝑛ℏΩ. Since the phonon frequency is a positive quantity, ℏ𝜔 𝑗 (𝒒) > 0, the Dirac Delta function ensures the process is only possible when the quasienergy is lowered during the process, i.e., 40 𝜀 𝒌 ′ 𝛼′ +𝑛ℏΩ < 𝜀 𝒌𝛼 , corresponding to a phonon emission. This kinematic constraint is imposed because phonons can only be emitted to, and not be absorbed from, a zerotemperature phonon bath. The second term is a source term, scattering an electron with quasienergy 𝜀 𝒌 ′ 𝛼′ + 𝑛ℏΩ into the state with quasienergy 𝜀 𝒌𝛼 . Once again, the Dirac Delta function ensures that the quasienergy is lowered during the process, i.e., 𝜀 𝒌 ′ 𝛼′ + 𝑛ℏΩ > 𝜀 𝒌𝛼 , such that a phonon is emitted. The occupation functions appearing in Eq. (4.8) account for Pauli blocking, ensuring that no scattering process can be completed if the initial state is empty or the final state is full. Electron-Electron Scattering The electron-electron collision integral is given by 4𝜋 1 ∑︁ ∑︁ ∑︁ ∑︁ ∑︁ ( 𝒌 3 ,𝛼3 ),( 𝒌 1 +𝒌 2 −𝒌 3 ,𝛼4 ) |V( 𝒌,𝛼),( 𝒌 ,𝛼 ) (𝑛, 𝑮)| 2 × 2 2 2 ℏ 𝑁 𝒌 2 ∈BZ 𝒌 3 ∈BZ 𝛼2 ,𝛼3 ,𝛼4 𝑛 𝑮  × 𝛿(𝜀 𝒌𝛼 + 𝜀 𝒌 2 𝛼2 − 𝜀 𝒌 3 𝛼3 − 𝜀 𝒌+𝒌 2 −𝒌 3 ,𝛼4 + 𝑛ℏΩ) (1 − 𝐹𝒌𝛼 )(1 − 𝐹𝒌 2 𝛼2 )𝐹𝒌 3 𝛼3 𝐹𝒌 1 +𝒌 2 −𝒌 3 ,𝛼4  − 𝐹𝒌𝛼 𝐹𝒌 2 𝛼2 (1 − 𝐹𝒌 3 𝛼3 )(1 − 𝐹𝒌 1 +𝒌 2 −𝒌 3 ,𝛼4 ) , el-el 𝐼 𝒌𝛼 [{𝐹𝒌𝛼 }] = (4.9) where 𝒌 3 ,𝛼3 ),( 𝒌 1 +𝒌 2 −𝒌 3 ,𝛼4 ) (𝑛) = V((𝒌,𝛼),( 𝒌 ,𝛼 ) 2 ∑︁ ∑︁ ∑︁ 2 {𝜈𝑖 } 2 +𝑛3 +𝑛4 𝑉𝒌,𝒌 ⟨𝜙𝑛−𝑛 |𝜈1 𝒌⟩⟨𝜙𝑛𝒌22 𝛼2 |𝜈2 𝒌 2 ⟩× 𝒌𝛼 2 ,𝒌 4 −𝒌 2 𝜈1 ,𝜈2 𝜈3 ,𝜈4 𝑛2 ,𝑛3 ,𝑛4 4 × ⟨𝜈3 𝒌 3 |𝜙𝑛𝒌33 𝛼3 ⟩⟨𝜈4 𝒌 4 |𝜙𝑛𝒌+𝒌 ⟩ 2 −𝒌 3 ,𝛼4 (4.10) is the photon-dressed electron-electron matrix element in the Floquet basis. The collision integral, Eq. (4.9), describes a two-electron scattering process. Note that, although the total momentum is conserved, the total quasienergy is conserved only up to an integer multiple of ℏΩ [see Dirac Delta function in Eq. (4.9)], which accounts for photoabsorption and emission, as will be discussed in Section 4.2. 4.2 Solving for the Steady State The following section is based on Refs. [75, 258]. To solve for the steady state condition, given by 𝐹¤𝒌𝛼 = 0, we first identify the microscopic collision processes that enter the FBE. For concreteness, we focus on a simple resonantly-driven semiconducting system, with the quasienergy band structure shown in Figure 4.1. Because the quasienergy is defined modulo ℏ𝜔, we can sketch two equivalent representations of the quasienergy bands. The first is the 41 (a) (b) k k Folded FBZ Unfolded Figure 4.1: The quasienergy of the driven system is defined modulo ℏ𝜔. In our discussion of the Floquet Boltzmann equation, it is convenient to consider two choices of the quasienergy bands. (a) The “folded” Floquet Brillouin Zone (FBZ) picture is analogous to the rotating frame in the rotating wave approximation, where states with band character corresponding to the valence band of the undriven system are shifted in energy by +ℏΩ. (b) The “unfolded” picture where the quasienergies are shifted such that quasienergy spectrum resembles the original bands, except with a gap opening at resonance. Dashed lines indicate ±ℏΩ/2. “folded” Floquet Brillouin zone (FBZ) picture analogous to the rotating frame in the rotating wave approximation, where states with band character corresponding to the valence band of the undriven system are shifted in energy by +ℏ𝜔 [Figure 4.1(a)]. In the folded representation, we sketch the original electronic band in black and the shifted valence band in blue. The second representation is the “unfolded” picture where the quasienergies are shifted such that quasienergy spectrum resembles the original bands, except with a gap opening at resonance [Figure 4.1(b)]. We will interchange between the two equivalent representations of the quasienergy structure, depending on which allows for a more convenient visualization the many-body scattering process in question. The weak electron-phonon and electron-electron interactions described by the FBE scatter electrons between single-particle Floquet states. These collision processes can be categorized into two types: • Regular scattering processes, sketched in Figure 4.2 for electron-phonon interactions [panel (a) and (b)] and electron-electron interactions [panel (c) and (d)]. These processes exist even in the equilibrium setting and restore electronic populations back to the thermal Fermi-Dirac distribution in the undriven bands. In the unfolded picture [panels (b) and (d)], it is clear that these 42 Electron-Phonon (a) Electron-Electron (b) (c) (d) k k k k Folded FBZ Unfolded Folded FBZ Unfolded Figure 4.2: Sketch of “regular” electron-phonon and electron-electron scattering processes that do not rely on photoemission and photoabsorption to conserve energy and momentum. In the folded picture [panels (a) and (c)], these scattering processes only connect the same Floquet copies [i.e. states sketched with the same color]. When the processes are sketched on the unfolded bands [see panels (b) and (d)], it becomes clear that no photoabsorption or emission is required to conserve energymomentum in the scattering processes. scattering processes conserve energy and momentum without absorption or emission of photons. • Floquet-Umklapp (FU) processes (photo-assisted scattering). These processes perturb the steady state distribution away from the equilibrium Fermi-Dirac distribution and rely on photon absorption or emission to achieve energymomentum conservation. First-order FU processes are sketched in Figures 4.3 and 4.4 for electron-phonon and electron-electron interactions, respectively, emitting or absorbing a single photon. We show the FU scattering processes in the unfolded picture [Figures 4.3(b)-(d), 4.4(b), and 4.4(d)] to emphasize that these first-order FU processes conserve energy and momentum only up to ±ℏΩ, which corresponds to an absorption or emission of a single photon (sketched by a blue squiggly line). Note that higher-order FU processes, which absorb or emit more than one photon, occur with suppressed probability for sufficiently weak drives [257, 258]. To approximate the electronic steady state in the presence of the scattering processes discussed above, we consider the FBE for the average density of electrons in each 43 (a) (b) (c) (d) k k k k Unfolded Folded FBZ Figure 4.3: Sketch of electron-phonon Floquet-Umklapp scattering processes that absorb or emit a single photon. In the folded picture [panel (a)], these scattering processes connect different Floquet copies [i.e., states sketched with different color]. When the processes are sketched on the unfolded bands [see panels (b-d)], it is clear that the scattering relies on an absorption or emission of a single photon (squiggly blue lines) to conserve energy and momentum. Floquet band, given by 𝑑2 𝒌 𝐹𝒌𝛼 (4.11) (2𝜋) 2 where 𝐴 denotes the area of the unit cell and 𝛼 = +, − indexes the upper (UFB) and lower Floquet bands (LFB), respectively, in the folded picture [see Figure 4.1(a)]. To further simplify the analysis, we focus on a charge neutral system, where a single variable, 𝑛 ≡ 𝑛+ = 1 − 𝑛− , parameterizes the density of electrons (holes) in the LFB (UFB). To construct a rate equation for 𝑛, we consider the collisions sketched in Figures 4.2, 4.3, and 4.4 that connect different Floquet bands. ∫ 𝑛𝛼 = 𝐴 Electron-Phonon Scattering The occupation function 𝑛 is modified by interband electron-phonon scattering processes in the folded picture, sketched by vertical lines in Figures 4.2 and 4.3. Scattering processes connecting the UFB band to the LFB band are proportional to the density of electrons in the UFB and holes in the LFB, both given by 1 − 𝑛. ph ph Therefore, such a process occurs with a rate given by 𝑛| ¤ in = 𝑊in (1 − 𝑛) 2 , where ph 𝑊in is an occupation-independent intrinsic scattering rate set by the electron-phonon ph matrix element [see Eq. (4.6)]. Similarly, we define an intrinsic scattering rate 𝑊out for collision processes from the LFB to the UFB, which occur with a rate given by ph ph 𝑛| ¤ out = −𝑊out 𝑛2 . 44 Floquet-Auger II Floquet-Auger I (a) (b) (d) (c) k k Folded FBZ k Unfolded Folded FBZ k Unfolded Figure 4.4: Sketch of electron-electron Floquet-Umklapp scattering processes that absorb or emit a single photon, categorized as Floquet-Auger I and II which transfer a single or pair of electrons across distinct Floquet bands. We sketch the scattering processes in the folded picture in panels (a) and (c) and the unfolded picture in panels (b) and (d). In the unfolded picture, it is clear that the scattering relies on an absorption or emission of a single photon (squiggly blue lines) to conserve energy and momentum. Electron-Electron Scattering Electron-electron scattering processes facilitate collision processes changing (interband) or preserving (intraband) the total particle density within each Floquet band. The interband scattering processes primarily arise from photo-assisted FU processes, which can be divided into two types: Floquet-Auger I (or “single” FloquetAuger) collisions, transferring a single electron between distinct Floquet bands; and Floquet-Auger II (or “double” Floquet-Auger) collisions, transferring two electrons between distinct Floquet bands, see Figure 4.4. These Floquet-Auger processes are the primary electron-electron scattering processes affecting the LFB occupation 𝑛. Electron-electron scattering processes of the Floquet Auger I and II type sketched in Figure 4.4 excite electrons from the LFB band to the UFB band, and therefore ee 3 contribute the sink rates to the LFB occupation given by 𝑛| ¤ ee 1−+ = −𝑊1 𝑛 (1 − 𝑛) and ee 4 ee ee 𝑛| ¤ ee 2−+ = −𝑊2 𝑛 , where 𝑊1 and 𝑊2 are intrinsic scattering rates independent of the occupation function. Owing to the particle-hole symmetry of the Floquet bands, reverse processes can be constructed that facilitate a net transfer of electrons from ee 3 the UFB band to the LFB band, which contribute source rates 𝑛| ¤ ee 1+− = 𝑊1 𝑛(1 − 𝑛) ee 4 and 𝑛| ¤ ee 2+− = 𝑊2+− (1 − 𝑛) to the LFB occupation. 45 Phenomenological Rate Equation Combining the electron-phonon and electron-electron scattering rates, the full rate equation for the average LFB occupation function 𝑛 is given by ph ph 𝑛¤ = 𝑛| ¤ in + 𝑛| ¤ out + 𝑛| ¤ ee ¤ ee ¤ ee ¤ ee 1−+ + 𝑛| 2−+ + 𝑛| 1+− + 𝑛| 2+− ph ph = 𝑊in (1 − 𝑛) 2 − 𝑊out 𝑛2 + 𝑊1ee [𝑛(1 − 𝑛) 3 − 𝑛3 (1 − 𝑛)] + 𝑊2ee [(1 − 𝑛) 4 − 𝑛4 ]. (4.12) To elucidate the role of the different processes on the steady state, we consider several limits ph ph • Without electron-phonon interactions 𝑊in , 𝑊out → 0, the steady state is given by 𝑛 = 1/2, which corresponds to an infinite-temperature state. This limit highlights the importance of phonon cooling in stabilizing steady states. ph ph • In the limit of strong phonon cooling, where 𝑊out ≫ 𝑊in , 𝑊1ee , 𝑊2ee , the steady state is roughly given by 𝑛 ≈ 0. This corresponds to an “ideal Floquet topological insulator (FTI) state” where the LFB is nearly fully-occupied. The drive-induced Berry curvature of Floquet systems is usually isolated near regions of the Brillouin zone surrounding the resonant Floquet gap, with opposite signs of Berry curvature above and below the gap [74]. Thus, because the ideal FTI state creates an imbalance of occupations of states above and below the Floquet gap, it gives rise to a finite anomalous Hall conductivity. 4.3 Survival of Floquet-Bloch States Under Continuous-Wave Driving So far, we have discussed how many-body scattering processes determine the occupation of Floquet states. However, for the Floquet states to remain coherent in the presence of these collision processes, the interband scattering rate of many-body collisions, denoted ℏ/𝜏inter , must be much smaller than the Floquet gap Δ [see condition 2 in Section 4.1]. Here, we show that steady states produced by continuous-wave (CW) irradiation can give rise to a Pauli-blocking effect suppressing the magnitude of ℏ/𝜏inter . Such a Pauli blocking effect enables CW lasers to induce coherent Floquet states, even though such CW laser sources are typically incapable of achieving the ultra-strong field amplitudes produced by ultrafast lasers (see, e.g., Ref. [1]). To provide an estimate of the interband scattering rate near a resonant Floquet gap, we cannot use the simplified analysis in Section 4.2, which only captures the momentum-averaged scattering processes across the Brillouin zone (BZ). Instead, 46 Figure 4.5: Sketch of an electron-electron collision process (green arrows) facilitating interband electronic scattering across the Floquet gap. we need to consider the collision processes in regions of the BZ close to the Floquet gap, sketched in Figure 4.5. Because electron-electron collisions typically occur on timescales much faster than electron-phonon scattering [258], we focus on interband electron-electron Auger collisions near the Floquet gap, sketched in Figure 4.5. Here, electrons in the initial Floquet states denoted 𝑠1 and 𝑠2 are scattered into the final states 𝑠3 and 𝑠4 . The resulting rate of electronic scattering out of the state 𝑠1 can be estimated as −1 𝜏inter ≈ 1 |𝑉𝑞 ⟨𝜙0𝑠1 |𝜙0𝑠4 ⟩⟨𝜙0𝑠2 |𝜙0𝑠3 ⟩| 2 𝐹𝑠2 (1 − 𝐹𝑠4 )(1 − 𝐹𝑠3 ), ℏ (4.13) where |𝜙𝑛𝑠 ⟩ denotes the 𝑛-th Floquet harmonic corresponding to the Floquet state 𝑠, 𝐹𝑠 denotes the steady state occupation, and 𝑉𝑞 denotes the bare electron-electron matrix element corresponding to the process with momentum transfer 𝑞. The electronic states near the Floquet gap are superpositions of the conduction and valence bands, and therefore have reduced spectral weight, i.e., |⟨𝜙0𝑠 |𝜙0𝑠 ⟩| ≈ 0.5. Additionally, photoexcitation processes, see Figure 4.3, produce densities of electronic occupations near the gap, which are relaxed by regular electron-phonon collision processes, see Figure 4.2. When these photoexcitation and phonon relaxation processes are balanced in a steady state, we can roughly estimate that the occupation functions 𝐹𝑠 are approximately half-filled near the gap, which can be more rigorously justified by numerical calculations, see Ref. [1]. Overall, we can estimate that |⟨𝜙0𝑠1 |𝜙0𝑠4 ⟩⟨𝜙0𝑠2 |𝜙0𝑠3 ⟩| 2 𝐹𝑠2 (1 − 𝐹𝑠4 )(1 − 𝐹𝑠3 ) ∼ 10−2 (4.14) 47 providing a roughly 100-times reduction in the bare interband scattering rate set by ℏ−1 |𝑉𝑞 | 2 . As we have demonstrated, the Pauli blocking effect due to photoexcited population near the gap plays a key role in suppressing the scattering rate in the steady state. Crucially, this reduced interband scattering rate enables coherent Floquet-Bloch states to survive in continuous-wave setups, even when the laser amplitudes are weaker than those used in ultrafast laser experiments. Part II Persistent Floquet Quantum Matter 48 49 Chapter 5 OPTICAL CONTROL OF SLOW TOPOLOGICAL ELECTRONS IN MOIRÉ SYSTEMS Floquet moiré materials possess optically-induced flat-electron bands with steadystates sensitive to drive parameters. Within this regime, we show that strong interaction screening and phonon bath coupling can overcome enhanced drive-induced heating. In twisted bilayer graphene (TBG) irradiated by a terahertz-frequency continuous circularly polarized laser, the extremely slow electronic states enable the drive to control the steady state occupation of high-Berry curvature electronic states. In particular, above a critical field amplitude, high-Berry-curvature states exhibit a slow regime where they decouple from acoustic phonons, allowing the drive to control the anomalous Hall response. Our work shows that the laser-induced control of topological and transport physics in Floquet TBG are measurable using experimentally available probes. 5.1 Introduction Time-periodic fields can drive materials into exotic non-equilibrium phases [15, 36, 75, 76, 80, 81, 83, 99, 167, 169, 214, 235, 258, 293, 294], with unconventional transport and optical characteristics [45, 64, 74, 140, 152, 189, 260, 304] controllable by external parameters. In laser-driven twisted bilayer graphene (TBG) [131, 165, 281, 291, 292], a flat-band regime with pronounced electron-electron interaction effects is accessible away from the magic angles [25]. Generating low-temperature Floquet states in such a regime requires cooling processes that compensate for strong drive-induced electron-electron heating. A common cooling solution involves coupling Floquet systems to low-temperature phonon baths [62, 63, 258]. We demonstrate that intrinsic electron-phonon coupling in TBG and Coulomb screening can stabilize low-temperature steady-states in Floquet TBG under terahertz (THz) frequency, circularly polarized laser drives. In this steady-state, the drive amplitude controls the filling of electronic states with large Berry curvature, resulting in a highly tunable anomalous conductivity 𝜎𝑥𝑦 [74, 185, 215, 253, 254] (Figure 5.1(a-b)). The ability to tune the Floquet steady-state results from the unique slow electron regime in TBG where phonons travel faster than—and decouple from—many flat band electronic states [77, 261]. 50 (a) laser (b) -1.5 -0.5 0.5 1.5 d (c) Figure 5.1: (a) Schematic experimental design. Circularly polarized laser induces non-trivial Berry-curvature in the narrow bands (see Figure 5.2(b-c)), resulting in an anomalous Hall conductivity 𝜎𝑥𝑦 . TBG lies on top of a dielectric and metallic gate that screen electron-electron interactions. (b) Anomalous Hall conductivity vs. drive amplitude E for 𝜁 ≈ 0.5 and various values of 𝜒 indicated on the scale (see below Eqs. 5.4 and 5.7 for definitions of 𝜁 and 𝜒). The 𝜎𝑥𝑦 features a rapid drop with E below the critical amplitude E ∗ (dashed line). Here, E0 = ℏ𝑣 𝐹 /(𝑒𝐿 2𝑀 ) ≈ 7.2×104 V/m. (c) Critical amplitude vs. 𝑐 ph /𝑣 0eff , where 𝑣 0eff = 𝑣 eff (0) is an effective electron velocity defined in the text. Enlarged red circle: E ∗ in (b). 5.2 The System We begin by constructing the time-periodic, interacting Hamiltonian for laser-driven TBG near the charge neutrality point and at a twist angle 𝜃. The single-particle Í (𝜉) (𝜉)† (𝜉) (𝜉)† effective Hamiltonian of undriven TBG is 𝐻ˆ 0 = 𝒌𝜈𝜉 𝐸 𝒌𝜈 𝑐ˆ 𝒌𝜈 𝑐ˆ 𝒌𝜈 , where 𝑐ˆ 𝒌𝜈 (𝜉) creates a Bloch state |𝜉𝜈𝒌⟩ of crystal momentum 𝒌, band 𝜈, and energy 𝐸 𝒌𝜈 , near valley index 𝜉 = ±1 of the single-layer graphene Brillouin zone [25, 149]. The index 𝜈 = ± labels the narrow central particle and hole bands (Figure 5.2(a, b)) with total bandwidth 𝑊, which are separated by a large energy gap from all other bands. We consider a circularly polarized laser of vector potential 𝑨(𝑡) = (E/Ω) [cos(Ω𝑡) 𝒙ˆ − sin(Ω𝑡) 𝒚ˆ ] with electric field amplitude E and angular-frequency Ω, which couples to electrons by minimal coupling 𝒌 → 𝒌 + 𝑒 𝑨(𝑡)/ℏ, resulting in the time-periodic Hamiltonian 𝐻ˆ 0 (𝑡). (𝜉) The periodic Hamiltonian 𝐻ˆ 0 (𝑡) gives rise to Floquet eigenstates |Φ 𝒌𝛼 (𝑡)⟩ with (𝜉) (𝜉) quasienergies 𝜀 𝒌𝛼 satisfying |𝜀 𝒌𝛼 | < 21 ℏΩ. We consider the regime 𝑊 ≤ ℏΩ < 2𝑊 corresponding to a single photon resonance within the central TBG bands. Specifically, we consider Ω = 5 meV/ℏ and TBG at a near-magic twist angle of 𝜃 = 1.13◦ whose Fermi velocity 𝑣 𝐹 ≈ 17 km/s (corresponding to 𝑊 = 5 meV in the Bistritzer-MacDonald model [25, 149]) is comparable to phonon speeds in TBG (𝜉) [148]. The drive mixes the two central bands 𝜈 = ±1, resulting in quasienergies 𝜀 𝒌𝛼 , with upper and lower Floquet bands denoted by 𝛼 = ± (Figure 5.2(d)) [131]. The drive opens off-resonant gaps of size Δ𝐾 ≈ 2𝑒 2 𝑣 2𝐹 E 2 /ℏΩ3 at the Dirac points 𝐾 and 51 (a) (c) (b) 25 meV -25 meV 0 0. 0. 32 25 5 Figure 5.2: (a) Zoom-in on schematic narrow bands in a moiré system. Drive with angular frequency Ω resonantly couples states along resonance rings (green curves). (b) Undriven spectrum of TBG along a line in the Brillouin Zone indicated by the orange curve in (c). Dashed frame encloses optically-active, narrow central bands (𝜉) 𝜈 = ±1. (c) Berry curvature B 𝒌+ in the upper Floquet band, with blue color intensity   (𝜉) (𝜉) proportional to tanh 2B 𝒌+ /𝐿 2𝑀 (color bar) so B 𝒌+ peaks are more visible. Dashed (𝜉) lines indicate areas enclosing B 𝒌+ peaks at the Dirac points and resonance ring. (d) Periodic quasienergy Floquet spectrum of the driven system, having two central bands shown in (a). The Floquet spectrum exhibits the upper (UFB, 𝛼 = +) and lower (LFB, 𝛼 = −) Floquet bands, separated by off-resonant gaps Δ𝐾 at the Dirac 𝐾, 𝐾 ′ points and a Rabi-like gap Δ 𝑅 along the resonance ring [130, 236, 239]. 𝐾 ′ of the moiré Brillouin zone and a Rabi-like gap of Δ 𝑅 ∼ 𝑉 along the resonance (𝜉) (𝜉) ring, which is the ring on the 𝒌-plane satisfying 𝐸 𝒌+ − 𝐸 𝒌− = ℏΩ (green rings in Figure 5.2(a, d)). Here, 𝑣 𝐹 is the Fermi velocity of the undriven band structure, 𝑉 is the energy scale of the drive, and the expression for Δ𝐾 comes from the Van-Vleck perturbative expansion [6, 130, 215, 239, 241, 284]. The key component for stabilizing Floquet many-body states is the electron coupling to low-temperature longitudinal TBG acoustic phonons: ∑︁ 𝜈𝜈 ′ 𝜉 (𝜉)† (𝜉) ˆ 𝐻el-ph = 𝑀 𝒌,𝒒,𝑮 𝑐ˆ 𝒌+𝒒+𝑮,𝜈′ 𝑐ˆ 𝒌𝜈 ( 𝑏ˆ †𝒒 + 𝑏ˆ −𝒒 ) + h.c. (5.1) 𝒌,𝒒,𝑮 𝜈,𝜈 ′ .𝜉 Here, 𝑮 is a moiré Brillouin zone reciprocal lattice vector, and √︁ √︁ 𝜈𝜈 ′ 𝜉 𝜉𝜈𝜈 ′ 𝑀 𝒌,𝒒,𝑮 = 𝐷 ℏ𝑐 ph 𝑞/( 2𝐴 𝑀 𝜌𝑐 ph )W𝒌,𝒒+𝑮 (5.2) is the matrix element with deformation potential 𝐷, moiré unit cell area 𝐴 𝑀 = √ 2
3𝐿 𝑀 /2, lattice vector length 𝐿 𝑀 = 𝑎/[2 sin 𝜃/2 ], monolayer graphene density 𝜌, and monolayer lattice vector length 𝑎 = 0.246 nm. The operator 𝑏ˆ †𝒒 creates an 52 acoustic phonon mode of momentum 𝒒 with amplitude 𝑞, speed 𝑐 ph , and energy ℏ𝑐 ph 𝑞. The speed of sound 𝑐 ph in TBG is roughly the same as that in monolayer graphene, but the small Brillouin zone in TBG folds the acoustic phonon dispersion 𝜉𝜈𝜈 ′ into many branches [148]. The form-factor W𝒌,𝒒+𝑮 ≡⟨𝜉𝜈′ 𝒌 + 𝒒 + 𝑮 |𝜉𝜈𝒌⟩ captures the decreasing coupling of electrons to folded phonon branches with large 𝑮 [119]. We also include electron-electron interactions: ∑︁ {𝜈𝑖 }𝜉 (𝜉)† (𝜉)† (𝜉) (𝜉) 𝐻ˆ el-el = 𝑉𝒌 1 ,𝒌 𝑐ˆ 𝑐ˆ 𝑐ˆ 𝑐ˆ , (5.3) 2 ,𝒒,𝑮 𝒌 1 +𝒒,𝜈1 𝒌 2 −𝒒,𝜈2 𝒌 2 ,𝜈3 𝒌 1 ,𝜈4 𝒌 1 ,𝒌 2 𝒒,𝑮 {𝜈𝑖 },𝜉 {𝜈 }𝜉 𝜉𝜈 𝜈 𝜉𝜈 𝜈 𝑖 1 4 2 3 where 𝑉𝒌 1 ,𝒌 = 𝑉𝒒+𝑮 W𝒌 1 ,𝒒+𝑮 W𝒌 2 ,−𝒒−𝑮 , with 𝑖 = 1, . . . , 4, contains the screened 2 ,𝒒,𝑮
Coulomb potential 𝑉𝒒 = 𝑒 2 /(2𝜖0 𝑞 𝐴 𝑀 )(1 + 𝜖 coth 𝑞𝑑 ) −1 for a gate separated from TBG by a dielectric of permittivity 𝜖 and thickness 𝑑, where 𝜖 0 is the vacuum permittivity (Figure 5.1(a)). We focus on electron dynamics in its Floquet basis, treating interactions 𝐻ˆ el-ph and 𝐻ˆ el-el as weak perturbations scattering electrons between single-particle Floquet (𝜉) (𝜉)† (𝜉) states [75, 76, 253, 257, 258]. The occupation probability 𝐹 (𝑡) = ⟨ 𝑓ˆ (𝑡) 𝑓ˆ (𝑡)⟩ 𝒌𝛼 𝒌𝛼 𝒌𝛼 is described by the Floquet-Boltzmann Equation (FBE) [93, 110, 257, 258], el-ph (𝜉) (𝜉) (𝜉) el-el 𝐹¤𝒌𝛼 (𝑡) = 𝐼 𝒌𝛼 [{𝐹𝒌𝛼 (𝑡)}] + 𝐼 𝒌𝛼 [{𝐹𝒌𝛼 (𝑡)}]. (𝜉)† (𝜉) (5.4) el-ph Here, 𝑓ˆ𝒌𝛼 (𝑡) creates a single-particle electron state |Φ 𝒌𝛼 (𝑡)⟩, and 𝐼 𝒌𝛼 and el-el are respectively the electron-phonon and electron-electron collision integrals, 𝐼 𝒌𝛼 evaluated by the Fermi golden rule (see Appendix A for FBE details). The (𝜉) (𝜉)† (𝜉) steady-state distribution yields 𝐹¤𝒌𝛼 = 0, and ⟨ 𝑓ˆ𝒌𝛼 (𝑡) 𝑓ˆ𝒌𝛼′ (𝑡)⟩ is supressed for ph ph 𝛼 ≠ 𝛼′ when 1/𝜏𝒌tot ≡ 1/𝜏𝒌el + 1/𝜏𝒌 ≪ Δ𝜀 𝒌 /ℏ, where 𝜏𝒌el and 𝜏𝒌 are the interband electron-electron and electron-phonon scattering times, respectively, and Δ𝜀 𝒌 = min𝑛∈Z |𝜀 𝒌+ − 𝜀 𝒌− + 𝑛ℏΩ| [110, 145, 257]. Because Δ𝜀 𝐾 = 2Δ𝐾 is minimal, the condition is equivalently 𝜁 ≡ ℏ/(2Δ𝐾 𝜏𝐾tot ) ≪ 1 (see Figure 5.4(d)). In Figs. 5.1(b) and 5.4(c-d), we show the maximal 𝜁 across fields E plotted in Figs. 5.1 and 5.3. 5.3 Transport Properties To probe the electronic dynamics induced by the laser, we study the anomalous conductivity in the steady-state of the system [39, 61, 63, 74, 185, 215, 253, 254, 280] ∫ 2𝑒 2 ∑︁ (𝜉) (𝜉) 𝑑 2 𝒌B 𝒌𝛼 𝐹𝒌𝛼 , (5.5) 𝜎𝑥𝑦 = ℏ 𝛼,𝜉=± 53 which averages the product of Berry curvature [74, 87, 239] ∫ Ω 2𝜋/Ω (𝜉) (𝜉) (𝜉) B 𝒌𝛼 = 𝑑𝑡 Im⟨𝜕𝑘 𝑥 Φ 𝒌𝛼 (𝑡)|𝜕𝑘 𝑦 Φ 𝒌𝛼 (𝑡)⟩, 𝜋 0 (5.6) (𝜉) and the steady-state fillings, 𝐹𝒌𝛼 . Without the drive, TBG has fragile topology with 𝜎𝑥𝑦 = 0 at charge neutrality [212, 233, 308]. The circularly polarized laser breaks time-reversal symmetry between the valleys 𝜉 = ±1, opens Haldane gaps in each valley, and produces nonzero 𝜎𝑥𝑦 . Our main finding is that 𝜎𝑥𝑦 can be controlled by the field strength. It features a rapid drop as a function of the amplitude of the drive, E, near the critical amplitude E ∗ (Figure 5.1(b)). This strong dependence on the external field indicates profound changes in the electronic steady-state distribution as the drive amplitude changes across E = E ∗ . Furthermore, this strong amplitude-dependence arises only when the undriven effective electronic velocity 𝑣 0eff is close to 𝑐 ph in TBG (Figure 5.1(c)), a condition unique to TBG near the “slow-electron” regime [77, 261]. 5.4 Phenomenological Analysis We explain the origin of the strong dependence of 𝜎𝑥𝑦 on the drive amplitude near E = E ∗ (Figure 5.1(b)) by focusing on key processes affecting 𝜎𝑥𝑦 , which involve momentum states (the 𝐾 and 𝐾 ′ points and resonance ring, see Figure 5.2(c)) with (𝜉) large Berry curvature B 𝒌𝛼 . We assume that the steady-state occupation of the upper (𝜉) (𝜉) Floquet band (UFB, 𝛼 = +) and valley index 𝜉 near 𝐾 are uniform, 𝐹𝒌+ = 𝐹𝐾+ , for 𝒌 ∈ S𝐾 , where S𝐾 is a small circle enclosing the full-width half maximum of the Berry curvature peak at 𝐾 (Figure 5.2(c)). The steady-state occupation emerges as a balance between the total incoming rate (𝜉) (𝜉) 𝐹¤𝐾+ | in into S𝐾 and outgoing rate 𝐹¤𝐾+ | out from S𝐾 . Single phonon emission connecting the UFB Sin (see Figure 5.3(a)) with S𝐾 is the dominant contribution to (𝜉) 𝐹¤𝐾+ | in . The two regions are connected by the phonon light-cone (see Fig 5.3(a)). (𝜉) (𝜉) (𝜉) (𝜉) This rate is 𝐹¤𝐾+ | ph,in ≈ R in (1 − 𝐹𝐾+ )𝐹in , where 𝐹in is the average UFB occupation in Sin , and R in is the average intrinsic scattering rate. Importantly, R in is proportional to the momentum-space area of Sin , denoted Ain , estimated by counting the UFB states that may scatter to S𝐾 by electron-phonon interactions. Hence, Sin is the intersection between the UFB and phonon light-cones originating anywhere within S𝐾 (Figure 5.3(a)). As Δ 𝑅 and Δ𝐾 widen with E, the Floquet bands become narrower [64, 131, 165], and Ain shrinks, vanishing at E = E ∗ (Figure 5.3(b)). The critical strength E ∗ is defined by 𝑣 eff (E ∗ ) = 𝑐 ph , where 54 (𝜉) (𝜉) 𝑣 eff (E) = max 𝒌 ′ (𝜀 𝒌 ′ + − 𝜀 𝑲+ )/| 𝒌 ′ − 𝑲| is the electronic velocity near the 𝐾 point. By estimating 𝑣 eff (E), one finds that E ∗ ∝ [1 − 𝑐 ph /𝑣 0eff ] 𝛾 for small 1 − 𝑐 ph /𝑣 0eff , where 𝛾 depends on the quasienergy structure and 𝑣 0eff ≡ 𝑣 eff (0). One can also show Ain ∝ max(E − E ∗ , 0) as E → E ∗ . (See Appendix A.) (𝜉) (𝜉) (𝜉) Similarly, the phonon-mediated outgoing rate is 𝐹¤𝐾+ | ph,out ≈ R out 𝐹𝐾+ (1 − 𝐹out ), (𝜉) where 𝐹out is the lower Floquet band (LFB, 𝛼 = −) average occupation in Sout , and ∫ R out is the average intrinsic rate, proportional to Aout = S 𝑑 2 𝒌, where Sout is out the momentum region enclosing intersections between the LFB with phonon light cones originating from states in S𝐾 (see Figure 5.3(a)). However, unlike Ain , Aout does not vanish as E → E ∗ and instead expands as E increases. Electron-electron interactions and photon-mediated Floquet-Umklapp (FU) processes introduce additional terms in the rate equation depending smoothly on E and (𝜉) roughly uniformly-spread in momentum. We thus include an incoming 𝐹¤𝐾+ | r,in = ph (𝜉) (𝜉) (𝜉) el Γin (1 − 𝐹𝐾+ ) and outgoing rate 𝐹¤𝐾+ | r,out = Γout 𝐹𝐾+ with Γin/out ≡ Γin/out + Γin/out , el(ph) where Γin/out , are rates of electron-electron (electron-phonon FU) processes. The strength of FU processes is weaker than R out by factors of ≈(𝑣 𝐹 𝑒E/Ω2 ) 2𝑛 , where |𝑛| > 1 is the number of photons emitted or absorbed [258]. FU processes also impart large phonon momentum transfers that the form-factor in Eq. C.6 suppresses. (𝜉) (𝜉) (𝜉) (𝜉) (𝜉) (𝜉) In the steady-state, 𝐹¤𝐾+ | in = 𝐹¤𝐾+ | ph,in + 𝐹¤𝐾+ | r,in and 𝐹¤𝐾+ | out = 𝐹¤𝐾+ | ph,out + 𝐹¤𝐾+ | r,out are equal, and (𝜉) R in 𝐹in + Γin (𝜉) . (5.7) 𝐹𝐾+ = (𝜉) (𝜉) R in 𝐹in + R out (1 − 𝐹out ) + Γin + Γout (𝜉) (𝜉) Note that 𝐹in , 1 − 𝐹out ≠ 0 due to electron (hole) excitations in the UFB (LFB) generated by FU processes. Since R in ∝ Ain , R in decreases as a function of E, shrinking to zero for E ≥ E ∗ (see Figure 5.3(b) for numerical verification). We (𝜉) expect a similar E-dependence of 𝐹𝐾+ and 𝜎𝑥𝑦 , yet smeared by additional scattering rates appearing in Eq. 5.7, as verified numerically in Figure 5.1(b). Additionally, ph (𝜉) el ≈ Eq. 5.7 elucidates the dependence of 𝐹𝐾+ on the ratio 𝜒 ≡ 𝜏𝐾el /𝜏𝐾 ≈ R out /Γout el (see Figure 5.1(b)), with 𝐹 (𝜉) → 0.5 as 𝜒 → 0. In Figs. 5.1(b) and R out /Γin 𝐾+ 5.4(b-c), we display 𝜒 evaluated at the amplitude E where 𝜁 is fixed. A similar rate equation can be derived for the occupation probability of holes in the LFB. Due to the emergent, approximate anti-unitary particle-hole symmetry [23, 270, 271] at charge neutrality that is preserved by the drive, the transition rates in the LFB are roughly similar to those in the UFB, leading to approximately 55 (a) (b) (c) (d) 0.99 0.974 0.958 Figure 5.3: (a) Schematics of the Floquet spectrum and one of the phonon lightcones originating from the area S𝐾 in the UFB. The intersection between the UFB (LFB) and all cones centered in S𝐾 form Sin (Sout ). As E → E ∗ , the area of Sin vanishes. (b-d) Numerical verification of the phenomenological model. (b) Area of Sin , Ain , vs. E for three values of 𝑐 ph /𝑣 0eff . (c) Average occupation in S𝐾 . (d) Anomalous Hall conductivity 𝜎𝑥𝑦 for same parameters as (b, c). At E ∗ (dashed (𝜉) lines), Ain , 𝐹𝐾+ , and 𝜎𝑥𝑦 sharply change. equal electron and hole occupations near the Dirac points in the UFB and LFB (𝜉) (𝜉) (𝐹𝐾+ ≈ 1 − 𝐹𝐾 ′ − ). Notice that the signs of the Berry curvatures near the Dirac points in the LFB and UFB are opposite, resulting in constructive contributions of electron and hole populations to 𝜎𝑥𝑦 . Thus, we can reproduce qualitatively the sharp change of 𝜎𝑥𝑦 with E in Figure 5.3 [131]. Occupations in the resonance ring vicinity (Figure 5.2(c)) yield a similar E-dependence, but with a much lower critical field (not visible for E plotted in Figs. 5.1 and 5.3) due to different effective electronic velocities near the resonance ring. 5.5 Numerical Analysis The results in Figure 5.3(b-d) utilized a simplified toy model describing TBG as a tight-binding hexagonal lattice, similar to graphene [42], but with parameters tuned to match 𝑣 𝐹 and the Brillouin zone size of TBG. This model misses some subtle details but captures the interplay between electron and phonon velocities and the large Berry curvature at the Dirac points and resonance ring. The model represents only the central 𝜈 = ±1 bands of the undriven bandstructure, but since the low drive angular frequency Ω is only resonant to these narrow bands, we can ignore the |𝜈| > 1 bands—valid when 𝜃 is near the magic angle where the |𝜈| > 1 and 𝜈 = ±1 bands are well-separated. In Appendix A, we present the numerical 56 analysis of a continuum model without electron-electron interactions [25, 149, 193], which yields qualitatively similar results, demonstrating that the controllable 𝜎𝑥𝑦 is insensitive to model details. In the toy model, 𝑣 0eff = 18.9 km/s, and we vary 𝑐 ph ∈ [17.9 km/s, 19.4 km/s] in Figure 5.1(c). In the range 𝑐 ph < 𝑣 0eff , the drive induces the regime 𝑐 ph > 𝑣 eff (E) for E > E ∗ . To capture the decaying overlap of the wavefunctions for momentum Umklapp transitions, in the toy model, we take √ 2 2 𝜉𝜈𝜈 ′ W𝒌,𝒒 → ⟨𝜉𝜈′ 𝒌 + 𝒒|𝜉𝜈𝒌⟩𝑒 −𝑙 𝑤 𝑞 /4 , with 𝑙 𝑤 ≈ 𝐿 𝑀 /(1.5 3) representing the radius of Wannier orbitals localized to TBG layer alignment sites [119]. First, we show how solving the FBE (Eq. 5.4) for the steady-state distribution verifies the phenomenological model. Consider the non-interacting limit by solving (𝜉) el-el = 0). The left-half column of Figure 5.4(a) Eq. 5.4 for 𝐹𝒌𝛼 with 𝜒 → ∞ (𝐼 𝒌𝛼 shows the non-interacting steady-state distributions for a phonon bath temperature of 1 K and 𝑐 ph = 0.99𝑣 0eff in the E > E ∗ and E < E ∗ cases. When E > E ∗ (left bottom quadrant), the Dirac points have reduced occupations (see zoom-in boxes) relative to when E < E ∗ (left top quadrant), because incoming scattering rates into S𝐾,𝐾 ′ are suppressed (verifying the phenomenological model). Figure 5.3(c) shows (𝜉) the occupation near the 𝐾 point, 𝐹𝐾+ , as a function of E for three values of 𝑐 ph /𝑣 0eff (𝜉) and verifies Ain , 𝐹𝐾+ , and 𝜎𝑥𝑦 sharply change at the same critical field E = E ∗ . (𝜉) Heating induced by FU processes causes 𝐹𝐾+ to slowly increase with E > E ∗ (see Eq. 5.7). Next, we quantify the strength of Coulomb screening necessary to stabilize the steady-state, which depends on the balance between electron-phonon cooling proel-el ≠ 0 by taking finite cesses and electron-electron heating processes. We include 𝐼 𝒌𝛼 𝜒. On the right-half column of Figure 5.4(a), we show the resulting steady-state (𝜉) occupations, which is slightly closer to the hot steady-state 𝐹𝒌± = 0.5 and has more smeared occupations than the non-interacting case (left half of Figure 5.4(a)). To quantify the effect of interactions on 𝜎𝑥𝑦 , note that, in Figure 5.1(b), 𝜎𝑥𝑦 drops less rapidly with E < E ∗ as 𝜒 decreases. We capture this behavior with the visibility parameter V ≡ −maxE<E ∗ |𝜕E 𝜎𝑥𝑦 |/[(𝑒 2 /ℎ)/E0 ]. Figure 5.4(b) demonstrates how V increases with 𝜒. Lastly, we relate 𝜒 and 𝜁 to physical parameters in TBG. Figure 5.4(c) shows the necessary gate distances 𝑑 and dielectrics 𝜖 to experimentally achieve various values of 𝜒, and Figure 5.4(d) shows the values of 𝜖 and deformation potentials 𝐷 satisfying 𝜁 < 1 for 𝑑 = 4 nm. One suitable dielectric is SrTiO3 with 𝜖 ∼ 1600 at Ω = 5 meV/ℏ angular frequencies [69, 243, 295]; note that surface optical phonons in SrTiO3 are of higher frequencies than Ω and would not interact V [4e2 /h] (a) 57 (b) 1 <latexit sha1_base64="WCdBMw0MbdqHbiiS92EsG9KdFLk=">AAAB/3icbVDLSsNAFL3xWesrKrhxM1gEVzUpRV0W3bisYB+QxDKZTtqhkwczE6HELvwVNy4UcetvuPNvnLRZaOuBgcM593LPHD/hTCrL+jaWlldW19ZLG+XNre2dXXNvvy3jVBDaIjGPRdfHknIW0ZZiitNuIigOfU47/ug69zsPVEgWR3dqnFAvxIOIBYxgpaWeeeiGWA0J5ll7glzk1Ol97Wzo9cyKVbWmQIvELkgFCjR75pfbj0ka0kgRjqV0bCtRXoaFYoTTSdlNJU0wGeEBdTSNcEill03zT9CJVvooiIV+kUJT9fdGhkMpx6GvJ/O0ct7Lxf88J1XBpZexKEkVjcjsUJBypGKUl4H6TFCi+FgTTATTWREZYoGJ0pWVdQn2/JcXSbtWtc+r9dt6pXFV1FGCIziGU7DhAhpwA01oAYFHeIZXeDOejBfj3fiYjS4Zxc4B/IHx+QONLJUo</latexit> K' 0.75 1000 1000 5 0. 0.5 1.0 0.7 500 5 0 500 -0.5 100 1 0.0 3 0.5 (d) 500 K 2 (c) K' K' 1 3 0 0. K 1.5 K 2.5 0.9 10 D [eV] 15 0.5 1.0 Figure 5.4: (a) Left column: steady-state occupation of the UFB when 𝜒 = ∞ (calculated on a 163 × 163 momentum grid). Right column: steady-state occupation when 𝜒 = 0.24 (calculated on a 73 × 73 momentum grid). Bottom row: strong-drive case (E = 4.3E0 > E ∗ ). Top row: weak-drive case (E = 0.97E0 < E ∗ ). Zoom-in boxes: the 𝐾, 𝐾 ′ points have reduced occupation when E > E ∗ relative to when E < E ∗ . (b) Visibility V vs. 𝜒. (c) Value of 𝜒 for various 𝜖 and gate distances 𝑑. (d) Value of 𝜂 for various 𝜖 and deformation potentials 𝐷, with 𝑑 = 4 nm. Points in (b, d): parameters used in Figure 5.1(b). with electrons in TBG via direct (non-FU) scattering processes [245]. 5.6 Conclusion TBG is a remarkable system whose Fermi velocity is comparable to the speed of sound. Upon THz-laser driving, the electronic population dynamics exhibits bottlenecks for electron-phonon scattering into high-Berry curvature Floquet states, strongly affecting the anomalous Hall transport. These bottlenecks can be sensitively controlled by the drive amplitude. If the undriven effective electron speed is faster than sound 𝑣 0eff > 𝑐 ph , a drive with E > E ∗ induces the opposite regime 𝑣 eff (E) < 𝑐 ph , weakening the electron-phonon coupling and suppressing the Hall conductivity (Figure 5.1(b)). We also find that a strong E-dependence of 𝜎𝑥𝑦 arises for efficient Coulomb screening by a close-by gate or a strong dielectric [49, 248, 289]. Experimental advances in Floquet engineering [185], and THz laser sources [52, 160], show that our predicitions should be accessible experimentally. Analysis of UV-visible or X-ray driven TBG is a subject of future work, which must account for optically-active dispersive bands [131, 291]. High-frequency drives could reduce heating, facilitating fewer electron-electron FU processes [258] while activating electron-phonon Umklapp cooling processes arising from tightly- 58 localized Wannier orbitals in TBG [119]. (In this work, these cooling processes are suppressed FU processes.) Another interesting direction involves developing a Hartree-Fock treatment for symmetry-broken phases in the steady-state of strongly coupled TBG [76]. We leave these exciting directions to future studies. 5.7 Acknowledgements We thank Netanel Lindner, Mark Rudner, Or Katz, Gaurav Gupta, Seamus O’Hara, Jason Alicea, Alex Thomson, Felix von Oppen, Kryštof Kolář, Étienne LantagneHurtubise, and Valerio Peri for valuable discussions. C.Y. gratefully acknowledges support from the DOE NNSA Stewardship Science Graduate Fellowship program, which is provided under cooperative agreement number DE-NA0003960. C.L. acknowledges support by the Gordon and Betty Moore Foundation’s EPiQS Initiative, Grant GBMF8682, start-up funds from Florida State University and the National High Magnetic Field Laboratory. The National High Magnetic Field Laboratory is supported by the National Science Foundation through NSF/DMR-1644779 and the state of Florida. G.R. and I.E. are grateful for support from the Simons Foundation and the Institute of Quantum Information and Matter, as well as support from the NSF DMR grant number 1839271. This work is supported by ARO MURI Grant No. W911NF-16-1-0361, and was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. 59 Appendix A APPENDIX A.1 Details of the Models In both the toy and continuum models, we take the undriven Hamiltonians 𝐻 ( 𝒌) and obtain the time-dependent Hamiltonian 𝐻 ( 𝒌, 𝑡) via minimal coupling 𝒌 → 𝒌 + 𝑒 𝑨(𝑡)/ℏ. Here, 𝑨(𝑡) = 𝐴[cos(Ω𝑡) 𝒙ˆ − sin(Ω𝑡) 𝒚ˆ ] (A.1) is the magnetic vector potential of the circularly polarized laser. We can expand the time-dependent eigenstates of the Hamiltonian in a Floquet-Bloch basis [241]: (𝜉) |𝜓 𝒌𝛼 (𝑡)⟩ = 𝑒 −𝑖𝜀 𝛼 𝑡/ℏ |Φ𝑚𝒌𝛼 (𝑡)⟩, (A.2) (𝜉) where |Φ𝑚𝒌𝛼 (𝑡)⟩ is periodic in time (|Φ𝑚𝒌𝛼 (𝑡)⟩ = |Φ𝑚𝒌𝛼 (𝑡 + 2𝜋/Ω)⟩), 𝜀 𝛼 are the quasienergies plotted in Figure 5.2(d), and 𝛼 enumerates the Floquet quasienergy bands. To determine the Floquet-Bloch basis, it is easiest to expand the timedependent |Φ𝑚𝒌𝛼 (𝑡)⟩ in terms of time-independent Fourier harmonics |𝜙𝑚𝒌𝛼 ⟩, ∑︁ 𝑚 |Φ 𝒌𝛼 (𝑡)⟩ = 𝑒 −𝑖𝑚Ω𝑡 |𝜙𝑚𝒌𝛼 ⟩, (A.3) 𝑚 take a Fourier transform the Hamiltonian, ∑︁ 𝑒 −𝑖𝑚Ω𝑡 𝐻 (𝑚) ( 𝒌), 𝐻 ( 𝒌, 𝑡) = (A.4) 𝑚 and solve the Schrödinger equation in the basis of Floquet harmonics: ∑︁ ′ ′ (𝜉) (𝜀 𝛼 + 𝑚ℏΩ)|𝜙𝑚𝒌𝛼 ⟩ = 𝐻 (𝑚−𝑚 ) ( 𝒌)|𝜙𝑚𝒌𝛼 ⟩. (A.5) 𝑚′ In the following subsections, we detail the exact form of the Floquet Hamiltonians. Tight binding Floquet toy Hamiltonian We use a rescaled, two-band tight binding model for graphene to replicate the flat conduction and valence bands of TBG. In the rescaled Hamiltonian ! 0 ℎ𝒌 𝐻toy ( 𝒌) = ∗ , (A.6) ℎ𝒌 0 60 ℎ𝒌 = 𝑊 ∑︁ 3 𝑒𝑖𝒌·𝜹 𝑗 , (A.7) 𝑗 we choose long hopping vectors √

𝜹 𝑗 = 𝐿 𝑀 / 3[sin 2𝜋𝑚/3 𝒙ˆ + cos 2𝜋𝑚/3 𝒚ˆ ], (A.8) with 𝐿 𝑀 = 0.246 nm/(2 sin 𝜃/2), and a narrow bandwidth 𝑊. The corresponding rescaled energies and Bloch states are 𝐸 𝜈 ( 𝒌) = 𝜈|ℎ 𝒌 |, (A.9) and ! 1 𝜈𝑒𝑖arg(ℎ𝒌 ) |𝜈𝒌⟩ = √ , 1 2 respectively, with 𝜈 = ±1 enumerating the narrow Bloch bands. (A.10) Following Ref. [42], we perform minimal coupling, which turns the functions ℎ 𝒌 into time-dependent quantities with Fourier transforms ∫ 2𝜋/Ω 1 (𝑛) ℎ 𝒌+𝑒 𝑨(𝑡)/ℏ 𝑒 −𝑖𝑛Ω𝑡 𝑑𝑡 ℎ𝒌 = 2𝜋/Ω 0 ∑︁ (A.11) = 𝑡𝑒𝑖𝒌·𝜹 𝑗 𝑒𝑖𝑛𝜙 𝑗 𝐽𝑛 (−Ẽ), 𝑗 where Ẽ is the dimensionless drive strength 𝑒𝐿 𝑀 𝑒𝐿 𝑀 E Ẽ = √ 𝐴 = √ ; 3ℏ 3ℏ Ω the phase angles are 𝜙0 = 𝜋/2, 𝜙1 = −5𝜋/6, and 𝜙2 = −𝜋/6; and ∫ 2𝜋 1 𝐽𝑛 (𝑧) = 𝑒𝑖𝑧 cos 𝜃 𝑒𝑖𝑛𝜃 𝑑𝜃. 2𝜋𝑖 𝑛 0 The Fourier-transformed Hamiltonian is ! (𝑛) 0 ℎ (𝑛) 𝒌 𝐻toy ( 𝒌) = ∗(𝑛) . ℎ𝒌 0 (A.12) (A.13) (A.14) Note that ℎ∗(𝑛) = 𝒌 ∑︁ 𝑡𝑒 −𝑖𝒌·𝜹 𝑗 𝑒𝑖𝑛𝜙 𝑗 𝐽𝑛 ( Ẽ) (A.15) 𝑗 is the Fourier transform of the conjugate of ℎ 𝒌 . In simulations, we generally truncate the Fourier Hamiltonian (Eq. A.5) to −12 ≤ 𝑚 ≤ 12, so that we account for a sufficient number of high-order Floquet-Umklapp processes in the Floquet-Boltzmann equation. We do not perform the gauge transformation ℎ (𝑛) → 𝑖𝑒 −𝑖𝒌·𝜹0 ℎ (𝑛) so as to 𝒌 𝒌 preserve the 𝐶3 symmetry of the matrix element in the Floquet-Boltzmann equation (see Eq. C.21). 61 Figure A.1: (a) The quasienergy band structure of the toy model with for the parameters used in the main text. (b) The quasienergy band structure of the continuum model at valley 𝜉 = +1. In both panels, the first Floquet Brillouin zone is shaded. See Section A.2 for details and justification for the parameters we have used. Continuum Model Floquet Hamiltonian The undriven continuum model for TBG [149] describes the bandstructure of TBG near the valley 𝜉 = ±1 of the monolayer graphene Brillouin zone. Its Hamiltonian 𝑈𝜉† ª ©𝐻 ( 𝒌) 𝐻𝜉 ( 𝒌) = ­ 1 ® 𝜉 𝑈𝜉 𝐻2 ( 𝒌) « ¬ 𝜉 𝐴1 𝐵1 𝐴2 𝐵2 𝑇 is diagonalized in the basis 𝜓𝑛𝒌 = (𝜓𝑛𝒌 , 𝜓𝑛𝒌 , 𝜓𝑛𝒌 , 𝜓𝑛𝒌 ) with ∑︁ 𝑋 𝑋 𝐶𝑛𝒌 (𝑮)𝑒𝑖𝑮·𝒓 𝜓𝑛𝒌 (𝒓) = 𝑒𝑖𝒌 mic ·𝒓 (A.16) (A.17) 𝑮 where 𝑋 = 𝐴𝑙 , 𝐵𝑙 represents sublattice 𝐴 or 𝐵 degree of freedom in layer index √ +1 −1 −1 𝑙 = ±1, 𝒌 mic = 𝒌 + (𝑲 +1 𝜉 + 𝑲 𝜉 )/2 + 𝒙ˆ 3𝜉/2|𝑲 𝜉 − 𝑲 𝜉 | is the microscopic momentum of the electrons, 𝒌 is the mini Brillouin zone momentum, and 4𝜋 𝑲 𝑙𝜉 = −𝜉 𝑅(−𝑙𝜃/2) 𝒙ˆ (A.18) 3𝑎 𝜉 for layer 𝑙 = ±1 and 𝑎 = 0.246 nm. In Eq. A.16, 𝐻𝑙 are the monolayer graphene Hamiltonians, which, in close vicinity of the 𝜉 = ±1 valleys, resemble Dirac cones: h i 𝜉 𝑙 𝐻𝑙 ( 𝒌) = −ℏ𝑣 ml 𝑅(𝑙𝜃/2)( 𝒌 − 𝑲 ) · (𝜉𝜎𝑥 , 𝜎𝑦 ) (A.19) mic 𝐹 𝜉 where 𝑅(𝜑) is the 2 × 2 rotation matrix, and 𝑣 ml 𝐹 is the monolayer Graphene Fermi velocity. The interlayer coupling is ! ! 𝑢 𝑢′ 𝑢 𝑢′ 𝜈 −𝜉 𝑖𝜉𝑮 1 ·𝒓 𝑈𝜉 = ′ + ′ 𝜉 𝑒 𝑢 𝑢 𝑢𝜈 𝑢 ! (A.20) 𝑢 𝑢′ 𝜈 𝜉 𝑖𝜉 (𝑮 2 +𝑮 3 )·𝒓 + ′ −𝜉 𝑒 𝑢𝜈 𝑢 62 Using minimal coupling, we obtain time-dependent monolayer graphene Hamiltonians, with Fourier transform n  𝜉 (𝑛) 𝐻𝑙 ( 𝒌) = −ℏ𝑣 𝑅(𝑙𝜃/2) [𝒌 mic − 𝑲 𝜉(𝑙) ]𝛿𝑛,0 𝑒1 E [(𝛿𝑛,1 + 𝛿𝑛,−1 ) 𝒚ˆ ℏ2 o − 𝑖(𝛿𝑛,−1 − 𝛿𝑛,1 ) 𝒙ˆ ] · (𝜉𝜎𝑥 , 𝜎𝑦 ). + (A.21) Then, 𝜉 (𝑛) † ©𝐻 ( 𝒌) 𝑈𝜉 𝛿𝑛,0 ª (A.22) 𝐻𝜉(𝑛) = ­ 1 ® 𝜉 (𝑛) 𝑈𝜉 𝛿𝑛,0 𝐻2 ( 𝒌) ¬ « is the Fourier transform of the continuum model Hamiltonian. For the continuum model, we truncate the Floquet Hamiltonian (Eq. A.16) to −6 ≤ 𝑚 ≤ 6. Upon diagonalizing the Floquet Hamiltonian, we obtain a large number of Floquet states per energy interval [−ℏΩ/2, ℏΩ/2]. We select two states per 𝒌-point whose spectral weights 𝐴𝛼0 ( 𝒌) = |⟨𝜙0𝒌𝛼 |𝜙0𝒌𝛼 ⟩| 2 are large (which makes their contribution to the Floquet-Boltzmann equation most important, see Section C). Quasienergy Bands In Section A.2, we provide and motivate the choices of physical parameters that we use in the main text. In Figure A.1, we preview the quasienergy bands for our choice of toy and continuum model parameters. A.2 Choice of Physical Parameters First, we present the physical parameters we use for the electronic Hamiltonian in the TBG continuum model (see Section A.1 for the Hamiltonian). We consider the non-interacting continuum model [25, 149] at a near-magic twist angle of 𝜃 = 1.13◦ . The bandwidth of the central bands at this angle is 𝑊 ≈ 5 meV, and a perturbative expansion of the Hamiltonian around the Brillouin zone Dirac points [25] estimates the Fermi velocity as 2 2 2 𝑣 𝐹 (𝜃) = 𝑣 ml 𝐹 (1 − 3𝛽 )/(1 + 3𝛽 (1 + 𝜂 )), (A.23) ′ ml 5 where 𝛽 = 𝑢′/(ℏ𝑘 𝜃 𝑣 ml 𝐹 ) and 𝜂 = 𝑢/𝑢 with 𝑣 𝐹 = 8 × 10 m/s, 𝑘 𝜃 = 4𝜋/(3𝐿 𝑀 ), 𝑢 = 0.0797 eV, and 𝑢′ = 0.0975 eV [25, 149]. Eq. A.23 predicts that the Fermi velocity at the chosen twist angle is 𝑣 𝐹 = 27 km/s. However, the derivation of 𝜉 Eq. A.23 approximates that 𝐻𝑙 is roughly 𝜃-independent and tends to overestimate 𝑣 𝐹 (see Figure 4 inset in [25]). We can obtain a better estimate by numerically 63 calculating the Fermi velocity along the path 𝐾-𝑀 in 𝒌-space of the 𝜈 = +1 band in the 𝜉 = +1 valley. (This is the direction of maximum Fermi velocity.) The estimate yields 𝑣 𝐹 = 17.5 km/s, and we hereafter use this value. In our Floquet Hamiltonian, we use a laser angular-frequency of Ω ≈ 𝑊/ℏ ≈ 5 meV/ℏ. Second, we present the parameters we use for the electronic Hamiltonian of the TBG two-band toy tight binding model (see Section A.1 for the Hamiltonian). We choose our toy model Fermi velocity, frequency, and twist angle to roughly match those of the continuum model. Specifically, we use a twist angle of 𝜃 = 1.13◦ and √ choose 𝑊 = 3.1 meV so that the Fermi velocity 𝑣 𝐹 = 𝑊 𝐿 𝑀 /(2 3ℏ) = 17 km/s roughly matches that of the continuum model at the same angle. In the toy model Floquet Hamiltonian, we choose Ω ≈ 5 meV/ℏ. Third, we discuss the parameters we use for the TBG phonons. For both the continuum and toy models, we consider phonons speeds in the range of 𝑐 ph ∈ [17.9 km/s, 19.4 km/s]. In the toy model, 𝑣 0eff = 18.9 km/s, and, in the continuum model, 𝑣 0eff = 19.5 km/s, so the range of 𝑐 ph we choose covers the regime 𝑐 ph < 𝑣 0eff , in which the drive induces the opposite regime 𝑐 ph > 𝑣 eff (E) when E > E ∗ . We also use the same phonon bath temperature of 𝑇ph = 1 K for the toy and continuum model calculations. Please see Section A.15 for details of the numerical 𝒌-point grid and Section A.12 for details of the toy model form factor. A.3 Anomalous Hall Conductivity Calculations for the Continuum Model In this section, we repeat the calculations in the main text on the TBG continuum model [25, 149]. We consider the non-interacting limit, setting 𝜖 → ∞ so that el-el = 0. 𝐼 𝒌𝛼 First, we discuss differences in the bandstructure and topology at valleys 𝜉 = + and 𝜉 = −. The circularly polarized laser opens a gap at the Dirac points, Δ𝐾 , effectively adding a mass term 𝜉Δ𝐾 𝜎𝑧 to the Hamiltonian (see Section A.1 and [131] for a derivation) in the vicinity of the Dirac points. Because the sign of the mass term depends on 𝜉, the 𝜉 = ±1 superlattice valley contributions to 𝜎𝑥𝑦 do not trivially cancel to zero. In fact, in reciprocal space, the Berry curvature and occupations near 𝜉 = +1 are simple 𝜋/3 rotations of those in 𝜉 = −1, so ∫ 𝑑 2 𝒌 (+1) (+1) 4𝑒 2 ∑︁ B 𝐹 . (A.24) 𝜎𝑥𝑦 = ℎ 𝛼=± MBZ (2𝜋) 2 𝒌𝛼 𝒌𝛼 64 (b) - (a) [km/s] 19.5 18.5 Figure A.2: (a) Left: the steady-state occupation of the lower Floquet band in valley 𝜉 = +1 of the continuum model [25, 149]. Right: the Berry curvature of the same band, which peaks near the Dirac points and the resonance ring. (b) The anomalous Hall conductivity 𝜎𝑥𝑦 as a function of drive strength E. In Figure A.2, we show the steady-state and 𝜎𝑥𝑦 for the continuum model calculation. 𝜉𝜈 ′ 𝜈 Note that we use the full form factor W𝒌,𝒒 = ⟨𝜉𝜈′ 𝒌 + 𝒒|𝜉𝜈𝒌⟩ as calculated from the continuum model wavefunctions (see Section A.12). A.4 Direct Variation of the Phonon Speed 𝑐 ph Throughout the main text, we use the drive strength E to control electron speeds. We could achieve similar results by keeping E fixed and varying 𝑐 ph instead. Figure A.3 shows the variation of 𝜎𝑥𝑦 as a function of 𝑐 ph . The curves resemble the dependence of 𝜎𝑥𝑦 on E in the main text (see, for e.g., Figure 1(b)). A.5 Symmetries and Berry Curvature Distributions of the Toy and Continuum Models In this section, we compare the symmetries of the toy and continuum models with and without the drive, and we show that the Berry curvature distributions of the models with the drive are consistent near the Dirac points. Symmetries The symmetries of the undriven BM continuum model are 𝐶2𝑧𝑇 = 𝜎𝑥 K, 𝐶3 = 𝑒𝑖2𝜋/3𝜎𝑧 , and 𝐶2𝑥 = 𝜎𝑥 𝜏𝑥 , where 𝜏𝑖 and 𝜎𝑖 are in the layer and sublattice degrees of freedom, 𝑖 = 0, 𝑥, 𝑦, 𝑧, and K is conjugation. At charge neutrality, undriven TBG also has an emergent, approximate unitary particle-hole symmetry 𝑃 = 𝑖𝜏𝑦 and anti- 65 unitary particle-hole symmetry P = 𝑃𝐶2𝑧𝑇 within each superlattice valley, which ensures 𝑃𝐻𝜉 ( 𝒌)𝑃−1 ≈ −𝐻𝜉 (−𝒌) and P𝐻𝜉 ( 𝒌)P −1 ≈ −𝐻𝜉 (−𝒌) (see Eq. A.16 for the definition of 𝐻𝜉 ( 𝒌)) [23, 270, 271]. We now discuss the effect of the drive on the symmetries of the BM model. The drive induces a dynamical Haldane mass term Δ𝐾 𝜉𝜎𝑧 (see Section A.5). It also resonantlycouples states around the resonance ring, where the effective Hamiltonian is 𝐻 𝑅 ( 𝒌) = 𝑉𝑅 ( 𝒌)|𝜉 − 𝒌⟩⟨𝜉 + 𝒌 | + 𝑉𝑅∗ ( 𝒌)|𝜉 + 𝒌⟩⟨𝜉 − 𝒌 | 𝜉 (A.25) as derived from degenerate perturbation theory, where 𝑉𝑅 ( 𝒌) = ⟨𝜉−𝒌 |𝐻𝜉(1) ( 𝒌)|𝜉+𝒌⟩ and |𝜉 ± 𝒌⟩ are the undriven single-particle Bloch states (see the main text for the definition) [130]. We can see that the drive breaks 𝐶2𝑧𝑇 symmetry by opening the Haldane gap. The drive also breaks 𝑃 symmetry because the drive-induced Haldane mass term Δ𝐾 𝜉𝜎𝑧 commutes with 𝑃. However, the drive preserves the anti-unitary P symmetry. One can see this by first noting that (𝐶2𝑧𝑇)𝜎𝑧 (𝐶2𝑧𝑇) −1 = −𝜎𝑧 , so PΔ𝐾 𝜉𝜎𝑧 P −1 = −Δ𝐾 𝜉𝜎𝑧 . Secondly, since P |𝜉−, ±𝒌⟩ = |𝜉+, ∓𝒌⟩, P𝐻 𝑅 ( 𝒌)P −1 = 𝑉𝑅 ( 𝒌)|𝜉+, −𝒌⟩⟨𝜉−, −𝒌 | + 𝑉𝑅∗ ( 𝒌)|𝜉−, −𝒌⟩⟨𝜉+, −𝒌 |. (A.26) Noting that P𝐻𝜉(1) ( 𝒌)P −1 = −𝐻𝜉(−1) (−𝒌) and |𝜉 + 𝒌⟩ = P −1 |𝜉−, −𝒌⟩, we find that 𝑉𝑅 ( 𝒌) = −⟨𝜉+, −𝒌 |𝐻𝜉(−1) (−𝒌)|𝜉−, −𝒌⟩ = −𝑉𝑅∗ (−𝒌). (A.27) Therefore, P𝐻 𝑅 ( 𝒌)P −1 = −𝐻 𝑅 (−𝒌), and the Hamiltonian is also particle-hole symmetric along the resonantly-coupled states. Lastly, the drive preserves 𝐶3 symmetry 3.38 3.86 4.35 4.83 Figure A.3: Anomalous Hall conductivity of the toy model as a function of the ratio 𝑐 ph /𝑣 0eff for three different drive field strengths E/E0 . The same electron-phonon decoupling process is visible as 𝜎𝑥𝑦 plateaus. 66 and 𝐶2𝑥 symmetry, which one can see by noting the following: −1 𝐶3 𝐻𝜉 ( 𝒌)𝐶3−1 = 𝐻𝜉 ( 𝒌), 𝐶2𝑥 𝐻𝜉 ( 𝒌)𝐶2𝑥 = 𝐻−𝜉 ( 𝒌), (A.28) −1 𝐶3 Δ𝐾 𝜉𝜎𝑧 𝐶3−1 = Δ𝐾 𝜉𝜎𝑧 , 𝐶2𝑥 Δ𝐾 𝜉𝜎𝑧 𝐶2𝑥 = −Δ𝐾 𝜉𝜎𝑧 , (A.29) and −𝜉 −1 = 𝐻 𝑅 ( 𝒌). 𝐶3 𝐻 𝑅 ( 𝒌)𝐶3−1 = 𝐻 𝑅 ( 𝒌), 𝐶2𝑥 𝐻 𝑅 ( 𝒌)𝐶2𝑥 𝜉 𝜉 𝜉 (A.30) Thus, the Hamiltonian near the Dirac points and resonantly-coupled states respect the 𝐶3 and 𝐶2𝑥 symmetries. Now, we discuss the symmetries of the toy model. In the undriven limit, the toy model has exact particle-hole symmetry 𝑃 = 𝜎𝑧 K. It also has the symmetries 𝐶2𝑧𝑇 = 𝜎𝑥 K, 𝐶 = 𝜎𝑧 (sublattice/chiral symmetry), and 𝐶3 = 𝑒𝑖2𝜋/3𝜎𝑧 . The drive opens a Haldane mass gap Δ𝐾 𝜉MBZ 𝜎𝑧 , where 𝜉MBZ = 1 (−1) for the mini Brillouin zone 𝐾 (𝐾 ′) point. We can now see that the drive breaks 𝐶2𝑧𝑇, 𝐶, and 𝑇 symmetry via the Haldane mass term, while preserving 𝑃 = 𝐶𝑇 symmetry. One can see that 𝑃 is preserved by noting that 𝑃Δ𝐾 𝜉MBZ 𝜎𝑧 𝑃−1 = −Δ𝐾 𝜉MBZ 𝜎𝑧 since conjugation K inverts the momentum and hence the sign of the mass term. Similar arguments as the continuum model case can be made to show that the Hamiltonian near the resonance ring respects 𝑃. Importantly, the drive preserves the emergent particle-hole symmetry in the continuum model while preserving the exact particle-hole symmetry in the toy model. As we note in the phenomenological analysis section of the main text, the emergent particle-hole symmetry ensures that the electron and hole scattering rates in the UFB and LFB are similar. Secondly, the drive breaks 𝐶2𝑧𝑇 symmetry in both models by opening a Haldane gap. In Section A.5, we show that the Haldane gap ensures the Berry curvature distributions of the models near the Dirac points are consistent. Berry Curvature The tunable conductivity 𝜎𝑥𝑦 relies only on the large Berry curvature and electronphonon scattering bottlenecks near the Dirac points of the mini Brillouin zone. In this section, we detail how the Berry curvature distributions for the toy and continuum models are consistent near the Dirac points, as the numerical calculations of Berry curvature demonstrate in Figure A.4. We now prove the agreement analytically. The Hamiltonian for the mini Brillouin zone Dirac cone in the toy model is given by toy 𝐻Dirac (𝒒) = ℏ𝑣 𝐹 𝒒 · (𝜉MBZ 𝜎𝑥 , 𝜎𝑦 ), (A.31) 67 where 𝒒 is the momentum measured from the 𝐾 or 𝐾 ′ point, 𝑣 𝐹 is the Fermi velocity of TBG, and 𝜉MBZ = + for the 𝐾 point and 𝜉MBZ = − for the 𝐾 ′ point in the mini Brillouin zone. The corresponding Hamiltonian for the continuum model is cont 𝐻Dirac (𝒒) = ℏ𝑣 𝐹 𝒒 · (𝜉𝜎𝑥 , 𝜎𝑦 ) (A.32) where 𝜉 is the superlattice valley index. Upon applying minimal coupling and the Van-Vleck perturbative expansion (see Section A.10 for details), one finds the effective Floquet Hamiltonians toy 𝐻Dirac,eff (𝒒) = ℏ𝑣 𝐹 𝒒 · (𝜉MBZ 𝜎𝑥 , 𝜎𝑦 ) + 𝜉MBZ Δ𝐾 𝜎𝑧 (A.33) cont 𝐻Dirac,eff (𝒒) = ℏ𝑣 𝐹 𝒒 · (𝜉𝜎𝑥 , 𝜎𝑦 ) + 𝜉Δ𝐾 𝜎𝑧 . (A.34) We note that in both models, the Berry curvature does not alternate signs between the 𝐾 and 𝐾 ′ points in mini Brillouin zone. Additionally, Eq. A.34 shows that the drive breaks time-reversal symmetry between the superlattice valleys 𝜉 = ±1 in the BM model, permitting nonzero 𝜎𝑥𝑦 when contributions to the conductivity from both superlattice valleys are combined. A.6 Full Phenomenological Model In this section, we derive a detailed phenomenological model that qualitatively reproduces the dependence of 𝜎𝑥𝑦 on E and the effect of interactions presented in the main text. Let us begin by adding more details to the phenomenological model for the 𝐾-point (a) (b) -100 0 100 - -100 0 100 Figure A.4: Comparing the Berry curvature distribution in the upper Floquet band in the (a) continuum model and (b) toy model at a drive amplitude of E/E0 ≈ 3.1. 68 occupation. We write (𝜉) el,FU el FU 𝐹𝐾+ = [R in 𝐹+ + Γin + Γin + R in 𝐹− ]/ [R in 𝐹+ + R out (1 − 𝐹− ) el,FU el,FU el el + Γin + Γin + Γout + Γout (A.35) FU FU 𝐹− + R out (1 − 𝐹+ )], + R in el, FU el FU , Γin/out where Γin/out and R in/out are the non-FU electron-electron, FU electronelectron, and FU electron-phonon scattering rates, respectively. Here, 𝐹𝛼 is the average occupation of Floquet band 𝛼 outside the resonance ring, and 𝐹+ = 1 − 𝐹− . We drop the superscript (𝜉) on the occupations for simplicity and work within a single superlattice valley. Let us now make the following approximations and definitions: el,FU el,FU el el Γin ≈ Γout ≡ Γ, and Γin ≈ Γout ≈ 𝑆Γ, (A.36) where FU processes are suppressed by a factor of 𝑆 ≡ (𝑉/ℏΩ) 2 , with 𝑉 ≈ 𝑣 𝐹 𝑒E/Ω. For phonon transitions, let us make the following definitions: R out ≡ R, and R in = 𝑟R, (A.37) FU /R FU , and approximate R FU ≈ where 𝑟 ≡ R in /R out . Let us also define 𝑟 FU ≡ R in out out FU FU 𝑆𝑎 R, where the factor 𝑎 > 1 accounts for the fact that the phase space area of states connected to the UFB 𝐾 point by FU processes is much larger than area of states connected to the UFB 𝐾 point by non-FU processes. We therefore obtain FU FU R out = 𝑆𝑎 FU R, and R in = 𝑆𝑎 FU𝑟 FU R. (A.38) Now, Eq. A.35 reduces to (𝜉) 𝐹𝐾+ = 𝑟 𝐹+ R + 𝑆𝐹− 𝑎 FU𝑟 FU R + (1 + 𝑆)Γ . (1 + 𝑟)𝐹+ R + 𝑆(1 + 𝑟 FU )𝐹− 𝑎 FU R + 2(1 + 𝑆)Γ (A.39) Let us further define 𝑥 ≡ Γ/R as a ratio of electron-electron to electron-phonon scattering rates and use 𝐹− ≈ 1 − 𝐹+ (ensured by emergent particle-hole symmetry, see Section A.5) to obtain (𝜉) 𝐹𝐾+ = 𝑟 𝐹+ + 𝑆(1 − 𝐹+ )𝑎 FU𝑟 FU + (1 + 𝑆)𝑥 . (1 + 𝑟)𝐹+ + 𝑆(1 + 𝑟 FU )(1 − 𝐹+ )𝑎 FU + 2(1 + 𝑆)𝑥 (𝜉) (A.40) We will determine the dependence of 𝐹𝐾+ on interaction strength 𝑥 at strong and weak drive amplitudes. 69 (𝜉) Figure A.5: The occupation 𝐹𝐾+ as predicted by the phenomenological model in ph Section A.6 for different ratios 𝜒 ≡ 𝜏𝐾el /𝜏𝐾 . Note that for large 𝜒, the occupation is lower than the non-interacting case at weak drive amplitudes, an effect of reduced Pauli blocking in the electron-phonon interactions. We now derive the phenomenological equation for 𝐹𝛼 . The rate equation is roughly 𝐹¤+ ≈Λin 𝐹− (1 − 𝐹+ ) − Λout 𝐹+ (1 − 𝐹− ) + Γ(1 + 𝑆)(1 − 𝐹+ ) − Γ(1 + 𝑆)𝐹+ (A.41) where Λin and Λout are electron-phonon scattering rates into and out of the UFB. Note that Λin ≈ 𝑆Λout since scattering processes described by Λin are FU processes. We also approximate Λout ≈ 𝑓𝑏 R, with a factor 𝑓𝑏 > 1, because we expect R, which is suppressed by strong electron-phonon scattering bottlenecks near the 𝐾 point, to be smaller than Λout , the total scattering rate into the UFB. We can then find the steady-state solution 𝐹+ in terms of 𝑥 and substitute the results into Eq. A.40. (𝜉) In Figure A.5, we show the occupation 𝐹𝐾+ as a function of E for different values ph ph of 𝜒 = 𝜏𝐾el /𝜏𝐾 ≈ 𝑥 −1 𝐹+ | E=E ∗ (c.f. 1/𝜏𝐾 ∼ R𝐹+ | E=E ∗ and 1/𝜏𝐾el ∼ Γ) where 𝐹+ | E=E ∗ is 𝐹+ evaluated at the drive ampltiude E ∗ , which we choose to be E ∗ = 2.5E0 . To generate the figure, we choose 𝑓𝑏 = 3 and 𝑎 FU = 28. Note that 𝑎 FU estimates that the total area of momentum states connected to A 𝐾 by electron-phonon FU processes covers roughly 1/6 of the Brillouin zone. We write the following E-dependent phenomenological equations for the ratios 𝑟 and 𝑟 FU that capture very roughly their dependence on E, inspired by Figure 4(b) in the main text. First, we approximate   2 𝑟 ≈ max (E−E ∗ )/(0.8E ) − 1, 0 . (A.42) 0 + 1 𝑒 Note 𝑟 ∼ 1 for E ≪ E ∗ and 𝑟 = 0 for E > E ∗ , capturing the behavior shown in 70 Figure 4(b) in the main text. Second, we choose 1/2 𝑟 FU ≈ 𝑒 (E−E ∗ )/(0.5E0 ) 1 + . +1 2 (A.43) Here, 𝑟 FU ∼ 1 for E ≪ E ∗ and decreases with E, but never reaches 0 (since the electrons are never decoupled from electron-phonon FU processes). Two features are notable in Figure A.5. First, the occupation decreases as a function of interaction strength for weak interactions (large 𝜒) and weak drive amplitudes, a result of interactions reducing Pauli blocking of the phonon processes from patch (𝜉) Sin to S𝐾 and from S𝐾 to Sout (i.e., increasing 𝐹in in Eq. 6 of the main text, (𝜉) el correspondingly suppressing 𝐹𝐾+ ). When interactions are strong (small 𝜒), Γin/out (𝜉) dominates, and 𝐹𝐾+ → 0.5. Second, the occupation increases slowly for E > E ∗ , since FU processes strengthen as (𝑉/ℏΩ) 2 grows with E. Both of these behaviors are visible in Figure 3(c) in the main text. A.7 Formal Definition, Numerical Evaluation, and Phenomenological Model of Ain As described in the main text, a patch Sin shaped as an elliptical annulus (see Figure 3(a)) with area Ain in momentum space vanishes as E → E ∗ . Here, we provide a formal definition of Ain and explain how we estimate its dependence on E numerically and analytically. Formal Definition Let us first define Ain formally. Consider a family of phonon cones centered throughout S𝐾 , the circular patch enclosing a 𝐾-point in the quasienergy spectrum (see Figure 3(a)). Suppose that a subset of the phonon cones are centered throughout a small quasienergy window 𝑑𝜀 𝒌+ . The 𝒌-space area of states 𝑑Ain containing intersections of the cones with the upper Floquet band is ∑︁ ∫ 𝑑Ain = 𝑑𝜀 𝒌+ 𝑑 2 𝒌 ′ 𝛿(𝜀 𝒌+ − 𝜀 𝒌 ′ + + 𝑠ℏ𝑐 ph | 𝒌 ′ − 𝒌 |). (A.44) 𝑠=± Next, we integrate over 𝜀 𝒌+ contained in S𝐾 to obtain ∫ ∫ 1 Ain = 𝑑A = × 𝑑2 𝒌 𝐷 (𝜀 𝒌+ ) 𝒌∈S𝐾 " # ∑︁ ∫ × 𝑑 2 𝒌 ′ 𝛿(𝜀 𝒌+ − 𝜀 𝒌 ′ + + 𝑠ℏ𝑐 ph | 𝒌 ′ − 𝒌 |) , 𝑠=± (A.45) 71 where 𝐷 (𝜀) = ∑︁ ∫ 𝛼 𝑑2 𝒌 𝛿(𝜀 − 𝜀 𝒌𝛼 ) (2𝜋) 2 (A.46) is the density of states in the quasienergy band structure. Exploiting the circular shape of S𝐾 , ∫ ∫ 𝑑2 𝒌 ≈ 𝑑 2 𝒌 Θ(| 𝒌 − 𝑲| − 𝑘 𝑝 ) (A.47) 𝒌∈S𝐾 where 𝑘 𝑝 is the radius of the circular area A 𝐾 of S𝐾 . Lastly, we calculate an approximate expression for 𝑘 𝑝 , the radius of A 𝐾 . In the vicinity of the Dirac cone, the Hamiltonian is 𝐻𝐾 ( 𝒌, 𝑡) = 𝒅 · 𝝈, (A.48) 2 where 𝒅 = ℏ𝑣 𝐹 𝜉 𝑘 𝑥 𝒙+ℏ𝑣 ˆ 𝐹 𝑘 𝑦 𝒚ˆ +𝜉Δ𝐾 E 𝒛ˆ. (See Section A.10 for a detailed derivation.) The 𝒛-component of the Berry curvature is 𝑧 B𝑘𝛼 =𝛼 Δ𝐾 𝑑𝑧 = 𝛼 2| 𝒅| 3 [(ℏ𝑣 𝐹 𝑘) 2 + Δ2𝐾 ] 3/2 (A.49) 𝑧 where 𝑑 𝑧 = 𝜉Δ𝐾 and 𝛼 = ±. At the half-maximum, B𝑘𝑧 𝑝 𝛼 = 0.5B0𝛼 , so 𝑘 𝑝 = (22/3 − 1) 1/2 Δ𝐾 . ℏ𝑣 𝐹 (A.50) Numerical Estimate To generate the values of Ain we present in Figure 3(b), we evaluate the integrals in Eq. A.45 on a finite-sized grid of 𝒌-points, smearing the step function by replacing Θ(| 𝒌 − 𝑲| − 𝑘 𝑝 ) → [𝑒 (| 𝒌−𝑲|−𝑘 𝑝 )/𝜎𝑘 + 1] −1 , where 𝜎𝑘 = 2𝜋/(𝐿 𝑀 𝑁) is the grid spacing between 𝒌-points on an 𝑁 × 𝑁 Monkhorst-Pack grid (see Section A.15). Thus, we approximate ∑︁ 1 × Ain ≈ [𝑒 (| 𝒌−𝑲|−𝑘 𝑝 )/𝜎𝑘 + 1] −1 𝐷 (𝜀 𝒌+ ) 𝒌 ∑︁ ∑︁    ′  × 𝛿(𝜀 𝒌+ − 𝜀 𝒌 ′ + + 𝑠ℏ𝑐 ph | 𝒌 − 𝒌 |)  .  𝑠=± 𝒌 ′    (A.51) For more information on how we approximate the Dirac Delta function on the grid, please see Section A.15. Note that we numerically-estimate the 𝐾-point occupation in a similar way, by calculating ∑︁ (𝜉) (𝜉) (A.52) 𝐹𝐾+ ≈ [𝑒 (| 𝒌−𝑲|−𝑘 𝑝 )/𝜎𝑘 + 1] −1 𝐹𝒌+ . 𝒌 72 (a) (b) Figure A.6: (a) The intersection S𝐾 (Figure 3(a)) as viewed on the Brillouin zone. The outer radius along the path 𝐾 𝑅 is ℎ 𝑏 (E). (b) Quasienergy (pink) along the path 𝐾 𝑅, with the phonon light cone (grey) that determines the outer radius of Ain . The intersections 𝑘 + and 𝑘 − between the cone and the upper Floquet band determines ℎ 𝑏 (E) = 𝑘 + − 𝑘 − . Phenomenological Model In this section, we prove that the intersection area Ain ∝ max(E ∗ − E, 0) as E → E ∗ . The shape of Ain is an elliptical annulus as shown in Figure 3(a). Let us use ℎ 𝑏 (E) and 𝑤 𝑏 (E) respectively to denote the outer major and minor axis radii of the elliptical annulus (see Figs. A.6(a) and A.7(a)). In the following sections, we begin by generating analytical estimates of ℎ 𝑏 (E) and 𝑤 𝑏 (E). Estimate of ℎ 𝑏 First, let us consider a slice of the upper Floquet band in 𝒌-space from the 𝐾 to the resonance ring (R) along the direction of ℎ 𝑏 (E), as we show in Figure A.6(b). Let us define a one-dimensional momentum component 𝑞 along the path 𝐾-𝑅. We sketch a phonon light cone (grey) originating from a point (yellow) in S𝐾 that determines the outer radius of Ain . The phonon cone intersects with the quasienergy at points 𝑘 + and 𝑘 − , and the outer radius of Ain is therefore ℎ 𝑏 (E) = 𝑘 + − 𝑘 − . First, consider the undriven limit E = 0, where the gaps Δ 𝑅 = 0 and Δ𝐾 = 0. We choose some point 𝑞 𝑚 such that 𝑘 − < 𝑞 𝑚 < 𝑘 + and series expand the energy 𝐸 (𝑞) of the undriven system around 𝑞 𝑚 : 1 𝐸 (𝑞) ≈ 𝐸 (𝑞 𝑚 ) + 𝐸 ′ (𝑞 𝑚 )(𝑞 − 𝑞 𝑚 ) + 𝐸 ′′ (𝑞 𝑚 )(𝑞 − 𝑞 𝑚 ) 2 2 2 = 𝑎2 𝑞 + 𝑎1 𝑞 + 𝑎0, (A.53) 73 (a) (b) Figure A.7: (a) Width of the intersection S𝐾 , 𝑤 𝑏 (E). (b) Circular coordinate system with arc length 𝑤 (increasing counterclockwise) that we use to determine 𝑤 𝑏 (E) = 𝑤 + − 𝑤 − . where 𝑎 2 = 𝐸 ′′ (𝑞 𝑚 )/2, 𝑎 1 = 𝐸 ′ (𝑞 𝑚 ) − 𝐸 ′′ (𝑞 𝑚 )𝑞 𝑚 , and 𝑎 0 = 𝐸 (𝑞 𝑚 ) − 𝐸 ′ (𝑞 𝑚 )𝑞 𝑚 + 𝐸 ′′ (𝑞 𝑚 )𝑞 2𝑚 /2. As we increase E, the gaps Δ𝐾 and Δ 𝑅 widen. Let us write the quasienergy in the vicinity of 𝑞 𝑚 as Δ𝐾 (A.54) 2 where 𝑓 (E) ≤ 1 is a scaling factor that decreases as E increases and accounts for band flattening due to Δ𝐾 and Δ 𝑅 . Let 𝜀(𝑞) ≈ 𝑓 (E)𝐸 (𝑞) + 𝑓 −1 = 1 − 𝑏 1 Ẽ − 𝑏 2 Ẽ 2 , (A.55) where 𝑏 1 ≥ 0 and 𝑏 2 ≥ 0 are constants dependent on the exact bandstructure (i.e., how the widening of Δ𝐾 and Δ 𝑅 with E affects the bandstructure near 𝑞 𝑚 ). The roots of the equation 𝐸 (𝑞) = Δ𝐾 /2 + ℏ𝑐 ph 𝑞 are 𝑘 ± , and we may write the equation as 𝑎 2 𝑞 2 + 𝑎 1 𝑞 + 𝑎 0 = 𝑓 ℏ𝑐 𝑠 𝑞, (A.56) from which we find that ℎ𝑏 = 𝑘 + − 𝑘 − = √︁ (𝑎 1 − 𝑓 ℏ𝑐 𝑠 ) 2 − 4𝑎 2 𝑎 0 . (A.57) Solving for E ∗ through the equation ℎ 𝑏 = 0, and then series expanding the expression (𝑎 1 − 𝑓 ℏ𝑐 𝑠 ) 2 − 4𝑎 2 𝑎 0 in powers of small E − E ∗ , we find that (𝑎 1 − 𝑓 ℏ𝑐 𝑠 ) 2 − 4𝑎 2 𝑎 0 ∼ √ E ∗ − E, so ℎ 𝑏 ∼ E ∗ − E. Estimate of 𝑤 𝑏 To estimate 𝑤 𝑏 (see Figure A.7(a)), we define a circular coordinate system shown in Figure A.7(b) whose origin is the 𝐾 point and arc length 𝑤 is zero along the 𝐾 𝑅 74 slice, increasing counterclockwise. The quasienergy 𝜀(𝑤) along the circle perimeter varies with 𝑤; let us approximate 𝜀(𝑤) ≈ ℏΩ/2 − (𝑑0 + 𝑑2 𝑤 2 ), (A.58) using some fitting parameters 𝑑0 and 𝑑2 . (We assume that 𝑤 = 0 is at local maximum of 𝜖 (𝑤), so there is no linear term in Eq. A.58.) Roughly, 𝑤 𝑏 = 𝑤 + − 𝑤 − , where we find 𝑤 + and 𝑤 − by finding the roots of the equation ℏΩ/2 − Δ(𝑤) = 𝑓 ℏ𝑐 𝑠 𝑞 𝑚 . (A.59) Here, once again, we use the factor 𝑓 in Eq. A.55 to account for band flattening as E increases from zero. So, √︁ 𝑤 𝑏 = 𝑤 + − 𝑤 − = 2 ( 𝑓 ℏ𝑐 𝑠 𝑞 𝑚 + ℏΩ/2 − 𝑑0 )/𝑑2 . (A.60) Solving for E ∗ by setting 𝑤 𝑏 = 0 and series expanding 𝑓 ℏ𝑐 𝑠 𝑞 𝑚 + ℏΩ/2 − 𝑑0 in √ powers of E, we find that 𝑤 𝑏 ∼ E ∗ − E. Estimate of Ain In the limit E → E ∗ , the elliptical annulus with finite thickness collapses into a filled ellipse. Thus, in the limit E → E ∗ , we estimate that Ain = 𝜋ℎ 𝑏 (E)𝑤 𝑏 (E) ∝ max(E ∗ − E, 0). A.8 Predicting E ∗ for the Toy Model Here, we use the quasienergy dispersion of the toy model to predict E ∗ . By writing an approximate, analytic expression for 𝑣 eff (E) (see Eq. 6), we can find E ∗ using the relation 𝑣 eff (E ∗ ) = 𝑐 ph . From Eq. 6, 𝑣 eff (E) = (𝜀 𝒌 ∗ + − 𝜀 𝑲+ )/| 𝒌 ∗ − 𝑲| for some appropriately-chosen 𝒌 ∗ (dropping the superlattice valley index for notational simplicity). One can find numerically that 𝒌 ∗ does not shift significantly with Ω or E. We write an ansatz  | 𝒌 ∗ − 𝑲| ℏ𝑣 𝐹  ′ 𝜀 𝒌 ∗ + ≈ ℏ𝑣 0eff | 𝒌 ∗ − 𝑲| − 𝑓1 Ẽ + 𝑓2′ Ẽ 2 , (A.61) 𝐿𝑀 Ω/(2𝑣 0eff ) where 𝑓1′ and 𝑓2′ are fitting constants dependent on the quasienergy bandstructure. Here, ℏ𝑣 𝐹 /𝐿 𝑀 is the order of magnitude energy scale of the resonance ring gap Δ 𝑅 . The dependence of 𝜀 𝒌 ∗ + on E arises predominantly from Δ 𝑅 . The dependence is stronger when 𝒌 ∗ is close to the resonance ring, and we encode this behavior in the ratio | 𝒌 ∗ − 𝑲|/Ω/(2𝑣 0eff ), where Ω/(2𝑣 0eff ) is the 𝒌-space distance between the 𝐾 75 (a) (b) Figure A.8: Comparing numerical evaluation of E ∗ (points) to an analytic fit to Eq. A.65. We use the same fitting parameters 𝑓2 = 0.778, 𝑓1 = 0, and 𝛿(𝑁) = 0.006 for both panels. point and the resonance ring. Separately, we know that 𝜀 𝑲+ = Δ𝐾 /2. We use Eq. 6 to infer  2ℏ𝑣 𝐹 𝑣 0eff  ′ Δ𝐾 ′ 2 . (A.62) − 𝑓 Ẽ + 𝑓 Ẽ 𝑣 0eff (E) = 𝑣 0eff − 1 2 2ℏ| 𝒌 ∗ − 𝑲| 𝐿𝑀Ω We know that 𝑣 0eff ∝ 𝑣 𝐹 . We also assume that | 𝒌 ∗ − 𝑲| does not change significantly with E, so it is independent of the drive and only dependent on the superlattice scale: | 𝒌 ∗ − 𝑲| ∝ 𝐿 −1 𝑀 . Thus, we can absorb some unknown coefficients into new coefficients 𝑓1′′ and 𝑓2′′ to obtain 𝑣 0eff (E) = 𝑣 0eff −  ℏ𝑣 2𝐹  ′′ 𝑓1 Ẽ + 𝑓2′′ Ẽ 2 . 𝐿𝑀Ω Upon solving for Ẽ ∗ from 𝑐 𝑠 = 𝑣 eff (E ∗ ), we find that √︄ √︃  𝐿𝑀Ω ∗ 0 2 Ẽ ≈ 1 − 𝑐 ph /𝑣 eff + 𝑓1 − 𝑓1 , 3 𝑓2 𝑣 𝐹 (A.63) (A.64) where 𝑓1 and 𝑓2 are new, rescaled fitting constants. Using the relation Ẽ = √ 𝑒𝐿 𝑀 E/( 3ℏΩ), we find √︃  ℏΩ3/2 ∗ 0 2 E ≈ 1 − 𝑐 ph /𝑣 eff + 𝑓1 − 𝑓1 , (A.65) 1/2 𝑓2 𝑒𝐿 1/2 𝑣 𝑀 𝐹 As 𝑐 ph → 𝑣 0eff , E ∗ ∝ (1 − 𝑐 ph /𝑣 0eff ) 𝛾 where 𝛾 = 1 (1/2) if 𝑓1 ≠ 0 (= 0). See Figure A.8 for a fit for two different frequencies Ω. 76 (a) (b) 0.99 0.96 Figure A.9: Comparing the dependence of 𝜎𝑥𝑦 on E for (a) the frequency considered in the main text and (b) a lower frequency where Floquet-Umklapp processes are stronger. Note that the frequency in panel (b) is inaccessible without generating two-photon resonances in the continuum model due to the peaked shape of the 𝜈 = ±1 bands near the Γ point. Finite grid size effects on an 𝑁 × 𝑁 Monkhorst-Pack grid (see Section A.15) generate a small numerical error 𝛿(𝑁) that enters A.65 as  3/2 √︃ ℏ𝐿 1/2 𝑀 Ω 0 2 1 − 𝑐 ph /𝑣 eff + 𝛿(𝑁) + 𝑓1 − 𝑓1 . E ≈ 𝑓2 𝑒𝐿 𝑀 𝑣 1/2 𝐹 ∗ (A.66) To see this, let us consider the details of the finite-sized grid. We impose energy conservation through a broadened Dirac Delta function (see Section A.15), which we model as a Gaussian function in energy with a tiny width √ √ 𝑊 . 2𝜎 ≈ 0.1 · 2 · 2𝑁/3 (A.67) (We motivate the choice of the prefactor of 0.1 in Section A.15.) Since we avoid the high symmetry 𝐾 point in our grids, the 𝒌-point with largest Berry curvature is, in fact, a point 𝒌 near point shifted away from 𝐾 by a small distance in momentum space of 0 1 Ω/(2ℏ𝑣 eff ) Ω |𝛿𝒌 | = | 𝒌 near − 𝑲| ≈ = 0 . (A.68) 2 2𝑁/3 4𝑣 eff (2𝑁/3) This point is shifted in quasienergy by ℏ𝑣 𝐹 |𝛿𝒌 | relative to the actual 𝐾 point. We √ can account for both of these effects by shifting 𝜀 𝑲+ → 𝜀 𝑲+ + 𝛿𝜀, with 𝛿𝜀 = 2𝜎 + ℏ𝑣 𝐹 |𝛿𝒌 | and solve 𝑣 eff (E ∗ ) = 𝑐 ph to find Eq. A.66 with 𝛿(𝑁) = 𝛿𝜀/(ℏ𝑣 0eff | 𝒌 ∗ − 𝑲|). 77 Figure A.10: Comparison of the fitted Δ 𝑅 and predicted Δ𝐾 in Equations A.72 and A.76 (solid lines) to those obtained from numerics (points), using ℏΩ = 5 meV in the toy model. Here, we fit Δ 𝑅 with factors of 𝑓1𝑅 = 0.04 and 𝑓2𝑅 = 0.0184 (see Eq. A.72). A.9 Different Frequencies Reducing Ω below the value considered above will increase the ratio (𝑣 𝐹 𝑒E/Ω2 ) 2 and in turn strengthen Floquet Umklapp processes, modifying the shape of the 𝜎𝑥𝑦 curve. We demonstrate this in Figure A.9(b) for an angular frequency Ω = 4.135 meV/ℏ. However, such a low-frequency regime is inaccessible in the continuum model (without generating two-photon resonances) due to the peaked shape of the continuum model 𝜈 = ±1 band near the Γ point, so we do not consider this lower (doubly-resonant) frequency regime in the main text. A.10 Gap Sizes In this section, we estimate the size of the Floquet-induced gaps Δ𝐾 and Δ 𝑅 . By the rotating wave approximation, the Floquet-induced gap at the resonance ring, Δ 𝑅 , is roughly proportional to the drive energy [239]. For a resonant drive that couples electronic states near the Dirac points, the drive energy is roughly 𝑣 𝐹 𝑒 𝐴/ℏ, (A.69) as predicted by minimal coupling 𝒒 → 𝒒 + 𝑒 𝑨(𝑡)/ℏ in the Dirac cone Hamiltonian 𝐻𝐾 (𝒒) = ℏ𝑣 𝐹 𝒒 · (𝜉𝜎𝑥 , 𝜎𝑦 ) (A.70) √ with 𝑣 𝐹 = 𝑊 𝐿 𝑀 /(2 3ℏ). (We always use perturbative drives that generally fall in the range of Ẽ < 1.) We expect that Δ𝑅 ≈ ℏ𝑣 𝐹 Ẽ. 𝐿𝑀 (A.71) 78 Such an approximation works well for low-frequency resonant drives that couple states near the Dirac points. However, resonant drives with higher frequencies, like those used in the main text, couple states closer to the Γ-points of the TBG energy dispersion where the bands are nonlinear in 𝑞. In such a case, higher order (e.g., O(Ẽ 2 )) contributions (from 𝑂 (𝑞 2 ) contributions of the bandstructure) to Δ 𝑅 become dominant. In the present example, the energy of the tight binding model for graphene is quadratic in momentum near the Γ point, so we write an ansatz Δ𝑅 ≈ ℏ𝑣 𝐹 𝑅 ( 𝑓 Ẽ + 𝑓2𝑅 Ẽ 2 ), 𝐿𝑀 1 (A.72) and fit 𝑓1𝑅 and 𝑓2𝑅 to match Δ 𝑅 obtained by numerically diagonalizing the Floquet Hamiltonian, as shown in Figure A.10. We can estimate the Floquet-induced 𝐾-point gap, Δ𝐾 , by considering the timedependent Dirac Hamiltonian 𝐻𝐾 (𝒒, 𝑡) = ℏ𝑣 𝐹 (𝜉𝑞 𝑥 𝜎 𝑥 + 𝑞 𝑦 𝜎 𝑦 ) + 𝑣 𝐹 𝑒 𝐴[𝜉 cos(Ω𝑡)𝜎 𝑥 − sin(Ω𝑡)𝜎 𝑦 ]. (A.73) and performing a Van Vleck expansion [140, 236, 239] to obtain an effective Floquet Hamiltonian [𝐻𝐾(−1) , 𝐻𝐾(1) ] 𝑒 2 𝑣 2𝐹 𝐴2 𝑧 (0) = 𝐻𝐾 + 𝜉 𝜎 𝐻𝐾,eff (𝒒) = 𝐻𝐾 + ℏΩ ℏΩ with 1 𝐻𝐾(𝑛) (𝒒) = 2𝜋/Ω ∫ 2𝜋/Ω 𝐻𝐾 (𝒒, 𝑡)𝑒 −𝑖𝑛Ω𝑡 𝑑𝑡. (A.74) (A.75) 0 From Eq. A.74, we can extract 2𝑒 2 𝑣 2𝐹 2 6ℏ𝑣 2𝐹 2 Δ𝐾 = 𝐴 = 2 Ẽ . ℏΩ 𝐿𝑀Ω A.11 (A.76) Floquet Boltzmann Equation Here, we present the full expression for the Floquet-Boltzmann equation [258], el-ph el-el [{𝐹 (𝑡)}]. The electron-phonon collision inte𝜕𝑡 𝐹𝒌𝛼 (𝑡) = 𝐼 𝒌𝛼 [{𝐹𝒌𝛼 (𝑡)}] + 𝐼 𝒌𝛼 𝒌𝛼 gral is 79 2𝜋 1 ∑︁ ∑︁ ∑︁ ∑︁ 𝒌 ′ 𝛼′ |G𝒌𝛼 (𝑛, 𝑗)| 2 ℏ 𝑁 ′ 𝑛 𝛼′ 𝑗 el-ph 𝐼 𝒌𝛼 [{𝐹𝒌𝛼 }] = 𝒌 ∈BZ × h 𝐹𝒌 ′ 𝛼′ (1 − 𝐹𝒌𝛼 )N (ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌)) − 𝐹𝒌𝛼 (1 − 𝐹𝒌 ′ 𝛼′ ) [1 + N (ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌))] × 𝛿(𝜀 𝒌 ′ 𝛼′ − 𝜀 𝒌𝛼 + ℏ𝜔 𝑗 (𝒒) + 𝑛ℏΩ) + 𝐹𝒌 ′ 𝛼′ (1 − 𝐹𝒌𝛼 ) [1 + N (ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌))] − 𝐹𝒌𝛼 (1 − 𝐹𝒌 ′ 𝛼′ )N (ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌)) i ′ ′ × 𝛿(𝜀 𝒌 𝛼 − 𝜀 𝒌𝛼 − ℏ𝜔 𝑗 (𝒒) + 𝑛ℏΩ)  (A.77) ′ ′ 𝒌𝛼 G𝒌𝛼 (𝑛, 𝑗) = √ √︃ ∑︁ ∑︁ 1 𝐷 𝜉𝜈 ′ 𝜈 ′ ′ ′ ℏ𝜔 ( 𝒌 − 𝒌) ⟨𝜙𝑛+𝑚 ⟨𝜈𝒌 |𝜙𝑚𝒌𝛼 ⟩ √︁ ′ ′ |𝜈 𝒌 ⟩W 𝑗 𝒌𝛼 𝒌,𝒌+𝑮 𝑗 𝐴Moiré 2𝜌𝑐 ph 𝑚 𝜈,𝜈 ′ (A.78) where 𝜌 = 7.61 × 10−7 kg/m2 is the 2D density of the graphene layers, 𝐷 is the deformation potential, and the acoustic phonon mode 𝑗 has frequency 𝜔 𝑗 (𝒒) = ℏ𝑐 ph |𝒒 + 𝑮 𝑗 | with {𝑮 𝑗 } being the set of all possible reciprocal lattice vectors. The function N (𝜀) = (𝑒 −𝜀/𝑘 𝐵𝑇ph − 1) −1 is the Bose-Einstein occupation of the phonon bath at temperature 𝑇ph . The electron-electron collision integral is 4𝜋 el-el 𝐼 𝒌𝛼 [{𝐹𝒌𝛼 }] = ℏ 1 ∑︁ ∑︁ ∑︁ ∑︁ ∑︁ ( 𝒌 3 ,𝛼3 ),( 𝒌 1 +𝒌 2 −𝒌 3 ,𝛼4 ) (𝑛, 𝑮)| 2 × |V( 𝒌,𝛼),( 𝒌 ,𝛼 ) 2 2 2 𝑁 𝒌 ∈BZ 𝒌 ∈BZ 𝛼2 ,𝛼3 ,𝛼4 𝑛 𝑮 2 3 × 𝛿(𝜀 𝒌𝛼 + 𝜀 𝒌 2 𝛼2 − 𝜀 𝒌 3 𝛼3 − 𝜀 𝒌+𝒌 2 −𝒌 3 ,𝛼4 + 𝑛ℏΩ)×   × (1 − 𝐹𝒌𝛼 )(1 − 𝐹𝒌 2 𝛼2 )𝐹𝒌 3 𝛼3 𝐹𝒌 1 +𝒌 2 −𝒌 3 ,𝛼4 − 𝐹𝒌𝛼 𝐹𝒌 2 𝛼2 (1 − 𝐹𝒌 3 𝛼3 )(1 − 𝐹𝒌 1 +𝒌 2 −𝒌 3 ,𝛼4 ) (A.79) 𝒌 3 ,𝛼3 ),( 𝒌 1 +𝒌 2 −𝒌 3 ,𝛼4 ) V((𝒌,𝛼),( (𝑛) = 𝒌 2 ,𝛼2 ) ∑︁ ∑︁ ∑︁ 𝜉𝜈 𝜈 𝜉𝜈 𝜈 1 4 2 3 𝑉𝒌 2 −𝒌 3 +𝑮 W𝒌 1 ,𝒒+𝑮 W𝒌 2 ,−𝒒−𝑮 × 𝜈1 ,𝜈2 𝜈3 ,𝜈4 𝑛2 ,𝑛3 ,𝑛4 2 +𝑛3 +𝑛4 4 × ⟨𝜙𝑛−𝑛 |𝜈1 𝒌⟩⟨𝜙𝑛𝒌22 𝛼2 |𝜈2 𝒌 2 ⟩⟨𝜈3 𝒌 3 |𝜙𝑛𝒌33 𝛼3 ⟩⟨𝜈4 𝒌 4 |𝜙𝑛𝒌+𝒌 ⟩. 𝒌𝛼 2 −𝒌 3 ,𝛼4 (A.80) We solve for 𝜕𝑡 𝐹𝒌𝛼 = 0 using the Newton-Raphson algorithm. To ensure charge Í neutrality, we add a Lagrange multiplier term 𝜆( 𝒌𝛼 𝐹𝒌𝛼 − 𝑁) to the FloquetBoltzmann equation, choosing some large constant 𝜆. 80 (b) (a) (c) continuum toy continuum toy continuum toy (e) (d) (f) continuum toy continuum toy continuum toy 𝜉𝜈 ′ 𝜈 Figure A.11: Comparing the average toy and continuum model form factors W𝒌,𝒒+𝑮 , with |𝒒 + 𝑮 | measured in units of 108 m−1 . (a-c) The intraband (𝜈 = 𝜈′) form factors at the 𝐾, 𝑀, and Γ points. (d-f) The interband (𝜈 ≠ 𝜈′) form factors at the 𝐾, 𝑀, and Γ points. A.12 Form Factor Details 𝜉𝜈 ′ 𝜈 Here, we discuss the details of the form factors W𝒌,𝒒+𝑮 used in the Boltzmann equation. In the continuum model, we calculate the form factor directly from the wavefunctions: ∑︁ 𝜉𝜈 ′ 𝜈 ′ 𝑋 ′ W𝒌,𝒒+𝑮 = 𝐶𝜈𝑋∗ (A.81) ′ 𝒌+𝒒 (𝑮 − 𝑮)𝐶𝜈𝒌 (𝑮 ). 𝑋,𝑮 ′ 2 2 For the toy model, we include, by hand, a suppression factor 𝑒 −𝑙 𝑤 |𝒒+𝑮 | /4 , which accounts for the moiré periodicity that the toy model is unable to capture [119]: 𝜉𝜈 ′ 𝜈 2 2 W𝒌,𝒒+𝑮 = ⟨𝜉𝜈′ 𝒌 + 𝒒|𝜉𝜈𝒌⟩𝑒 −𝑙 𝑤 |𝒒+𝑮 | /4 . (A.82) √ We choose 𝑙 𝑤 = 𝐿 𝑀 /(1.5 3) so that the form factor dependence on |𝒒 + 𝑮 | captures that of the continuum model. To check the agreement between the toy and continuum model form factors, we calculate the form factors on a 𝒌-grid, with a set of reciprocal lattice vectors {𝑮}, 𝜉𝜈 ′ 𝜈 and plot the average value of W𝒌,𝒒+𝑮 as a function of |𝒒 + 𝑮 |, for 𝒌 at the 𝐾, Γ, and 𝑀 points of the mini Brillouin zone. The results, shown in Figure A.11, show that the toy model form factor captures the general features of that of the continuum model. We can show analytically that the toy and continuum model form factors agree when 𝒌 and 𝒌 + 𝒒 lie within the same 𝐾 or 𝐾 ′ valley in the mini Brillouin zone. These 81 low-momentum transfer processes are crucial to the tunable 𝜎𝑥𝑦 presented in the text. Let us write 𝒌 ≡ 𝑲 + 𝒑 (and 𝒌 + 𝒒 = 𝑲 + 𝒑 + 𝒒), where |𝒒| and | 𝒑| are small enough such that the energy dispersion still resembles a Dirac cone at momenta 𝒌 and 𝒌 + 𝒒. By utilizing the eigenfunctions of the Dirac cone Hamiltonian, one can show that the form factors are   1 𝜉𝜈 ′ 𝜈 ′ 𝒑 · ( 𝒑 + 𝒒) W𝑲+ 𝒑,𝒒 = 1 + 𝜈𝜈 . (A.83) 2 | 𝒑|| 𝒑 + 𝒒| The formula holds true regardless of the chirality of the Dirac nodes. Note that form factors representing direct (non-FU) scattering transitions between different 𝐾 and 𝐾 ′ valleys in the mini Brillouin zone are not relevant since such scattering transitions are kinematically prohibited due to the slow-electron regime. A.13 Validity of the Diagonal Density Matrix Approximation In general, one needs to keep track of all coherences between the Floquet states, (𝜉)† (𝜉) ⟨ 𝑓ˆ𝒌𝛼 (𝑡) 𝑓ˆ𝒌 ′ 𝛼′ (𝑡)⟩, to fully-characterize the steady-state of a Floquet system. Translation symmetry suppresses the coherences for 𝒌 ≠ 𝒌 ′. The 𝛼 ≠ 𝛼′ interband ph coherences are suppressed for 𝜏𝒌tot ≫ ℏ/Δ𝜀 𝒌 , where 1/𝜏𝒌tot = 1/𝜏𝒌el + 1/𝜏𝒌 , 1/𝜏𝒌el ph and 1/𝜏𝒌 are the interband electron-electron and electron-phonon scattering rates, respectively, and Δ𝜀 𝒌 = min𝑛∈Z |𝜀 𝒌+ − 𝜀 𝒌− + 𝑛ℏΩ|. In this section, we will explain how we numerically estimate the scattering rates. Formal Definition of Scattering Times Following Ref. [257], we define the interband scattering rates (fixing the initial Floquet band 𝛼) as 1 = ph 𝜏𝒌 h 2𝜋 1 ∑︁ ∑︁ ∑︁ ∑︁ 𝒌 ′ 𝛼′ |G𝒌𝛼 (𝑛, 𝑗)| 2 (1 − 𝐹𝒌 ′ 𝛼′ ) [1 + N (ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌))]× ℏ 𝑁 ′ 𝑛 𝛼′ ≠𝛼 𝑗 𝒌 ∈BZ × 𝛿(𝜀 𝒌 ′ 𝛼′ − 𝜀 𝒌𝛼 + ℏ𝜔 𝑗 (𝒒) + 𝑛ℏΩ) i + (1 − 𝐹𝒌 ′ 𝛼′ )N (ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌))𝛿(𝜀 𝒌 ′ 𝛼′ − 𝜀 𝒌𝛼 − ℏ𝜔 𝑗 (𝒒) + 𝑛ℏΩ) , (A.84) ∑︁ ∑︁ ∑︁ 1 4𝜋 1 ∑︁ ∑︁ 𝒌 3 ,𝛼3 ),( 𝒌 1 +𝒌 2 −𝒌 3 ,𝛼4 ) = |V((𝒌,𝛼),( (𝑛, 𝑮)| 2 × 𝒌 ,𝛼 ) 2 el 2 2 ℏ 𝑁 𝒌 ∈BZ 𝒌 ∈BZ 𝛼2 ,𝛼3 ,𝛼4 𝑛 𝑮 𝜏𝒌 2 3 𝛼3 or 𝛼4 ≠𝛼 × 𝛿(𝜀 𝒌𝛼 + 𝜀 𝒌 2 𝛼2 − 𝜀 𝒌 3 𝛼3 − 𝜀 𝒌+𝒌 2 −𝒌 3 ,𝛼4 + 𝑛ℏΩ)𝐹𝒌 2 𝛼2 (1 − 𝐹𝒌 3 𝛼3 )(1 − 𝐹𝒌 1 +𝒌 2 −𝒌 3 ,𝛼4 ). (A.85) 82 (a) (b) 0 0.5 0 1 0.5 1 Figure A.12: The color represents the ratio ℏ/(𝜏𝒌tot Δ𝜀 𝒌 ) at different points in momentum space (a) for the upper Floquet band and (b) for the lower Floquet band for the case 𝜒 ≈ 2.8 and 𝜁 ≈ 0.5. Numerical Calculation of Scattering Rates In Figure A.12, we show the ratio ℏ/(𝜏𝒌tot Δ𝜀 𝒌 ) in the regimes E/E0 = 0.966 < E ∗ /E0 ph and E/E0 = 2.898 > E ∗ /E0 for the case 𝜒 = 𝜏𝐾el /𝜏𝐾 ≈ 2.8 and 𝜁 = ℏ/(2𝜏𝐾tot Δ𝐾 ) ≈ 0.5, where 𝜁 is the maximum value across the range of E considered in Figure 1(b) in the main text, and 𝜒 is evaluated at the drive amplitude at which 𝜁 is fixed. One sees that ℏ/(𝜏𝒌tot Δ𝜀 𝒌 ) ≪ 1 for most of the Brillouin zone and ℏ/(𝜏𝒌tot Δ𝜀 𝒌 ) < 1 where the interband gaps are the smallest; by analysis of the Floquet-Redfield equation in Refs. [257, 259], the diagonal density matrix was shown to be a good approximation in this regime. Note that the definitions in Eqs. C.24 and C.25 calculate the electron (rather than hole) scattering times, and hence Pauli blocking results in different scattering times for the upper and lower Floquet bands; the scattering rates quoted in the main text take the maximum rate. Separately, we also note that the toy model underestimates the resonance gap (see Figure A.1) relative to the continuum model, and therefore overestimates ℏ/(𝜏𝒌tot Δ𝜀 𝒌 ) around the resonance ring. A.14 Dielectric Function of Strontium Titanate The dielectric function for SrTiO3 is 𝜖 (Ω) = 𝜖∞ 3 𝜔 2 − Ω2 Ö 𝐿𝑗 𝑗=1 𝜔𝑇2 𝑗 − Ω2 (A.86) at angular frequency Ω, with the experimentally-determined longitudinal and transverse optical phonon frequencies, 𝜔 𝐿 𝑗 and 𝜔𝑇 𝑗 , respectively, given in Ref. [69, 243, 295]. One finds that |𝜖 (5 meV)| = 1682. 83 (b) (a) Figure A.13: Convergence of anomalous conductivity with grid size for (a) 𝑁 (mod 3) = 1 and (b) 𝑁 (mod 3) = 2. Due to the positioning of grid points near the 𝐾 point, the results at low grid resolutions show significant disagreement. A.15 Monkhorst-Pack Grid, Numerical Integration, and Convergence In this section, we describe the methods we use to discretize the momentum Brillouin zone. We perform the Boltzmann equation integrals, introduced in Equations C.20 and C.22, over an 𝑁 ×𝑁 Monkhorst-Pack (MP) set of grid points [193], with 𝒌-points 𝒌 𝑚,𝑛 = 𝑚𝑮 1 + 𝑛𝑮 2 , 𝑁 (A.87) odd 𝑁, and 𝑚, 𝑛 = 0, . . . , 𝑁 − 1. Specifically, we avoid values of 𝑁 (mod 3) = 0 that generate a 𝒌-point exactly at the high-symmetry point of 𝐾, because such grids converge poorly when the drive strength is weak and Floquet-induced gap Δ𝐾 is small. Energy and Momentum Conservation Here, we discuss in detail how we impose momentum and energy conservation on this MP grid. The space of MP 𝒌 vectors are closed under addition and subtraction (modulo a reciprocal lattice vector), so conservation of momentum (e.g., 𝒌 + 𝒌 2 − 𝒌 3 in Eq. C.22), is simple to implement. We impose energy conservation via a smeared Dirac Delta function   2 2   1.04766𝑒 −𝜀 /2𝜎 /(2.5066283𝜎),  𝛿(𝜀) =    0,  where we have chosen numerical factors so that ∫ ∞ 𝛿(𝜀)𝑑𝜀 = 1. −∞ if |𝜀| < 2𝜎, (A.88) otherwise, (A.89) 84 E ∗ [105 V/m] 10 5 0 1.05 1.10 θ [◦ ] [km/s] 12 15 1.15 18 21 Figure A.14: The requirement that the laser drive strength E is perturbative, i.e. a fraction of electron bandwidth 𝑒E 𝐿 𝑀 < 𝑊, narrows the range of E values that can be used. As a result, the range of 𝑐 ph whose E ∗ is visible is limited as well - we postulate that they are pushed to higher drive strengths E. The smearing parameter 𝜎 is one-tenth of the maximum quasienergy spacing between nearest-neighbor MP 𝒌-points (𝜉) (𝜉) 𝜎 = 0.1 max |𝜀 𝒌𝛼 − 𝜀 𝒌 ′ 𝛼 |, ′ ⟨𝒌,𝒌 ⟩,𝛼 (A.90) where ⟨𝒌, 𝒌 ′⟩ restricts 𝒌 ′ to be a nearest-neighbor of 𝒌, and we have tuned the prefactor of 0.1 so that upon calculating the steady-state without Floquet-Umklapp (𝜉) processes, we obtain a Fermi-Dirac distribution, 𝐹𝒌𝛼 = (𝑒 𝜀𝒌 𝛼 /𝑘 𝐵𝑇ph + 1) −1 with temperature 𝑇ph of the phonon bath [90]. Convergence of Conductivities In Figure A.13, we show the convergence of the Hall conductivity 𝜎𝑥𝑦 with grid size, using ℏΩ = 5 meV. In the main text, we use a 163 × 163 MP grid for non-interacting calculations, and a 73 × 73 grid for interacting calculations. A.16 Berry Curvature Calculations We follow the Berry curvature calculation presented in [87], defining U(1) link variables ⟨𝛼( 𝒌, 𝑡)|𝛼( 𝒌 + 𝝁ˆ , 𝑡)⟩ (A.91) 𝑈 𝜇 ( 𝒌, 𝑡) = |⟨𝛼( 𝒌, 𝑡)|𝛼( 𝒌 + 𝝁ˆ , 𝑡)⟩| 85 where 𝜇 = 𝑥, 𝑦, 𝝁ˆ = 𝑮 𝜇 /𝑁, and |𝛼( 𝒌, 𝑡)⟩ are the Bloch vectors (i.e., |𝜓 𝒌𝛼 (𝑡)⟩ = 𝑒 −𝑖𝒌·𝒓 |𝛼( 𝒌, 𝑡)⟩). The Berry curvature is " # 𝑈𝑥 ( 𝒌, 𝑡)𝑈𝑦 ( 𝒌 + 𝒙ˆ , 𝑡) (2𝜋) 2 B 𝒌𝛼 (𝑡) = 2 arg (A.92) 𝑈𝑥 ( 𝒌 + 𝒚ˆ , 𝑡)𝑈𝑦 ( 𝒌, 𝑡) 𝑁 𝐴𝑀 and we use the time-averaged Berry curvature 1 B 𝒌𝛼 ≡ 2𝜋/Ω ∫ 2𝜋/Ω B 𝒌𝛼 (𝑡)𝑑𝑡 (A.93) 0 in transport calculations. A.17 The Drive Amplitude Perturbative Regime at Different Twist Angles We have treated the laser drive as a perturbation to the undriven TBG Hamiltonian, which restricts the range of field strengths E we can use to a weak perturbative regime. This also narrows the range of phonon speeds 𝑐 ph that will generate a critical field strength E ∗ in the perturbative regime, hence the narrow range of 𝑐 ph we have considered in, e.g., Figure 1(c). For various twist angles, we estimate the range of drive strengths E that are perturbative in the unshaded region of Figure A.14 and overlap in solid lines the predicted value of E ∗ for different speeds of sound. The shaded, non-perturbative regime corresponds to drive energy scales 𝑣 𝐹 𝑒E/Ω greater than a fraction, e.g., 0.3, of the bandwidth 𝑊. Here, we follow the analysis in [25] to estimate the undriven Fermi velocity √︂  2 (1 − 3𝛼2 )/(1 + 3𝛼2 (1 + 𝜂2 )) × 𝑣 ml + 𝑣 2min , (A.94) 𝑣 𝐹 (𝜃) = 𝐹 where 𝑣 min = 104 m/s is a manually set minimum Fermi velocity of the undriven flat bands, and we use the same parameters as in Section A.2. We also adjust Ω such that Ω/𝑣 𝐹 (𝜃) is constant and equal to those considered in Figs. 1-4. 86 Chapter 6 SIGNATURES OF FLOQUET ELECTRONIC STEADY STATES IN GRAPHENE UNDER CONTINUOUS-WAVE MID-INFRARED IRRADIATION Light-induced phenomena in materials can exhibit exotic behavior that extends beyond equilibrium properties, offering new avenues for understanding and controlling electronic phases. So far, non-equilibrium phenomena in solids have been predominantly explored using femtosecond laser pulses, which generate transient, ultra-fast dynamics. Here, we investigate the steady non-equilibrium regime in graphene induced by a continuous-wave (CW) mid-infrared laser. Our transport measurements reveal signatures of a long-lived Floquet phase, where a non-equilibrium electronic population is stabilized by the interplay between coherent photoexcitation and incoherent phonon cooling. The observation of non-equilibrium steady states using CW lasers stimulates further investigations of low-temperature Floquet phenomena towards Floquet engineering of steady-state phases of matter. 6.1 Introduction Since the inception of quantum theory, light-matter interaction has been a significant source of fascinating discoveries and innovative technologies. In the last few years, Floquet engineering, the use of light to control the properties of a material, emerged as a focus of intense research [59, 124, 140, 167, 215, 216, 225, 239, 256]. This upsurge has been mainly driven by theoretical efforts [17, 63, 84, 153, 216, 239, 242, 253], with a few key experiments in the condensed matter realm, including the detection of Floquet-Bloch states through time- and angle-resolved photoemission spectroscopy [174, 187, 301, 329], second harmonic generation [260], as well as the observation of light-induced shifts of exciton resonances in WS2 [143, 263] and light-induced Hall effect in graphene [185]. These experiments predominantly explored the transient phenomena induced by ultrafast pulsed lasers, uncovering some of the physics predicted by the Floquet theory [120]. Other studies explored the continuous-wave (CW) regime by inducing discrete Andreev-Floquet bound states in Josephson junctions under CW microwave irradiation [108, 220]. However, so far, solid-state experiments in the CW regime have not accessed the Floquet physics of delocalized Bloch states due to challenges related to population transfer effects 87 Figure 6.1: (a) Device layout with two contacts for longitudinal bias, 𝑉Bias , and two contacts for measurements of the transverse voltage, 𝑉𝑦 . The device has a square geometry with a 5-𝜇m side and an indium tin oxide (ITO) top gate with an Al2O3 dielectric sublayer, for applying a gate voltage 𝑉Gate . (b) Drive-modified Floquet bands of the graphene Dirac cone, which exhibits a Floquet gaps of size at the resonance energies 𝜀 = ±ℏΩ/2, where ℏ is the reduced Planck constant and Ω is the angular frequency of the laser. (c) Density of states of the Floquet bands calculated numerically for a circularly and linearly polarized laser of power density 𝑃 = 3 mW/𝜇m2 . (d) Photo-induced change of source-drain current (Δ𝐼) as a function of Fermi energy 𝐸 𝐹 for circularly and linearly polarized laser irradiation, as measured at a constant source-drain voltage of 6 mV. Δ𝐼 is measured with a lockin amplifier using a chopper modulation as a reference. The circles represent data points and the solid lines are obtained by adjacent point averaging. The dotted lines mark the 𝐸 𝐹 values corresponding to ±ℏΩ/2, with the related uncertainty indicated by the gray stripes. The uncertainty is attributed to the gate efficiency calibration and the detail is described in Appendix B. The dips in Δ𝐼 at energies ±ℏΩ/2 are much broader than the Floquet gap Δ(see c) and arise from the non-equilibrium steady-state distribution of electrons. The measurements were performed on sample A at a cryostat temperature of ∼ 3.5 K, laser photon energy of 117 meV, and laser power density of 1.1 mW/𝜇m2 . and heating. In this work, we demonstrate that under intermediate intensity CW mid-infrared (mid-IR) irradiation [see Figure 6.1(a)], the electronic population in graphene forms a non-equilibrium steady state with signatures of the underlying single-particle Floquet physics. In particular, the resulting electronic steady state relies on the Rabi-like Floquet gaps and a suppressed density of states (DOS) around electronic energies of half the photon energy in graphene’s Floquet band structure [see Figure 6.1(b) and (c)]. Importantly, we demonstrate that photoinduced transport can be used as a probe of the emergent Floquet steady state in the system. 88 Specifically, we report on signatures of these steady Floquet effects in the gate voltage dependence of the laser-induced longitudinal photoresponses of graphene samples at cryogenic temperatures ranging from 2.5 K to 50 K. 6.2 Devices and photoconductive measurements Our graphene devices were fabricated from large-area epitaxial graphene grown on SiC. The typical electron mobility of our material is about 5000 cm2 /Vs for the carrier concentration range studied in this work. (Figure B.7 shows an image of a cluster with 4 devices.) Each device was a 5 𝜇m × 5 𝜇m graphene square, with two source-drain contacts for the longitudinal electrical transport, two contacts to measure the transverse voltage, and a separate top gate electrode, as depicted in Figure 1a. The top gate comprised a 90-nm-thick dielectric Al2 O3 layer and a gate electrode of sputtered 110-nm-thick indium tin oxide (ITO). The properties of our epitaxial graphene, the details of the fabrication process, and the optical properties of the gate electrode are outlined in Appendix B. Our laser operates with a photon energy of ℏΩ = 117 meV, deliberately chosen to be below the optical phonon generation threshold in graphene (160-200 meV), in contrast to previous experiments that utilized photon energies above this threshold [185]. Importantly, this approach minimizes photo-induced phonon generation, guaranteeing acoustic phonons remain at low cryogenic temperatures and providing cooling channels for photoexcited electrons. Overall, we measured three top-gated graphene devices on two different chips: samples A and B on the first chip, and sample C on the second chip. The devices were cooled in a closed-cycle optical cryostat with a base temperature of 2.5 K and were irradiated with a 10.6-𝜇m-wavelength CW CO2 laser, providing up to 25 W of linearly polarized light. The laser polarization was controlled with a 𝜆/4 plate. The radiation was delivered using high-power-withstanding molybdenum mirrors, and it was focused on the sample to a spot with a waist of about 50 𝜇m using a ZnSe lens with a focal distance of 50.8 mm. The maximum power used to irradiate the sample was 16 W, yielding a power density P up to 2.6 mW/𝜇m2 after considering energy attenuation in the cryostat window and in the top gate of the graphene devices. Further details of the experimental setup are described in Appendix B. We performed photoconductive measurements on the Floquet system by measuring the longitudinal current using a lock-in amplifier, with the reference signal to the lock-in provided by a chopper. The source-drain bias was mV unless otherwise 89 noted. The chopper modulated the laser beam with a frequency of around 23 Hz and a duty cycle of ∼ 10%. This procedure provided a direct measurement of the photocurrent, denoted and defined as the difference in the source-drain current between the irradiated and equilibrium systems. To explore the doping dependence of the photocurrent, we also extracted the equilibrium Fermi energy, 𝐸 𝐹 , as a function of the applied gate voltage using the classical Hall effect at different gate voltages, as described in detail in Appendix B. The experimentally measured photocurrent Δ𝐼 [see Figure 6.1(d)] exhibits dips as a function of near doping close to the Floquet gaps for both circular and linear polarization of the laser beam. We note that due to the size of the sample chosen to optimize the coupling to the incident beam [146], the circular polarization might be distorted by the proximity to metallic electrodes [24] (see Appendix B for details). Nonetheless, the photocurrent dips are robust to changes in polarization and are sensitive only to the electronic photoexcitation rate set by the laser power density and frequency. Importantly, the broadness of the dips, significantly exceeding the width of the Floquet gaps in the graphene density of states [see Figure 6.1(d)], cannot be explained with single-particle Floquet physics and instead indicates strong non-equilibrium electronic population effects. 6.3 Theory of Floquet Steady States Under CW illumination, the electronic population forms a non-equilibrium steady state distribution predominantly stabilized by electron-phonon scattering processes [see Figure 6.2(a)]. To understand the photo-assisted population dynamics, we focus on the low-energy band dispersion of graphene, which exhibits two Dirac cones with Fermi velocity corresponding to the graphene 𝐾 and 𝐾 ′ valleys [see Figure 6.2(a)]. Here, 𝒌 denotes the crystal momentum, and 𝛼 enumerates the energy bands. In our theoretical analysis, we focus on circular laser polarization for concreteness. We expect the longitudinal photocurrent to be qualitatively the same for linear polarization parallel to the current. The primary effect of a circularly polarized laser drive is the opening of dynamical gaps at the resonance energies with a size [33, 140, 215]. An additional Haldane gap opens at the Dirac point1, but the gap is too small to be resolved for the power densities used in the experiment, and is therefore not shown in Figure 6.2(a) citation of the electron to a virtual state (arrow 1) through photon absorption followed by the phonon-emission process relaxing the electron to the conduction band (arrow 2). The electronic population excited by this process is subsequently spread across a small window of energies assisted by 90 Figure 6.2: (a) Key scattering processes between the graphene 𝐾 and 𝐾 ′ valleys facilitated by photons (arrow 1), surface acoustic phonons (arrows 2, 3), and graphene acoustic phonons (arrow 4) that contribute to the steady state. (b) Steady state distribution (black solid curve) for 𝑃 = 1.9 mW/𝜇m2 and equilibrium distribution (blue dashed curve) of electrons at doping 𝐸 𝐹 = 0, where the longitudinal photoconductivity is enhanced relative to equilibrium. Here, 𝜀 denotes the quasienergy and 𝐹𝒌𝛼 denotes the electronic occupation. (c) Steady state distribution (black solid curve) of electrons at doping 𝐸 𝐹 = 0.35ℏΩ and 𝑃 = 0.8 mW/𝜇m2 , where the photoconductivity is suppressed relative to an equilibrium distribution expectation (blue dashed curve). (d) Same as (c) but for a larger power density 𝑃 = 1.9 mW/𝜇m2 , where the steady state distribution exhibits additional electron density above the Floquet gap. e Steady state distribution (black solid curve) of electrons for 𝑃 = 1.9 mW/𝜇m2 at doping 𝐸 𝐹 = 0.75ℏΩ, where the photoconductivity is approximately identical to that of an equilibrium distribution (blue dashed curve), with a slightly raised effective temperature due to multi-photon heating processes. The horizontal dotted lines in (b–e) mark 𝜀 values corresponding to ±ℏΩ/2. the emission of low-energy acoustic phonons (arrow 3) and can be subsequently relaxed through additional acoustic phonon emission processes (arrow 4) back into the valence band. Ultimately, in the steady state, the excited electronic occupation in the conduction band is set by the balance between phonon-assisted photo-excitations (arrows 1-3) and phonon-assisted cooling processes (arrow 4). In Figure 6.2(a) and throughout our theoretical calculations, we include two phonon branches: a slow surface acoustic phonon branch with Rayleigh wave speed 1.3 km/s [arrows 2 and 3 in Figure 6.2(a)] and a faster graphene acoustic phonon branch with speed 11 km/s [arrow 4 in Figure 6.2(a)] [50]. The surface acoustic phonon speed used in our simulations captures the order of magnitude of the Rayleigh wave speed of the SiC [210]. To find the steady state electronic population 𝐹𝒌𝛼 , we solved numerically 91 the full Floquet-Boltzmann equation, which includes all the microscopic electronphonon scatterings between Floquet-Bloch states in the laser-driven graphene (see Appendix B for details) [74, 257, 258, 312]. The resulting steady state occupations at various electronic dopings are plotted in Figure 6.2(b)-(e), and the photocurrent is calculated from the steady state occupations using linear response theory in the weak source-drain bias regime. In our theory, we also account for charge puddles and a small, intensity-dependent lattice temperature difference between the driven and undriven systems. We estimate the conductivity in the presence of charge puddles using a phenomenological formula detailed in Appendix B. Remarkably, the steady state distributions denoted by black curves in Figure 6.2(b)(e) exhibit multiple step-like features associated with large |𝜕𝐹𝒌𝛼 /𝜕𝜀 𝒌𝛼 |, resembling effective Fermi surfaces, which significantly affect transport. The electron-like effective Fermi surfaces (𝜕𝐹𝒌𝛼 /𝜕𝜀 𝒌𝛼 < 0) contribute to a longitudinal current parallel to the direction of an applied electric field, while hole-like effective Fermi surfaces (𝜕𝐹𝒌𝛼 /𝜕𝜀 𝒌𝛼 > 0) suppress the overall photocurrent by contributing current antiparallel to the applied field. This is in strict contrast to an equilibrium distribution, where a single Fermi surface appears in the electronic distribution [see blue dashed curves in Figure 6.2(b)-(e)], and only electrons near the Fermi surface contribute to the longitudinal transport. The Fermi surfaces in the distribution predominantly arise from the scattering processes 1-4 sketched in Figure 6.2(a). Most notable is the non-equilibrium distribution for an electron-doped system 0 < 𝐸 𝐹 < ℏΩ/2 [see Figure 6.2(c) and (d)], where the equilibrium Fermi surface located at finite density of states (see blue dashed curve) is separated into three electron-like Fermi surfaces (see black solid curve) which are located near the dips of the density of states and therefore contribute very few electronic carriers. In particular, two electron-like Fermi surfaces are centered at the Floquet gaps, where electronic group velocities and density of states are reduced, and another is positioned at zero energy, where the density of states vanishes. The reduced concentration of electronic carriers gives rise to a suppressed conductivity in the driven system. This phenomenon persists for both weak and strong driving amplitudes, as shown in Figure 6.2(c) and (d), respectively. For doping above resonance 𝐸 𝐹 > ℏΩ/2, processes 1–4 are Pauli-blocked, giving rise to an equilibrium-like distribution [see Figure 6.2(e)]. 6.4 Optically-controlled photoresponse Having understood the interplay of the Floquet bands, steady state occupation, and photocurrent Δ𝐼, we now compare the theoretically calculated and experimentally 92 Figure 6.3: (a) Photo-induced change of source-drain current (Δ𝐼) as a function of 𝐸 𝐹 under irradiation at various peak power densities, as measured on sample C at a constant source-drain voltage of 6 mV and a cryostat temperature of 3.2 K. The photon energy is ℏΩ = 117 meV. The laser beam is circularly polarized. The circles represent data points and the solid lines are the fitted curve by a Gaussian-like function described in the Methods. The dotted line marks the 𝐸 𝐹 value corresponding to −ℏΩ/2, with the related uncertainty indicated by the gray stripes. (b) Theoretically predicted Δ𝐼 as calculated from the Floquet Boltzmann equation. The dotted line marks the 𝐸 𝐹 value corresponding to −ℏΩ/2. (c) Depth 𝑎 0 of the photocurrent dip and associated error, as calculated from a Gaussian-like fit (see details in the Methods section). For both the theoretical and experimental data, 𝑎 0 decreases with the power density 𝑃 for large 𝑃 due to enhanced heating processes in the Floquet steady state. (d) Experimental and theoretical FWHM of the photocurrent dip and the Floquet gap size (red dotted line), Δ, as a function of the power density. The FWHM exceeds Δ, indicating the emergence of photoexcited electrons in the nonequilibrium Floquet steady state. The detailed error analysis of FWHM is described in Appendix B. The uncertainty in the laser power density due to power fluctuation in (c) and (d) are estimated by the change in the laser power before and after each transport measurement. measured value of Δ𝐼 and discuss the tunability of the photoresponse by the laser power. In Figure 6.3(a), we focus on the experimentally measured photocurrent dip at negative 𝐸 𝐹 and plot the photocurrent for various laser power densities. In Figure 6.3(b), we show the theoretically calculated photocurrent for the same power densities. Let us discuss a few salient features captured by the theoretical model. A key feature is the broad width of the photocurrent dip, which we estimate by fitting the photocurrent to a Gaussian-like function detailed in the Methods section. The amplitude of the Gaussian 𝑎 0 quantifies the depth of the photocurrent dip, while the full width at half maximum (FWHM) quantifies its width. Notably, our theoretical prediction of the decrease in 𝑎 0 with the power density 𝑃, agrees with the experimental observation [see Figure 6.3(c)], indicating a power dependence of 93 Figure 6.4: (a) Longitudinal photocurrent as a function of at a fixed source-drain voltage of 6 mV, with laser spot in focus and defocused. For all three curves, the laser power is ∼ 3.6 W, corresponding to a peak power density of 0.6 mW/𝜇m2 at focus. Under defocused irradiation, with the laser spot slightly moved to the side, the power density drops by ∼ 3 times (gray curves). Circular polarization of 10.6 𝜇m wavelength laser radiation. The cryostat temperature stabilized at ∼ 3.4 K. b Theoretically calculated photocurrent for weak drives, exhibiting shallower dips near resonance as a function of decreasing driving power, in agreement with (a). (c) Temperature dependence of the longitudinal photocurrent as a function of 𝐸 𝐹 . The photocurrent decreases, and the dips at 𝐸 𝐹 = ±ℏΩ/2 disappear at high temperatures. The laser beam is circularly polarized. (d) Theoretically calculated photocurrent for laser power density 1.4 mW/𝜇m2 , the photocurrent dip becomes less visible at higher temperatures. The dotted lines in (a)–(d) mark the 𝐸 𝐹 value corresponding to ±ℏΩ/2, where the related uncertainty is indicated by the gray stripes in (a) and (c). the steady state distribution. The reduction of 𝑎 0 with 𝑃 arises due to the increase of the photoexcited carrier density above the Floquet gap, which scales as ∼ 𝑃1/2 , effectively increasing the photocurrent [cf. the shift of the effective Fermi surface near ℏΩ/2 in Figure 6.3(c) and (d)] [33]. The FWHM of the photocurrent dip observed in the experiment and predicted by the theory are both broader than the size of the Floquet gap, see Figure 6.3(d), indicating a Floquet-induced electronic population inversion in the steady state. We note that Δ𝐼 > 0 for 0 < 𝐸 𝐹 < ℏΩ/2 due to charge puddle and lattice heating effects (see Appendix B for more details). Finally, we note that the horizontal offset in the theoretically calculated photocurrent shown in Figure 6.3(b) relative to the experimental data in Figure 6.3(a) may arise from changes in the total electronic carrier density in the sample during laser illumination, which were not accounted for in the experimental assignment of the chemical potential. To further verify that the transport signatures observed in the experiment arise from 94 photo-induced dynamics, we explore the experimentally measured photo-response for out-of-focus lasers, which reduce the irradiation power density by roughly a factor of three. Figure 6.4(a) demonstrates that the visibility of the photocurrent dip is significantly suppressed as the power density is reduced. This behavior, reproduced in theory [see Figure 6.4(b)], reflects the suppressed probability of drive-induced photoexcitation processes under weak laser irradiation. Next, we explore the lattice temperature dependence of the electronic steady state and conductivity. Figure 6.4(c) and (d), respectively, show the measured and the theoretically predicted Δ𝐼 as a function of 𝐸 𝐹 , for several values of the lattice temperature. At higher temperatures, the distribution of the photoexcited electrons spreads over larger energy support, relaxing the sharp energy-momentum bottlenecks in the steady state distribution [see processes 1-4 in Figure 6.2(a)]. As a result, the dips in Δ𝐼 become less pronounced, virtually disappearing around 20 K. The discrepancy of the temperature dependence around charge neutrality between the experimental and theoretical plots is attributed to the finite-temperature physics of the charge puddles, which could modify the relaxation time of electrons at different temperatures under weak source-drain bias [163], which is not captured in the theory. The strong lattice temperature dependence of both the theoretically calculated and experimentally observed photocurrent, however, highlights the role of low phonon temperatures in stabilizing low-temperature electronic phenomena of Floquet effects in our system. 6.5 Discussion The photoconductivity dip discussed above has been reproduced for other samples, which we present in the Appendix B. Different samples with different charge inhomogeneities showed similar widths of the photocurrent dip. This observation is consistent with the emergence of a Floquet steady state, which is predicted to be weakly sensitive to disorder. The positions of the dips in the electron and hole regions of the gate dependence are also slightly asymmetric, similar to particle-hole asymmetric results reported in McIver et al. [185]. So far, we have not discussed the transverse photoconductivity of the system. Our transverse voltage measurements displayed a laser helicity-dependent component on Hall ∼ 10−3 𝑒 2 /ℎ, the 𝜇𝑉 scale, which amounts to a Hall conductivity of the order of 𝜎𝑥𝑦 Hall ∼ 𝑒 2 /ℎ, see Figure B.11. This range of values undershoots the prediction of 𝜎𝑥𝑦 from our theoretical model, which is mainly designed as a simple model for the longitudinal component of the photocurrent. We believe that a key reason for this 95 discrepancy is the distortion of the circular polarization by the metallic contacts, leading to regions of the sample with nearly linear polarization [24] (see Appendix B for details). In these regions, the Berry curvature can have large off-diagonal elements, reducing the photoresponse. Additionally, our theoretical calculation of the Hall voltage does not account for electron-electron interactions and charge puddles. Electron-electron scattering could suppress the Hall conductivity by raising the effective temperature of the steady-state electron-like Fermi surfaces [see Figure 6.2(b)–(e)] near the resonant Floquet gaps [258, 312]. Charge puddles could further suppress the Hall voltage by producing regions in the sample with low conductivity, which exhibit weak longitudinal source drain current and heavily suppressed local Hall voltage. More details of the transverse photoresponse measurements are provided in Appendix B. In summary, we have explored the transport properties of graphene driven by a CW mid-infrared laser. In such metallic systems, the Floquet dressing of single-particle bands by resonant periodic drives is intertwined with many-body effects, such as electron scattering and population dynamics. Notably, we demonstrate that the interplay between these processes at intermediate driving intensities and cryogenic temperatures facilitates the formation of low-entropy electronic steady states. These steady states emerge from a cascade of photo-assisted electron-phonon scattering events that effectively cool the photoexcited electrons to low entropy states. Our findings suggest that, in graphene on SiC, these processes are dominated by the emission of surface acoustic phonons at the graphene-SiC interface, along with acoustic phonons in the graphene itself. Furthermore, the formation of these steady states is crucially dependent on the underlying single-particle Floquet physics, such as the presence of Floquet gaps in the single-particle spectrum and scattering into replica Floquet bands. We interpret the observed pronounced dip in longitudinal conductivity in our samples as evidence of Floquet steady states. The characteristics of this dip—its position, depth, and width as functions of doping—align closely with predictions from Floquet many-body theory. Our results indicate a promising approach to perform Floquet engineering in metallic systems for sustained operation, potentially leading to the creation of Floquet steady-state phases, such as driveinduced symmetry broken phases [76], laser-induced flat bands [3, 131, 165], and optically-controlled topological transport [74, 312]. 96 6.6 Methods Experiment The CW source was a 10.6 𝜇m-wavelength, air-cooled Synrad J48-2 CO2 laser, providing up to 25 W of linearly polarized light, modulated with a chopper (Scitec Instruments 300CD) with 27 Hz modulation frequency. The circular polarization of the laser radiation was controlled by a custom-made zero-order 𝜆/4 plate, provided by Optogama UAB. For the delivery of the optical beam, we used molybdenum mid-IR high power mirrors. The samples were cooled down to 2.5 K with a closed-cycle cryostat Oxford Instruments Optistat AC-V14. For optical access to our samples, we used a 2-mm thick ZnSe window. The beam delivery was done using an Edmund Optics cage focusing system mounted on a custom-made mechanical attachment to the cryostat. For beam focusing, we used a 50.8-mm focal distance, 1-inch diameter, ZnSe lens. Focusing and alignment of the mid-IR beam was controlled by a system of micrometers [127]. The estimation of the beam diameter at the lens focus is based on a Gaussian beam with our experimental parameters (18-19 mm laser beam waist before lens and 50.8 mm lens focal distance), and it yields ∼35-38 𝜇m. Experimentally, we were able to achieve focusing down to a 50 𝜇m spot. This was confirmed during the fine alignment of the laser beam: the magnitude of was changing from the noise level to a maximum signal and then back to the noise level (about 10-12-times drop of signal magnitude) within 4-5 readout thimble divisions of in-plane alignment micrometers (e.g., within 40-50 𝜇m distance). The maximum applied laser power in our experiments was 16 W, corresponding to a peak power density ∼ 8 mW/𝜇m2 for ∼ 50 𝜇m-diameter beam spot. Taking into account the transmission coefficients of the cryostat window and the gate electrode, we estimate the applied laser intensity as ∼ 2.6 mW/𝜇m2 . The measurements of photoinduced currents and voltages were performed using a home-made bias box, an HP 6177 C DC current/voltage source (for generation of gate voltages), a National Instruments BNC-2110 junction box, an Ametek 5110 lock-in amplifier, a DL Instruments 1211 current preamplifier, and a custom-made differential voltage preamplifier. The data collection system was controlled using a customized LabVIEW-based program. The calibration of gate efficiencies of our samples (calibration of gate voltage in units of electron Fermi energy) was performed using a 9 Tesla Quantum Design Physical Property Measurement System at the NHMFL in Tallahassee, FL. The 97 details about gate efficiency calibration are provided in Appendix B. Device Fabrication For the device fabrication we adapted the process developed by Yang et al. [313]. to electron-beam lithography (EBL) [73] with additional lithography steps to deposit and pattern a top gate. The details of the device fabrication are provided in Appendix B. The epitaxial graphene on SiC was purchased from Graphene Waves. Fitting We estimate the full width at half maximum (FWHM) of the photocurrent dip by   fitting the photocurrent to the function Δ𝐼fit = 𝑎 0 exp −(𝑉𝑔 − 𝐵) 2 /2𝜎 2 + 𝐶𝑉𝑔 + 𝐷, where 𝑎 0 , 𝐵, 𝜎, 𝐶, and 𝐷 are fitting parameters, and 𝑉𝑔 is the gate voltage. We √ extract the full width half maximum using the relation FWHM = 𝐸 𝐹 (𝐵+ 2 ln 2𝜎) − √ √︁ 𝐸 𝐹 (𝐵 − 2 ln 2𝜎), where 𝐸 𝐹 (𝑉𝑔 ) = ℏ𝑣 𝐹 𝜋𝑘 (𝑉𝑔 − 𝑉𝑑 ). Here, 𝑘 is estimated from Hall measurements, and 𝑉𝑑 is estimated to be the gate voltage at which 𝜎𝑥𝑥 is minimized (see Appendix B). 6.7 Acknowledgements We acknowledge support from NSF (projects DMR CMP #2104755, DMR CMP #2104770, and OSI #2329006), ANID FondeCyT (Chile) through grant number 1211038, and the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (PHY-2317110). C.Y. gratefully acknowledges support from the DOE NNSA Stewardship Science Graduate Fellowship program, which is provided under cooperative Agreement No. DE-NA0003960. G.R. and I.E. are grateful to the AFOSR MURI program, under agreement number FA9550-22-1-0339, as well as the Simons Foundation. Part of this work was done at the Aspen Center for Physics, which is supported by the NSF grant PHY-1607611. F.N. gratefully acknowledges support from the Carlsberg Foundation, grant CF22-0727. C.L. was supported by start-up funds from Florida State University and the National High Magnetic Field Laboratory. The National High Magnetic Field Laboratory (NHMFL) is supported by the National Science Foundation through NSF/DMR1644779, NSF/DMR-2128556 and the State of Florida. L. E. F. F. T. acknowledges partial support from the EU Horizon 2020 research and innovation program under the Marie-Sklodowska-Curie Grant Agreement No. 873028 (HYDROTRONICS Project), and of The Abdus Salam International Centre for Theoretical Physics and the Simons Foundation. The authors thank Dr. Michael Jackson, Dr. Yanfei Yang, 98 Eli Adler, DaVonne Henry, Amjad Alqahtani, and Thy Le for helpful discussions, Taylor Terrones, Michael Chavez, and Leon Der for help with our experimental setup, and Dr. David Graf for help with experiments at NHMFL. 99 Appendix B APPENDIX B.1 Sample Irradiation with a Continous-Wave Mid-Infrared Laser Beam The optical setup, with the components described in the Materials and Methods section, is shown in Figure B.1. A pair of molybdenum mid-infrared (Mid-IR) highpower mirrors were used for the delivery of the laser beam. For the pre-alignment of the orientation of the molybdenum mirrors and for the initial positioning of ZnSe lens closer to the position in focus, we used a visible laser (𝜆 = 520 nm) collimated with the Mid-IR irradiation, with a camera to monitor the laser spot on the sample (see the inset in Figure B.1). After this initial alignment, the visible laser and the beamsplitters were removed, and the fine adjustment of the beam positioning with the Mid-IR laser was performed by using the photo-generated current in our devices as a reference. The illumination caused a change in the source-drain current due to the photo-generated hot electrons, as described in the main text. An example of current change under 10.6-𝜇m-wavelength laser illumination is Figure B.2 below. We adjust the beam position by maximizing the photocurrent response. The devices were irradiated with a 10.6-𝜇m-wavelength CW CO2 laser, providing up to 25 W of Figure B.1: Scheme of the sample illumination system. Schematic illustration of the optical setup used for delivering and focusing the mid-infrared irradiation. The beamsplitters are used only during the preliminary alignment of optical elements. The inset image shows the visible laser spot on the sample used for the preliminary alignment of the molybdenum mirrors and the ZnSe focusing lens. 100 Figure B.2: Measurement sequence in time. Experimentally measured cryostat temperature (top panel) and longitudinal current (bottom panel) with laser blocked and unblocked intervals. Note that the actual graphene lattice temperature, 𝑇 (𝑡), is expected to be further elevated relative to the cryostat temperature. When the laser is blocked, the sample cools down to 𝑇bg , giving rise to a strong bolometric effect in the longitudinal current. The laser illumination is blocked/unblocked in the optical path between the chopper and the sample (light coral stripes indicate when laser is unblocked), the average laser power density is 0.125 mW/𝜇m2 and the peak power density is 1.25 mW/𝜇m2 , ℏΩ = 117 meV (10.6 𝜇m wavelength). linearly polarized light. The laser beam polarization was controlled with a 𝜆/4 plate. Experimental samples were designed to maximize efficiency of the beam-sample coupling for the mid-IR irradiation [146]. At the same time, the half-wavelength size of the graphene devices and the attachment of the metallic electrodes to its edges result in predominantly linear polarization of the incident electromagnetic field near the edges of the graphene sample, regardless of the initial polarization of the laser source [24]. The presence of linearly polarized fields in some parts of the sample is expected to influence helicity-dependent effects. B.2 Measurement Sequence The laser beam was modulated by a chopper (the modulation frequencies were in the range of 10-50 Hz) with a duty cycle of ∼ 10%. During the on period, the system forms a non-equilibrium electronic state characterized by a photocurrent 𝐼dr that relaxes during the off period to an equilibrium state with temperature 𝑇eq and corresponding photocurrent 𝐼eq (𝑇eq ). The relaxation occurs over the phonon 101 Figure B.3: Laser intensity and temperature dependence of the longitudinal conductance. Gate voltage dependence of the source-drain current at different temperatures with laser blocked (gray) and under irradiation at the base temperature (red), with 𝑃 = 0.6 mW/𝜇m2 and with source-drain voltage fixed at 50 mV. scattering time, typically on the picosecond scale. The measured photocurrent is, therefore, given by Δ𝐼 = |𝐼dr − 𝐼eq (𝑇eq )|, which is detected directly using a lock-in amplifier, with the reference signal to the lock-in provided by a chopper. Figure B.2 shows the theoretically predicted photocurrent as a function of time, switching between 𝐼dr and 𝐼eq (𝑇eq ), within the duty cycle. Due to the illumination, the system temperature 𝑇eq is elevated above the background temperature 𝑇bg . To test the temperature change and the corresponding bolometric effect, we blocked the beam in the optical path between the chopper and the sample for extended times of a few seconds. During a timescale denoted 𝜏𝑟 , the system’s temperature relaxed back to 𝑇bg corresponding a reduction of the current to the value 𝐼eq (𝑇bg ). We note that the background itself is slightly heated by the laser, leading to gradual changes in 𝑇bg . The bottom panel of Figure B.2 shows the longitudinal (source-drain) current measured as a function of time and at constant source-drain voltage while the laser beam was blocked or unblocked. The time dependence of the current can be fit to the function 𝐼eq (𝑇 (𝑡)), where 𝑇 (𝑡) = 𝑇bg + (𝑇eq − 𝑇bg )𝑒 −𝑡/𝜏𝑟 within one unblocking-blocking interval. The relation of the photoresponse Δ𝐼 to effects of electron heating may be illustrated further in the following supplementary experiment. Figure B.3 shows the current vs. gate voltage curves of a graphene device at three different temperatures and at a fixed source-drain bias voltage without any laser irradiation (gray curves). The temperature dependence of the electrical resistance in graphene is determined by 102 Figure B.4: Mid-Infrared transmission spectrum of ITO. Example of spectrum measured from our typical 110-nm thick ITO layer in the range around 10.6 𝜇m wavelength (corresponding to the wavenumber 943 cm−1 ). different intrinsic and extrinsic sources of scattering, including phonons, defects in the crystal lattice or deformation in the graphene sheet (wrinkles or steps), as well as impurities in the substrate or on the graphene surface [26, 37, 41, 60, 266]. The presence of electron- hole puddles and quantum corrections also complicate the dependence of the electrical resistance on carrier density and temperature, especially at low temperature [101, 181, 184, 306]. For our graphene samples, in the whole range of gate voltage that we measured around the Dirac point, the current increases when the temperature increases. B.3 Transmission property of ITO top gate for Mid-IR irradiation The Mid-IR radiation actually delivered to the graphene is attenuated by the optical elements. Our ZnSe focusing lens has an anti-reflective (AR) coating, while the cryostat window has no AR coating and transmits 70% of the 10-𝜇m wavelength laser beam. The other important source of attenuation is the Indium-Tin Oxide (ITO) material of the top gate in the experimental samples. Depending on growth conditions, post-treatment, and many other factors, the optical transmission of this material may vary significantly. Measured or theoretically expected values of MidIR transmission of ITO range from less than 10% to more than 50% for material thicknesses around 100 nm, depending on the electron concentrations [48, 106]. For an accurate estimation of attenuation in our case, we fabricated and tested ITO samples using the exact same growth conditions and thickness of the material as those we used in our experimental graphene devices. The infrared transmission spectra were measured using a Nicolet Nexus 8700 Fourier-Transform spectrometer 103 Figure B.5: Longitudinal photoresponse vs. carrier density at different B fields. Longitudinal voltage photoresponse as a function of magnetic field and carrier density, under chopper-modulated irradiation with circular polarization, at a laser intensity of 20 𝜇W/𝜇m2 and photon energy ℏΩ = 117 meV. The sample is biased at a fixed current 𝐼𝑥 of 1 𝜇A, and the photoinduced change in source-drain voltage Δ𝑉𝑥𝑥 is measured with a lock-in amplifier. equipped with a KBr beamsplitter and a DTGS photodetector. A typical transmission spectrum of a 110-nm-thick ITO layer is shown in Figure B.4. Transmission measurements were performed at multiple locations on two different samples, yielding an average transmission of 45%. B.4 Cyclotron Resonance of the Top-Gated Epitaxial Graphene Samples We performed cyclotron resonance measurements on an epitaxial graphene sample on SiC at 1.6 K using an 18/20 T general purpose superconducting magnet at NHMFL in Tallahassee, FL. We used a 1 MOhm resistor connected in series with the sample to keep the source-drain current at a constant value of about 1 𝜇A throughout the measurements. The sample was illuminated by a CW CO2 laser and a quarter waveplate, as described above. Unlike the zero-field measurement, the laser was delivered through a Mid-IR fiber and focused onto the sample with a ZnSe lens. As shown in Figure B.5, the magnetoresistance reveals a resonance magnetic field value of 11.5 ± 0.2 T that corresponds to a transition between the 0th and the 1st Landau levels, or √︁ (B.1) Δ𝐸 = 𝑣 𝐹 2ℏ𝑒𝐵/𝑐 = ℏΩ where ℏΩ = 117 meV and 𝐵 = 11.5 ± 0.2 T. We can, therefore, extract the Fermi velocity in our epitaxial graphene as 𝑣 𝐹 = (0.960 ± 0.008) × 106 m/s. The cyclotron 104 Figure B.6: Gate efficiency calibration. (a) (Black) Source-drain voltage and (blue) transverse voltage from the Hall effect measurements for 𝑉𝑔 = 7 V and a constant bias (source-drain) current 𝐼sd = 1 𝜇A. The green and red lines are examples of the set of “4 points” used for extracting the actual transverse voltage. (b) (Black) Conductance at different gate voltages. (Blue) Carrier density at different 𝑉𝑔 from Hall measurements. (Red) Linear fitting of carrier density at different 𝑉𝑔 . resonance peak is determined to have an FWHM of ∼ 8 meV, which is about two times broader than that reported in exfoliated graphene samples [46, 244]. Such broadening can be attributed to the charge puddles and traps in graphene introduced during the fabrication process. We can also extract the magnetic length at the resonance field 𝑙 𝐵 ∼ 7.6 nm, which is smaller than our estimated mean free path (𝑙MFP ) for carrier density 𝑛𝑐 ∼ 2.9 × 1012 cm−2 , about 𝑙MFP ∼ 30 nm. B.5 Gate Efficiency Calibration The Hall effect measurements were performed on sample B at 4 K to determine the carrier density at different gate voltages using a Quantum Design PPMS® system. A 1 MOhm resistor was connected in series with the device to keep the current around 1 𝜇A throughout the measurements. The typical Hall voltage and source-drain voltage curves as a function of the magnetic field at a fixed gate voltage are plotted in Figure B.6(a), where the Hall voltage revealed a non-linear behavior as well as an offset. We note that these source-drain voltages are two-terminal measurements. Unlike four-terminal measurements, two-terminal measurements yield curves that depend on the geometry of the samples [2]. In addition, a slight misalignment of the Hall terminals may lead to a purely geometric (not Hall-effect-related) contribution to the Hall voltage from the bias current that is proportional to the source-drain voltage. We estimated this misalignment as ∼ 300 nm along the source-drain current direction in the presented sample. To exclude the contribution from such misalignment, we used the “four-point” approach described below to extract the linear relationship between 105 the Hall voltage and the applied magnetic field: we randomly picked four data points around the bottom of the source-drain voltage curve at different fields, but with the same source-drain voltage value, so that the misalignment would contribute equally to the Hall voltages among these four points, and we extracted the carrier density from a linear fitting of those four points. For each gate voltage, three sets of the “four points” were randomly picked. Figure B.6(a) shows two sets of four points, in red and green, respectively, for measurements at a fixed gate voltage 𝑉𝑔 = −7 V. For each set of 4 points the following linear fitting was performed: 𝑉Hall [V] = 𝑘 [VT−1 ]𝐵[T] + 𝑉Hall offset [V] (B.2) where the slope 𝑘 links to the charge carrier density 𝑛𝑐 with the following relationship: 𝐼 [A] 𝑛𝑐 [cm−2 ] = 10−4 × (B.3) 𝑒𝑘 [VT−1 ] where 𝐼 was kept around 1 𝜇A as mentioned above and 𝑒 = 1.602 × 10−19 C. Table B.1 summarizes the data obtained from the fittings and the carrier concentrations obtained at different gate voltages. The extracted values of carrier concentrations at different gate voltages are plotted in Figure B.6(b). From a linear fit of those points, we obtained the following “calibration curve” to relate the carrier density to the gate voltage: 𝑛𝑐 [cm−2 ] = −𝑘 (𝑉𝑔 − 𝑉𝑑 ) [V] (B.4) where 𝑘 = 1.63 × 1011 ± 𝛿𝑘, 𝛿𝑘 = 9 × 109 , 𝑉𝑑 = 𝑘𝑛0 = 10.7 V ± 𝛿𝑉𝑑 , and √︁ 𝛿𝑉𝑑 ≈ 𝑉𝑑 (𝛿𝑛0 /𝑛0 ) 2 + (𝛿𝑘/𝑘) 2 . Here, 𝑛0 = 1.74 × 1012 and 𝛿𝑛0 = 9.37 × 1010 , with the uncertainties estimated from the 68% confidence interval of the linear fit. The following relationship between the carrier density 𝑛𝑐 and Fermi energy 𝐸 𝐹 in graphene was then applied to determine the corresponding 𝑉𝑔 for 𝐸 𝐹 = ℏΩ/2: √ 𝐸 𝐹 = ℏ𝑣 𝐹 𝜋𝑛𝑐 , (B.5) where ℏ is the reduced Plank constant, 𝑣 𝐹 is the Fermi velocity extracted from the cyclotron resonance measurements as 0.96 × 106 m/s is our top-gated epitaxial graphene samples. For the first crossings, the Fermi energy is half the photon energy, i.e., 𝐸 𝐹 = ±58.5 meV, and the corresponding gate voltage was calculated to be about 1.6 V from the charge neutrality point, as summarized in Table B.2. B.6 Device Fabrication The top-gated graphene FETs were fabricated using epitaxial graphene on SiC purchased from Graphene Waves. To prevent sample contamination from photoresists, 106 Table B.1: Fitting results of Hall measurements at different gate voltages Slope (×10−4 )/VT−1 Set 1 Set 2 Set 3 Ave. slope (×10−4 )/VT−1 -3 V -5.290 -5.250 -5.283 -5.274 Gate voltage (V) -5 V -7 V -15 V -6.244 -9.074 7.458 -6.247 -9.024 7.521 -6.267 -9.029 7.412 -6.252 -9.042 7.464 -17 V 6.294 6.369 6.281 6.315 Table B.2: 𝑉𝑔 corresponds to 𝐸 𝐹 = ℏΩ/2 𝐸 𝐹 /meV 𝑛𝑐 /cm−2 𝑉𝑔 /V −ℏΩ/2 −58.5 −2.51 × 1011 −12.2 CNP 0 0 −10.7 +ℏΩ/2 +58.5 +2.51 × 1011 −9.1 a thin Pd/Au layer (5 nm Pd + 15 nm Au) was deposited on the graphene by electronbeam evaporation before further processing [73, 313]. The contacts were fabricated first to prevent the charging effect during the electron beam lithography (EBL) process. Contact patterns were written following a standard EBL process using a Zeiss SUPRA55-VP system on a methyl methacrylate/polymethyl methacrylate (MMA/PMMA) bilayer e-beam resist, followed by magnetron sputtering deposition of a total of 1.5 nm Ti and 400 nm Au. The graphene was then patterned into Hall bars following a process described in Figure B.7(a), where a layer of PMMA defined by EBL was used as the etch mask during dry etching (Ar plasma, 50 s.c.c.m., 150 W, Oxford Plasmalab 80), and the EBL pattern was designed in such a way that only the channel area shown in Figure B.7(b) remained unexposed. Prior to the top gate fabrication, diluted aqua regia (DAR, HNO3:HCl:H2O = 1:3:4) was used to remove the Pd/Au protection layer. The samples were annealed in vacuum afterwards to remove the residues from the DAR treatment as well as adsorbates from ambient exposure (H2 O, O2 , etc.). Since the graphene grown on SiC is intrinsically n-doped due to the formation of the buffer layer during the graphene growth, before the atomic layer deposition (ALD) growth of an Al2 O3 layer, we doped the samples with nitric acid vapors in a standard fume hood, following a procedure developed by Mhatre et al. [188]. to bring the Fermi level closer to the charge neutrality point. As shown in Figure B.7(c), a 90nm Al2 O3 dielectric layer was grown on top of graphene by atomic layer deposition (Beneq TFS 200). The Al2 O3 masking of the contact pads was removed by buffered hydrofluoric acid using a layer of Shipley 1813 patterned by photolithography as 107 Figure B.7: Scheme of key steps in top-gated graphene device fabrication. (a) Patterning of graphene Hall bars with a thin Pd/Au protection layer. (b) Optical image of the graphene devices prior to removing the metal protection layer and the top gate fabrication. (c) Illustration of top gate fabrication using Al2 O3 as the dielectric layer and ITO as the gate contact. (d) Optical image of the top-gated devices. The dashed line marks the area of devices shown in (b). the etch mask. The top gate contact was patterned by photolithography, followed by sputtering of 110-nm indium tin oxide (ITO). Figure B.7(b) shows a cluster of four top-gated devices. B.7 Characterization of Single-Particle Dynamics via Floquet States We first derive the driven graphene Hamiltonian near the valleys 𝜉 = +1, −1 corresponding to the 𝐾 and 𝐾 ′ valleys of graphene. The Dirac Hamiltonian in valley 𝜉 is given by 𝐻 𝜉 ( 𝒌) = ℏ𝑣 𝐹 𝒌 · (𝜎𝑥 , 𝜉𝜎𝑦 ) where 𝒌 = (𝑘 𝑥 , 𝑘 𝑦 ) is the electronic momentum and 𝜎𝑥 and 𝜎𝑦 are the Pauli matrices. We include the irradiation by the circularly polarized laser of vector potential 𝑨(𝑡) = 𝐴(cos Ω𝑡, sin Ω𝑡) via mini- 108 mal coupling: 𝐻 𝜉 ( 𝒌, 𝑡) = 𝐻 𝜉 ( 𝒌 + 𝑒 𝑨(𝑡)/ℏ). The time evolution generated by 𝜉 𝜉 𝐻 𝜉 ( 𝒌, 𝑡) has a complete set of solutions of the form |𝜓 𝒌𝛼 (𝑡)⟩ = 𝑒 −𝑖𝜀𝒌 𝛼 𝑡/ℏ |Φ 𝒌𝛼 (𝑡)⟩, 𝜉 𝜉 where |Φ 𝒌𝛼 (𝑡)⟩ = |Φ 𝒌𝛼 (𝑡 + 𝑇)⟩ is the Floquet-Bloch state, 𝜀 𝒌𝛼 is the quasienergy, and 𝑇 = 2𝜋/Ω denotes the laser driving period. (Here, we drop the 𝜉 index in 𝜀 𝒌𝛼 , because the quasienergy spectrum is identical in both valleys.) As a result of their time periodicity, the Floquet-Bloch states can be decomposed as Í 𝜉 𝑛𝜉 𝑛𝜉 |Φ 𝒌𝛼 (𝑡)⟩ = 𝑚 𝑒 −𝑖𝑚Ω𝑡 |𝜙 𝒌𝛼 ⟩, with |𝜙 𝒌𝛼 ⟩ denoting the Floquet harmonics that can be found by solving the Floquet Schrodinger equation (see Ref. [241]). The FloquetBloch states form a complete basis of stationary solutions to the time-evolution generated by 𝐻 𝜉 ( 𝒌, 𝑡), which are taken onto themselves after each driving period, up to a phase controlled by the quasienergy 𝜀 𝒌𝛼 . Note that 𝜀 𝒌𝛼 is only defined up to an integer multiple of the photon energy ℏΩ, compensated by a redefinition of 𝜉 |Φ 𝒌𝛼 (𝑡)⟩ via multiplication by factor(s) of 𝑒𝑖Ω𝑡 . We use the convention where 𝜀 𝒌𝛼 𝑛𝜉 𝑛𝜉 0𝜉 0𝜉 is chosen such that max𝑛 |⟨𝜙 𝒌𝛼 |𝜙 𝒌𝛼 ⟩| = |⟨𝜙 𝒌𝛼 |𝜙 𝒌𝛼 ⟩|. Laser illumination with a circularly polarized laser opens dynamic gaps in the Floquet quasienergy spectrum. The gap opening at the resonance energies around 𝜀 𝒌𝛼 = ±ℏΩ/2, with a size Δ𝐸 1 linear in the laser field amplitude, controls the Rabi frequency of the resonant inter-band transitions induced by the laser drive. At the Dirac point (𝜀 𝒌𝛼 = 0), the band degeneracy is lifted by a second-order resonance coupling between the conduction and valence bands, with a gap opening quadratic in the laser field amplitude [33, 140, 215]. In the low power regime, Δ𝐸 1 is, therefore, larger than the bandgap at the Dirac point. Illustrating this, Figure B.8(a) shows the theoretical prediction for the quasienergy bandgap Δ𝐸 1 as a function of laser power density and photon energy. For the photon energy and power density used in our experiment, Δ𝐸 1 is predicted to be in the range of ∼ 1-10 meV, while the gap around the Dirac point, too small to be resolved in our measurements, cannot be clearly discerned in the numeric calculations. B.8 Steady State Simulations of Laser-Illuminated Graphene 𝜉 To determine the non-equilibrium steady state occupation 𝐹𝒌𝛼 of the Floquet bands [3, 74, 257, 258], we consider the Floquet-Boltzmann equation (FBE) for electronphonon collisions, which describes the flow of electronic occupations into Floquet el-ph,𝜉 el-ph,𝜉 𝜉 𝜉 𝜉 states. The FBE is given by 𝜕𝑡 𝐹𝒌𝛼 = 𝐼 𝒌𝛼 [{𝐹𝒌𝛼 }], where 𝐼 𝒌𝛼 [{𝐹𝒌𝛼 }] is the electron-phonon collision integral. To estimate the collision integral, we consider electronic coupling to graphene-SiC surface acoustic phonons and graphene longitudinal acoustic phonons, indexed by 𝑗 = 0 and 1, respectively. Assuming weak 109 Figure B.8: Characteristics of the single-particle quasienergy spectrum. (a) Singleparticle quasienergy bandgap Δ𝐸 1 opening at 𝜀 𝑘𝛼 = ±ℏΩ/2, with 𝜀 𝑘𝛼 = 0 corresponding to the Dirac point. The axes indicate the laser power density and photon energy. (b) Predicted light-induced modifications of the time-averaged density of states of graphene (see section J for definition) as a function of Fermi energy shift from the Dirac point for different power densities, with circular polarization (top) and linear polarization (bottom). Notice that the laser-induced gap at the Dirac point, which is a second-order process, is not resolved in the experiment for the chosen laser intensities. (See the main text for details.) The vertical dotted lines mark the 𝐸 𝐹 , value at ±ℏΩ/2. scattering rates relative to the Floquet gap Δ𝐸 1 , we use Fermi’s golden rule to calculate the electron-phonon collision integral el-ph,𝜉 𝐼 𝒌𝛼 [{𝐹𝒌𝛼 }] ≈ 1 2𝜋 ∑︁ ∑︁ ∑︁ 𝒌 ′ 𝛼′ 𝜉 ′ |𝐺 𝒌𝛼𝜉 (𝑛, 𝑗)| 2 × 𝑁 ℏ 𝜉 ′ =±1 ′ 𝛼′ , 𝑗,𝑛 𝒌 ∈BZ 𝜉′ 𝜉′ × [{𝐹𝒌 ′ 𝛼′ (1 − 𝐹𝒌𝛼 )N (ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌)) − 𝐹𝒌𝛼 (1 − 𝐹𝒌 ′ 𝛼′ ) [1 + N (ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌))]} 𝜉 𝜉 × 𝑝(𝜀 𝒌 ′ 𝛼′ − 𝜀 𝒌𝛼 + ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌) + 𝑛ℏΩ) 𝜉′ 𝜉′ + {𝐹𝒌 ′ 𝛼′ (1 − 𝐹𝒌𝛼 ) [1 + N (ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌))] − 𝐹𝒌𝛼 (1 − 𝐹𝒌 ′ 𝛼′ )N (ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌))} 𝜉 𝜉 × 𝑝(𝜀 𝒌 ′ 𝛼′ − 𝜀 𝒌𝛼 − ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌) + 𝑛ℏΩ)], (B.6) where 𝜔 𝑗 ( 𝒌 ′ − 𝒌) = 𝑐 ph | 𝒌 ′ − 𝒌 | is the frequency dispersion of the phonon branch 𝑗, 𝑗 𝑐 ph is the speed of sound, 𝑁 is the number of discretized 𝒌-points on the 2D grid, and N (𝜀) = (𝑒 −𝜀/𝑘 𝐵𝑇 − 1) −1 is the occupation of a phonon mode with energy 𝜀, evaluated at temperature 𝑇. The matrix element ∑︁ ∑︁ 𝒌 ′ 𝛼′ 𝜉 ′ 𝑛+𝑚,𝜉 𝑚𝜉 𝐺 𝒌𝛼𝜉 (𝑛, 𝑗) = 𝑀 𝑗 𝒌 𝒌 ′ ⟨𝜙 𝒌𝛼 |𝜙 𝒌𝛼 ⟩ (B.7) 𝑗 𝑚 𝜈𝜈 ′ 110 describes the electronic coupling to the 𝑗-th phonon branch, where √︃ 1 𝐷𝑗 𝑀 𝑗 𝒌 𝒌 = √ √︁ ℏ𝜔 𝑗 ( 𝒌 ′ − 𝒌), 𝑗 𝐴 2𝜌𝑐 ph ′ (B.8) 𝐴 is the unit cell size of graphene, 𝜌 is the density of graphene, and 𝐷 𝑗 is the deformation potential corresponding to the phonon branch 𝑗. To impose energy conservation on electron-phonon scattering processes, we use the smeared DiracDelta function 2 2 𝑒 −𝜀 /2𝜎 𝜃 (2𝜎 − |𝜀|) (B.9) 𝑝(𝜀) = 1.05 2.51𝜎 where 𝜃 (𝑥) is the Heaviside step function and the numerical prefactors ensure ∫∞ normalization ( −∞ 𝑝(𝜀)𝑑𝜀 = 1). Here, the phenomenological level broadening of 𝜎 = 3 meV describes the phonon and electronic spectral broadening due to disorder in the system. Our calculations assume that the speeds of the phonon branches are given by 𝑐0ph = 1.3 km/s and 𝑐1ph = 11 km/s and their deformation potentials satisfy 𝐷 0 /𝐷 1 = 0.18. el-ph,𝜉 To obtain the steady state occupations, we solve for 𝐼 𝒌𝛼 [{𝐹𝒌𝛼 }] = 0 using the Newton-Raphson method. Our calculations are performed on a discretized square momentum grid covering the region 𝑘 𝑥 , 𝑘 𝑦 ∈ [−0.17 nm−1 , 0.17 nm−1 ] with 105 equally-spaced grid points in the 𝑘 𝑥 and 𝑘 𝑦 directions. We focus only on low-energy electronic states, where the Floquet occupations can be approximated as rotationally 𝜉 symmetric about the Dirac node, i.e., 𝐹𝒌𝛼 ≈ 𝐹𝑘𝛼 , with 𝐹𝑘𝛼 depending only on the momentum magnitude 𝑘 = | 𝒌 | and band index 𝛼. To impose the electronic particle Í number, we add a Lagrange multiplier term 𝜆( 𝒌𝛼 𝐹𝒌𝛼 − 𝑔𝑁)/𝑁 to the FBE with a large constant 𝜆, and we vary the doping by changing 𝑔. The corresponding √︁ Fermi energies 𝐸 𝐹 = ℏ𝑣 𝐹 𝑛/𝜋 in equilibriu mcan be calculated from the electronic ∫ +1 as determined using the numerically-obtained steady density 𝑛 = 𝑑 2 𝒌/(2𝜋) 2 𝐹𝒌+ state. B.9 Transport in the Floquet Steady State To calculate the longitudinal photocurrent Δ𝐼 from the steady state, we use linear response theory to estimate the longitudinal conductivity in the driven system: ∑︁ ∫ 𝑑𝑘 ∫ 𝑑𝐹𝑘𝛼 2 𝜎𝑥𝑥 ≈ 𝑒 𝑑𝜃 𝑘𝜏(𝑘)𝑣 2𝑥 , (B.10) 2𝜋 𝑑𝜀 𝑘𝛼 𝛼 where 𝑣 𝑥 = |∇ 𝒌 𝜀 𝒌𝛼 | cos 𝜃 and 𝜏(𝑘) ∝ 𝑘 for relaxation processes dominated by scattering from charge impurities [37]. The longitudinal photocurrent is given by 111 Figure B.9: Temperature dependence of the longitudinal conductivity at 𝐸 𝐹 = 0 without irradiation, which indicates charge puddle effects. 0 )𝑉 , where 𝑉 is the source-drain voltage and 𝜎 0 is the longitudinal Δ𝐼 = (𝜎𝑥𝑥 −𝜎𝑥𝑥 sd sd 𝑥𝑥 conductivity in the equilibrium system, obtained by replacing 𝑣 𝑥 = 𝑣 𝐹 cos 𝜃 and 𝐹𝑘𝛼 with the Fermi-Dirac distribution. In addition to the above-mentioned effects, charge puddles also modify the longitudinal conductivity. In particular, for doping near charge neutrality, where a clean graphene sheet normally exhibits low conductivity, the charge puddles introduce regions with larger chemical potential and greater mobility, thereby enhancing the conductivity [163]. To roughly capture this behavior, we employ a simple model to relate the conductivity 𝜎 in clean graphene to that with charge puddles, 𝜎𝑝 = 𝜎 + 𝑐(𝑇), similar to that used in Ref. [60]. Here, 𝑐(𝑇) is a function which captures the temperature-dependence of the experimentally-measured conductivÍ14 ity at charge neutrality, fit using 𝑐(𝑇) = 𝑖=0 𝑐𝑖 𝑇 𝑖 to the experimental data for 𝑇 < 10 K, and fit using the function 𝑐(𝑇) = 𝑑1 + 𝑑2𝑇 1/2 for 𝑇 > 10 K, where 𝑐𝑖 , 𝑑1 , and 𝑑2 are fitting parameters (see Figure B.9). Due to heating effects from the drive, the temperature during the duty cycle of the chopper differs when the laser is blocked by the chopper 𝑇eq and when the laser is unblocked 𝑇dr . We account for the temperature mismatch and charge puddle effects by shifting the photocurrent by a magnitude [𝑐(𝑇dr ) − 𝑐(𝑇eq )]𝑉sd in our simulations. We find that this temperature mismatch accounts for the highly temperature and amplitude dependent photocurrent Δ𝐼 magnitude at charge neutrality 𝐸 𝐹 = 0 observed in the experiment. In particular, in our theory, we assume both 𝑇dr − 𝑇eq ≈ 7.3 K/(mW/𝜇m2 )𝑃 and 𝑇eq = 3.5 K + 1.83 K/(mW/𝜇m2 )𝑃 increase linearly as a function of the laser power density 𝑃. The coefficients chosen allow our theory to roughly capture the 112 dependence of Δ𝐼 at charge neutrality as 𝑇eq and 𝑃 are varied in the experiment. We emphasize that the fitting procedures for the variables 𝑐(𝑇), 𝑇dr − 𝑇eq , and 𝑇eq do not affect the width, depth, or shape of the photocurrent dip predicted in our theory; rather, it only shifts the Δ𝐼 by a constant independent of electronic doping. B.10 Density of States and Berry Curvature Calculation The time-averaged density of states (defined as the time-dependent spectral function averaged over one period) was calculated numerically through the formula ∑︁ DOS(𝜀) = 𝛿(𝜀 𝑘𝛼 − 𝜀) (B.11) 𝑘𝛼 where the broadening parameter 𝜎 of the Gaussian function 𝛿(𝜀) is chosen to be much smaller than the resonant Floquet gap. The transverse conductivity was calculated from the anomalous conductivity in the 2 Í ∫ 𝜉 𝑥𝑖 , where 𝐵 𝜉 is the time-averaged Berry steady state 𝜎𝑥𝑦 = 2𝑒ℏ 𝛼 𝑑 2 𝒌 𝐵 𝒌𝛼 𝐹𝒌𝛼 𝒌𝛼 curvature, given by20 " # ∫ 2𝜋/Ω 𝑈𝑥 ( 𝒌, 𝑡)𝑈𝑦 ( 𝒌 + 𝒆 𝑥 , 𝑡) 1 (2𝜋) 2 𝜉 𝐵 𝒌𝛼 = arg 𝑑𝑡 , (B.12) 2𝜋/Ω 0 𝑁𝑥 𝑁 𝑦 𝐴 𝑈𝑥 ( 𝒌 + 𝒆 𝑦 , 𝑡)𝑈𝑦 ( 𝒌, 𝑡) where 𝑈 𝜇 ( 𝒌, 𝑡) = ⟨𝑢 𝒌𝛼 (𝑡)|𝑢 𝒌+𝒆 𝜇 ,𝛼 (𝑡)⟩ |⟨𝑢 𝒌𝛼 (𝑡)|𝑢 𝒌+𝒆 𝜇 ,𝛼 (𝑡)⟩| 2 , (B.13) 𝐺 𝒆 𝜇 = 𝑁 𝜇𝜇 𝜇, ˆ 𝜇 = 𝑥, 𝑦, 𝑁 𝜇 is the number of grid points in the 𝜇-direction, 𝐺 𝜇 is the length of the discretized momentum grid along the 𝜇-direction, and |𝑢 𝒌𝛼 (𝑡)⟩ are the 𝜉 Bloch vectors, defined by |𝜓 𝒌𝛼 (𝑡)⟩ = 𝑒𝑖𝒌·𝒓 |𝑢 𝒌𝛼 (𝑡)⟩. B.11 Transverse Hall Voltage An expected consequence of the Floquet states described above is the appearance of a Hall signal [3, 74, 215]. We measure the transverse voltage in samples under irradiation with both circular and linear polarization, as shown in Figures B.10B.13. Floquet theory predicts that a nonzero transverse voltage originates only from nontrivial band topology, inconsistent with the observation of a transverse signal for linear polarization. However, the extraneous, helicity-independent transverse signal has been observed before under linearly polarized Mid-IR irradiation at zero sourcedrain bias in McIver et al. [185] and attributed to an effective extra bias voltage caused by band bending fields at the contacts, a possible result of asymmetries introduced by charge puddles. The helicity-independent component may also be a 113 Figure B.10: Transverse voltage response under different conditions. Transverse voltage as a function of 𝐸 𝐹 , under light irradiation (ℏΩ = 117 meV) with left-hand circular polarization. The dotted lines mark the values corresponding to 𝐸 𝐹 = ±ℏΩ/2, with the related uncertainties indicated by the gray stripes. (a) Transverse voltage measured at different bias voltage values. The cryostat temperature is 3.1 K, and the laser beam is circularly polarized. (b) Transverse voltage measured under different irradiation power. The curves are vertically shifted for clarity. The cryostat temperature ranges from 3.0-3.3 K depending on laser intensity. The laser beam is circularly polarized. The circles represent data points, and the solid lines are obtained by adjacent point averaging. (c) Transverse voltage measured at different temperatures. For both (a) and (c), the laser intensity is 0.8 mW/𝜇m2 and for (b) and (c) the bias voltage is 50 mV. The measurements are from sample C. product of photovoltaic effects generated by optical transitions in highly irradiated graphene [34, 70] or anisotropies in the Floquet band spectrum near the resonant Floquet gap [33], which may appear due to the distorted circular polarization near the metallic contacts (see Section B.1). The weak polarization-dependent Hall voltage exhibits signs of a nontrivial 𝐸 𝐹 dependence, shown in Figure B.11(b) for graphene sample A, displaying peaks starting to emerge around doping 𝐸 𝐹 = ±ℏΩ/2. In Figure B.11(c), we show the corresponding Hall conductivity. Upon inverting the sign of the source-drain bias 𝑉sd , the Hall voltage peaks reverse sign. Such a helicity-dependent Hall voltage contribution may be an indicator of the emergence of Floquet chiral edge states in our samples. The theoretically calculated Floquet Hall conductivity, see Figure B.11(d), also exhibits peaks near doping 𝐸 𝐹 = ±ℏΩ/2. However, the magnitude of the Hall conductivity observed in the experiment is much weaker than the theorypredicted levels. There are several possible reasons for this discrepancy. One 114 Figure B.11: Transverse voltage response under different conditions. (a) Transverse voltage as a function of 𝐸 𝐹 , under light irradiation (ℏΩ = 117 meV) with different polarizations of the laser. (b) helicity-dependent Hall voltage in Sample A. (c) Hall conductivity in Sample A. Here, the laser intensity is 1.4 mW/𝜇m2 , bias sourcedrain voltage is 6 mV, and cryostat temperature is 3.4 K. (d) Theoretically calculated Floquet Hall conductivity with nontrivial 𝐸 𝐹 -dependence arising from the tunable electronic steady state. The dotted lines in (a)-(d) mark the values corresponding to 𝐸 𝐹 = ±ℏΩ/2, with the related uncertainties indicated by the gray stripes in (a)(c). In (a)-(c), the circles represent data points, and the solid lines are obtained by adjacent point averaging. possible contribution to the suppressed Hall voltage is the distorted polarization of the laser beam near the metallic contacts (see Section B.1 for details), which can create an anisotropic Floquet gap size in momentum space, as was already mentioned in the main text. Charge puddles could also play an important role in reducing the measured Hall response. The photocurrent Δ𝐼 flows through the most conducting puddles with larger chemical potential. Insulating regions with depleted photocurrent in this case would generate a much smaller Hall voltage than predicted, reducing the overall average Hall voltage. B.12 Reproducibility of the Results The photoresponse features and the signatures of Floquet-Bloch band formation are reproducible with multiple sweeps on the same device. As discussed in the main text and above, we characterized the transverse conductivity of the devices for different polarizations of laser irradiation. The results for samples A, B and C are shown in Figures B.11, B.12, and B.13. For all three samples, the longitudinal photocurrent Δ𝐼 showed dips around 𝐸 𝐹 = ±ℏΩ/2 for circularly and linearly polarized light. Such symmetric dips around the Dirac point are consistent with the emergent nonequilibrium Floquet steady states in graphene. 115 B.13 Effect of Hall contact misalignment on the transverse signal measured under illumination We made some estimates to quantify the contribution to the transverse voltage due to the geometric effect of the contact misalignment and compared it to the transverse voltage 𝑉𝑦 measured under illumination. First, we measured the transverse voltage 𝑉Misalignment as a function of the source-drain voltage 𝑉sd and the sourcedrain current 𝐼sd in the zero magnetic field, with no laser illumination and at room temperature. The measured misalignment for sample A is plotted in Figure B.14(a). From this measurement we extracted the quantity 𝑅Misalignment /𝐼sd ∼ 99.1 Ω. Figure B.14(b) shows the estimated contribution to the transverse voltage from the photoinduced current change Δ𝐼, 𝑉Misalignment = 𝑅Misalignment Δ𝐼, in the same graph with the measured transverse photoresponse. The comparison shows that this misalignment contribution is much smaller than the transverse signal measured under illumination. Similar misalignment contributions were determined in all the measured samples. Figure B.12: Floquet signatures in the longitudinal photocurrent in additional devices. The dotted lines mark the 𝐸 𝐹 values corresponding to ±ℏΩ/2, with the related uncertainty described by the gray stripes. (a) Longitudinal photocurrent as a function of 𝐸 𝐹 under different polarization illumination in sample B. Parameters: Laser power density 1.4 mW/𝜇m2 and bias voltage 6 mV. (b) Longitudinal photocurrent as a function of 𝐸 𝐹 , under different polarization illumination in sample C. Parameters: Laser power density 1.6 mW/𝜇m2 , bias voltage 6 mV, and cryostat temperature 3.4 K. 116 Figure B.13: Transverse voltage response in additional devices. Transverse voltage as a function of 𝐸 𝐹 , under irradiation with different polarizations in sample C [panel (a)] and the difference between the transverse voltage response from circular polarizations of the laser beam with opposite chirality extracted from the curves in panel (a) [see panel (b)], and upon reversing the source-drain bias 𝑉sd . Laser power density 0.5 mW/𝜇m2 , cryostat temperature 3.1 K. The dotted lines mark the 𝐸 𝐹 values corresponding to ±ℏΩ/2, with the related uncertainty described by the gray stripes. B.14 Linearity in the device response The source-drain bias voltage dependence of the photoresponse is an important characteristic that can help determine its underlying physical mechanisms and separate intrinsic contributions from graphene or from the graphene-electrode boundaries. Figure B.15(a) shows a typical 𝐼-𝑉 curve (sample C). The clear linear relationship between the bias voltage and current indicates the ohmic contact nature of our graphene-electrode interface. Figure B.15(b) demonstrates the bias voltage dependence of the longitudinal photoresponse at a few different fixed 𝑉𝑔 , which reveals a linear relationship between the sample response and the bias voltage. Moreover, no clear photoresponse signal was observed at 𝑉sd = 0 for all measured gate voltages. Furthermore, as presented in Figure B.15(c), the normalized photoresponse curves collected at different 𝑉sd as a function of Fermi energy further confirm the linearity of our sample’s photoresponse. B.15 Chopper frequency dependence of the photoresponse To better understand our samples’ photoresponse, we measured the longitudinal photoresponse as a function of the chopper modulation frequency. As one can see 117 Figure B.14: Effect of Hall contacts misalignment. (a) (Red) 𝑉Misalignment and (black) 𝑉sd measured at the same source-drain current. Room temperature. (b) Misalignment contribution to the measured transverse voltage for (top) circular and (bottom) linear irradiation. Solid lines represent the transverse voltages measured by the lock-in amplifier; dotted lines represent the contribution resulting from misalignment in the Hall contacts. The experimental data from Figure B.11 are shown for reference. in Figure B.16, as the chopper frequency is increased, the signal first increases and then rapidly drops. Such non-monotonic behavior can be attributed to the heating and cooling processes of electrons that are associated with the chopping of the laser beam. At low frequencies, electrons are heated with a longer exposure time to the irradiation during each chopping cycle, while for high frequencies, although the exposure time is shorter, the electrons do not have enough time for cooling. As a result, for both too low and too high frequencies, a higher effective electron temperature is expected. Furthermore, as discussed in the main text, an elevated effective electron temperature will not only lead to the fading of Floquet features but will also suppress the overall signal strength. Therefore, for better detection of the Floquet features, one has to carefully control the chopper frequency and maintain the electron temperatures below the size of Floquet gaps. All results presented in this work were collected with a chopper frequency around 23 Hz. B.16 Error Analysis In this section, we provide details of the error estimates in Fermi level position, specifically for 𝐸 𝐹 = ℏΩ/2 as well as the full width at half maximum (FWHM) analysis for sample C, as presented in Figure 6.3(c)-(d) in the main text. The electronic carrier density in the sample was estimated using the relation 𝑛 = 𝑘 (𝑉𝑔 − 𝑉𝑑 ) (B.14) 118 Figure B.15: Bias dependence of sample response. (a) I-V curve from sample C at different values of 𝑉𝑔 . (b) Photo-induced current change Δ𝐼 as a function of bias voltage at different values of 𝑉𝑔 . (c) Normalized longitudinal photoresponse Δ𝐼 as a function of 𝐸 𝐹 at different bias voltages, under circularly polarized irradiation, with a power density of 1.5 mW/𝜇m2 . The curves are vertically shifted for clarity. The cryostat temperature is 3.4 K. The dotted lines mark the 𝐸 𝐹 values corresponding to ±ℏΩ/2, with the related uncertainty described by the gray stripes. where 𝑘 ≈ 1.63 × 10−15 m−2 /V ± 𝛿𝑘 and 𝛿𝑘 = 9 × 1013 m−2 /V were estiamted from the gate efficiency calibration. The voltage 𝑉𝑑 ≈ 0.55 V ± 𝛿𝑉𝑑 was estimated to be the gate voltage at which 𝜎𝑥𝑥 is minimized, and the uncertainty 𝛿𝑉𝑑 ≈ 0.8 V was estimated from the gate efficiency calibration. The corresponding uncertainties in √ the Fermi energy 𝐸 𝐹 = ℏ𝑣 𝐹 𝜋𝑛 are given by √︃ (B.15) 𝛿𝐸 𝐹 ≈ (𝐸 𝐹 /2) (𝛿𝑘/𝑘) 2 + [𝛿𝑉𝑑 /(𝑉𝑔 − 𝑉𝑑 )] 2 . The FWHM of the conductivity dip [see Figure 6.3(d)] FWHM = Δ𝐸 𝐹 = 𝐸 𝐹 (𝑉𝑔1 ) − √ √ 𝐸 𝐹 (𝑉𝑔2 ), where 𝑉𝑔1 = 𝐵 + 2 ln 2𝜎 and 𝑉𝑔2 = 𝐵 − 2 ln 2𝜎, was estimated by performing a least squares fit of the experimental data to the function h i 2 2 Δ𝐼fit = 𝑎 0 exp −(𝑉𝑔 − 𝐵) /2𝜎 + 𝐶𝑉𝑔 + 𝐷. (B.16) We estimated the uncertainty in the FWHM using the relationship v u #2 " #2 t 2  2  2 " 𝜕Δ𝐸 𝐹 𝜕Δ𝐸 𝐹 𝜕Δ𝐸 𝐹 𝜕Δ𝐸 𝐹 𝜕Δ𝐸 𝐹 𝛿Δ𝐸 𝐹 = 𝛿𝑣 𝐹 + 𝛿𝑉𝑑 + 𝛿𝑘 + 𝛿𝑉𝑔1 + 𝛿𝑉𝑔2 𝜕𝑣 𝐹 𝜕𝑉𝑑 𝜕𝑘 𝜕𝑉𝑔1 𝜕𝑉𝑔2 (B.17) 119 Figure B.16: Longitudinal photoresponse as a function of chopper frequency. Photoinduced current change measured in sample C under 1.5 mW/𝜇m2 of circularly polarized irradiation. The sample is biased at a fixed 6 mV. The cryostat temperature is 3.4 K. Error bars indicate the maximum deviation in the lock-in amplifier reading during the measurement. where 𝛿𝑣 𝐹 ≈ 0.008 × 106 m/s as determined by cyclotron resonance measurements. The uncertainties 𝛿𝑉𝑔1 and 𝛿𝑉𝑔2 were estimated using the relation √︃ √ (B.18) 𝛿𝑉𝑔1 ≈ 𝛿𝑉𝑔2 ≈ 𝛿𝐵2 + ( 2 ln 2𝛿𝜎) 2 where 𝛿𝐵 and 𝛿𝜎 are the uncertainties in the fitting parameters 𝐵 and 𝜎, respectively, as determined from the 68% confidence interval of the least squares fit. Part III Floquet Physics Beyond Optical Drives and Beyond Low-Dimensional Systems 120 121 Chapter 7 QUANTIZED ACOUSTOELECTRIC FLOQUET EFFECT IN QUANTUM NANOWIRES External coherent fields can drive quantum materials into non-equilibrium states, revealing exotic properties that are unattainable under equilibrium conditions—an approach known as “Floquet engineering.” While optical lasers have commonly been used as the driving fields, recent advancements have introduced nontraditional sources, such as coherent phonon drives. Building on this progress, we demonstrate that driving a metallic quantum nanowire with a coherent wave of terahertz phonons can induce an electronic steady state characterized by a persistent quantized current along the wire. The quantization of the current is achieved due to the coupling of electrons to the nanowire’s vibrational modes, providing the lowtemperature heat bath and energy relaxation mechanisms. Our findings underscore the potential of using non-optical drives, such as coherent phonon sources, to induce non-equilibrium phenomena in materials. Furthermore, our approach suggests a new method for the high-precision detection of coherent phonon oscillations via transport measurements. 7.1 Introduction New phonon sources have recently enabled on-demand access to coherent phonon beams across a broad spectrum of frequencies [100, 103, 116, 123, 206, 226, 286, 297]. These phonon excitations in solids [18, 19, 89, 125, 162, 175, 179, 275] can induce new optical properties [135, 282], strong correlation physics [16, 134, 142, 164, 178, 192], tunable magnetic properties [18, 177, 206, 211], and acoustoelectric effects [126, 133, 154, 173, 221, 234, 262, 302]. The unique characteristics of coherent phonons—e.g., finite momentum, low energy, and electron coupling— enable Floquet engineering of non-equilibrium spatial-temporal electronic phenomena [40, 91, 114, 224]. Notably, terahertz (THz) phonons [20, 22, 35, 77] could become tools for the dynamic manipulation of narrow bandwidths materials, such as moiré systems [77, 131, 312]. Recent proposals demonstrated that coherent phonon beams can be used for Floquet engineering of nontrivial band topology [40, 224]. We demonstrate that a continuous propagating wave of coherent THz phonons can drive a quantum wire into a non-equilibrium Floquet steady state, resulting in a 122 persistent charge current [see Figure 5.1(a)]. Furthermore, under optimal conditions of doping and driving strength, this current can achieve a quantized value, 𝐽 = 𝑒𝜔/𝜋, where 𝜔 represents the phonon driving angular frequency and 𝑒 is the electron charge [see Figure 5.1(b)]. In the adiabatic limit, 𝜔 → 0, the quantization of current aligns with the principles of a topological pump [7, 29, 32, 47, 71, 82, 111, 159, 201, 201, 278, 279]. Remarkably, we find that quantized transport can be maintained over a wide range of THz frequencies, even beyond the adiabatic limit. This suggests a robust mechanism underpinning the quantization, stabilized by the non-equilibrium electronic steady state, which is set by the coupling to a bath of low-temperature thermal phonons and electron-electron interactions [28, 78, 200, 229, 298]. (b) (a) anti-reflective material absorbing material (c) J phonon source (d) 5 0 -5 Figure 7.1: (a) Schematic experimental setup. A THz-frequency coherent phonon wave (with atomic displacements sketched by orange arrows) of momentum 𝑞 and angular frequency 𝜔 propagates from the phonon source on the right to the absorbing material on the left. A screening material of dielectric constant 𝜖 is placed below the CNT. Black sinusoid represents the coherent phonon potential. Electrons (black dots) generate quantized current 𝐽 = 𝑒𝜔/𝜋 when confined to the troughs of the potential, achieved when incoherent phonon relaxation (ep) is much faster than electron-electron heating (ee). (b) Current 𝐽 vs. Floquet gap Δ/ℏ𝜔 for different 𝜖 (see inset). Vertical line denotes Δ = ℏ𝜔. Inset: Δ∗ vs. 𝜖, where Δ∗ is the minimal Floquet gap at which 𝐽 is nearly quantized, set as 0.96𝑒𝜔/𝜋. (c) Band structure of a (10, 0) armchair CNT. Inset: Fermi energy lies near the band bottom of the lowest positive-energy band, and the electronic density 𝑛𝑒 is chosen to be commensurate to 𝑞. (d) Quasienergy spectrum of the driven system. Blue shading on the 𝛼 = 0 band indicates the optimal filling resulting in quantized current. Incoherent phonon scattering transitions (black arrows) relax electrons into the 𝛼 = 0 band. A device for robust generation of quantized current on demand through coherent phonon illumination has numerous potential applications in metrology [95, 129, 209, 123 222, 273, 326], electronics, and quantum computing [21, 57, 88, 316]. Furthermore, this device can be used to characterize and detect coherent phonons through transport measurement. Traditionally, the detection of coherent phonons has relied on optical methods such as reflectivity measurements [72, 96, 157, 314, 315]. Leveraging current quantization to probe the phonon field offers a more direct and sensitive method of coherent phonon detection. In this work, we focus on an experimental setup of a single-walled carbon nanotube (CNT) [79, 105, 128, 191, 247, 249] coupled to a continuous source of coherent phonons. The phonon waves propagate from the right to the left end of the CNT, where they are absorbed by an absorbing material, see Figure 7.1(a). Anti-reflective material with appropriate thickness suppresses reflected phonon waves that destructively interfere with the coherent phonon mode. The electronic steady state in the CNT is formed from the balance between interactions with the coherent phonon wave, incoherent phonons of the CNT, and free electrons. Our model considers a detailed microscopic description of the phononic spectrum of the CNT that serves as the low-temperature heat bath for the phonons and the electron-electron interactions. 7.2 Phonon-Driven Carbon Nanotube. To analyze the steady-state properties of this setup, we use an effective model of a † single-walled CNT. We define 𝝍ˆ 𝑘,𝑠 ≡ 𝜓ˆ †𝑘,𝐴,𝑠 𝜓ˆ †𝑘,𝐵,𝑠 as the creation operator of a Bloch state on the CNT, where 𝜓ˆ †𝑘, 𝑗,𝑠 creates an electron of crystal momentum 𝑘 on sublattice 𝑗 = 𝐴, 𝐵, and 𝑘 is the Bloch momentum in the direction 𝒌ˆ ∥ along the tube axis. The index 𝑠 = 0, . . . , 𝑁 − 1 enumerates the discrete electronic momenta along the direction 𝒌ˆ ⊥ around the circumference of the tube. Here, 𝑁 is the number of graphene unit cells contained in a length |𝑻| along the tube axis, and the CNT is periodic along the tube axis by translations of vector 𝑻. The electronic momentum 𝑘 ∈ [−𝜋/|𝑻|, 𝜋/|𝑻|] is approximately continuous for a long tube. Corresponding eigenenergies of the electronic states are given by 𝐸 𝑠𝜈 (𝑘), where 𝜈 = +, − denote the conduction and valence bands, respectively, see Figure 7.1(c). We focus on a semiconducting CNT whose Fermi surface in equilibrium lies near the bottom of the lowest conduction band [see inset of Figure 7.1(c)]. We omit the index 𝑠, considering only the lowest conduction and highest valence bands described by 𝜈 = +, − with eigenenergies 𝐸 𝜈 (𝑘) and eigenstates |𝜈𝑘⟩. The two ∫ † bands are described by the effective Hamiltonian 𝐻ˆ e = 𝑑𝑘/2𝜋 𝝍ˆ 𝑘 𝐻e (𝑘) 𝝍ˆ 𝑘 , where the 𝑥 axis is aligned along a bond between carbon atoms. To describe electron- 124 Í (𝜆) (𝑡), phonon interactions in the CNT, we consider the Hamiltonian 𝐻ˆ ep (𝑡) = 𝜆 𝐻ˆ ep ∫ † (𝜆) (𝜆) 2 (𝜆) ˆ ˆ ˆ ˆ ˆ where 𝐻ep (𝑡) = 𝑑 𝒓 𝝍 𝒓 𝑉 (𝒓, 𝑡) 𝝍 𝒓 and 𝑉 (𝒓, 𝑡) describe electronic coupling to a phonon mode indicated by 𝜆, 𝒓 = (𝑥, 𝑦) is the spatial coordinate along the tube, ∫ † † ˆ ˆ 𝝍ˆ 𝒓 = 𝑃−1 𝑑𝑘/(2𝜋)𝑒 −𝑖(𝛿𝑘 𝒌 ⊥ +𝑘 𝒌 ∥ )·𝒓 𝝍ˆ 𝑘 , 𝛿𝑘 is the momentum along 𝒌ˆ ⊥ , and 𝑃 is the perimeter of the CNT [77, 296]. [See Appendix C for details.] We assume that a phonon source [see Figure 7.1(a)] generates a coherent wave of phonons in one of the modes denoted by 𝜆 = 𝜆 0 that propagates through the CNT, while other phonons are in low-temperature thermal equilibrium. The phonon mode 𝜆 0 has momentum 𝒒 = 𝑞 𝒌ˆ ∥ , angular frequency 𝜔, and finite displacement
expectation value ⟨𝒖ˆ (𝜆0 ) (𝒓, 𝑡)⟩ = 𝑢 0 cos 𝒒 · 𝒓 − 𝜔𝑡 𝒌ˆ ∥ , where 𝑢 0 is the displacement amplitude and 𝒖ˆ (𝜆0 ) (𝒓, 𝑡) is the displacement operator of the mode. While we consider coherent longitudinal phonon modes, coherent transverse modes should induce similar behavior. We assume that the CNT is sufficiently short compared to the coherent phonon decay length such that 𝑢 0 is approximately constant [see Appendix C]. The electronic dynamics can be divided to coherent components described by the (𝜆 0 ) time- and spatially-periodic Hamiltonian 𝐻ˆ 0 (𝑡) = 𝐻ˆ e + 𝐻ˆ ep (𝑡) and incoherent Í (𝜆) components due to coupling to thermal phonon modes, 𝐻ˆ b (𝑡) = 𝜆≠𝜆0 𝐻ˆ ep (𝑡). For simplicity, we assume that the wavelength of the coherent phonon mode is commensurate with the periodicity of the CNT along the tube axis. The single-particle Hamiltonian 𝐻ˆ 0 (𝑡) can be diagonalized by the Floquet-Bloch states, Í (𝑛) |𝜓 𝑘𝛼 (𝒓, 𝑡)⟩ = 𝑒 −𝑖( 𝒌·𝒓+𝜀 𝑘 𝛼 𝑡/ℏ) 𝑛∈Z 𝑒 −𝑖𝑛(𝒒·𝒓−𝜔𝑡) |𝜙 𝑘𝛼 ⟩ [91, 114, 224, 241]. Here, 𝜀 𝑘𝛼 is the quasienergy satisfying ∑︁ (𝑚) (𝑚 ′ ) (𝜀 𝑘𝛼 + 𝑚ℏ𝜔)|𝜙 𝑘𝛼 ⟩ = 𝐻e (𝑘 + 𝑚𝑞) + 𝑉𝑚−𝑚 ′ |𝜙 𝑘𝛼 ⟩ (7.1) 𝑚 ′ ≠0 where 𝛼 enumerates the Floquet bands and 𝑉𝑛 are the Fourier harmonics of ⟨𝑉ˆ (𝜆0 ) (𝒓, 𝑡)⟩, Í i.e., ⟨𝑉ˆ (𝜆0 ) (𝒓, 𝑡)⟩ = 𝑛≠0 𝑒 −𝑖𝑛(𝒒·𝒓−𝜔𝑡) 𝑉𝑛 . The quasienergy spectrum 𝜀 𝑘𝛼 arises from replicas of the original energy bands 𝐸 𝜈 (𝑘) shifted in energy and momentum by 𝑚ℏ𝜔 and 𝑚𝑞, respectively, where 𝑚 ∈ Z [see light grey, dashed curves in Figure 7.1(d)]. At the crossings between the Floquet replicas corresponding to 𝑘 = 𝑘 ∗ +𝑚𝑞, the quasienergy spectrum exhibits Rabi-like gaps of size Δ ≈ |𝑉1 |, see solid curves in Figure 7.1(d) [see Appendix C]. Remarkably, the quasienergy satisfies the periodicity condition 𝜀 𝑘𝛼 = 𝜀 𝑘+𝑞,𝛼 − ℏ𝜔, which is the basis for the quantized current 125 presented in this work. Specifically, the current is given by [74] ∫ 2𝑒 ∑︁ 𝑞 𝑑𝑘 𝑑𝜀 𝑘𝛼 𝐽= 𝐹𝑘𝛼 . ℏ 𝛼 0 2𝜋 𝑑𝑘 (7.2) † (𝑡) 𝑓ˆ𝑘𝛼 (𝑡)⟩ is the occupation of the Floquet-Bloch state |𝜓 𝑘𝛼 (𝒓, 𝑡)⟩ Here, 𝐹𝑘𝛼 (𝑡) = ⟨ 𝑓ˆ𝑘𝛼 † (𝑡). The factor of two in Eq. (7.2) accounts for spin decreated by operator 𝑓ˆ𝑘𝛼 generacy. When only the 𝛼 = 0 band is fully-occupied, 𝐽 = 𝑒𝜔/𝜋, resulting in quantized current. Our goal is to determine 𝐹𝑘𝛼 (𝑡) in the steady state, which is set by incoherent electronic dynamics due to electron-phonon and electron-electron scattering. 7.3 Floquet-Boltzmann Equation. To model the dynamics under electron-phonon scattering, we consider the microscopic Hamiltonian 𝐻ˆ b (𝑡) for electronic coupling to incoherent bath phonons, as defined above. Our model considers only the longitudinal acoustic phonon mode of speed 𝑐 ph , momentum 𝑝, and energy ℏ𝑐 ph 𝑝, which dominates the electron-phonon scattering near the Fermi surface of the CNT. We assume that the CNT incoherent phonons are coupled to an external heat bath, remaining in thermal equilibrium at temperature 𝑇 and thermalizing quickly with the heat bath phonons and environment. Additionally, we consider scattering due to electron-electron interactions, given by the Hamiltonian ∫ 𝑑𝑘 1 𝑑𝑘 2 𝑑𝑝 ˆ 𝐻ee = V𝑘 1 ,𝑘 2 ( 𝑝) 𝑐ˆ†𝑘 1 +𝑝,+ 𝑐ˆ†𝑘 2 −𝑝,+ 𝑐ˆ𝑘 2 ,+ 𝑐ˆ𝑘 1 ,+ , (7.3) (2𝜋) 3 where V𝑘 1 ,𝑘 2 ( 𝑝) = 𝑈 ( 𝑝)W𝑘 1 ,𝑝 W𝑘 2 ,−𝑝 /(2𝜖), W𝑘,𝑝 ≡ ⟨+, 𝑘 + 𝑝|+, 𝑘⟩ is the formfactor, 𝑈 ( 𝑝) is the Coulomb potential, 𝜖 is the dielectric constant of the surrounding screening medium, and 𝑐ˆ†𝑘 𝜈 creates an electron in the eigenstate |𝜈𝑘⟩ of 𝐻ˆ e . We consider scattering within the 𝜈 = + band only, because the Fermi level [see inset of Figure 7.1(c)] is energetically well-separated from other bands, restricting scattering to small momentum transfers 𝑝 ≲ 𝑞 near the Fermi surface. For small 𝑝, we approximate 𝑈 ( 𝑝) ≈ 1 eV, consistent with estimates in Refs. [176, 325]. Under electron-phonon and electron-electron scattering, the electrons form a steady state distribution with occupations 𝐹𝑘𝛼 (𝑡) determined by the Floquet-Boltzmann equation (FBE) [93, 110, 257, 258], b ee 𝐹¤𝑘𝛼 (𝑡) = 𝐼 𝑘𝛼 [{𝐹𝑘𝛼 (𝑡)}] + 𝐼 𝑘𝛼 [{𝐹𝑘𝛼 (𝑡)}] (7.4) 126 (a) (c) 5 1 4 3 (d) 2 1 (b) 2 3 4 100 500 2000 5 (e) Figure 7.2: (a) Floquet bands upon driving by a coherent phonon wave. Dashed light purple arrows (1-3): dominant electron-phonon intraband and cooling processes relaxing electrons into the 𝛼 = 0 band. Solid red arrows (4-5): electron-phonon heating processes exciting electrons into the 𝛼 = 1 band, which are kinematically suppressed when Δ ≫ 𝛿𝜀. (b) Pairs of zigzag and squiggly red arrows: electronelectron scattering exciting electrons into the 𝛼 = 1 band. Dashed light purple arrow: electron-phonon process relaxing excited electrons into S+ . (c)-(e) Occupation 𝐹𝑖 of the patch S𝑖 for 𝑖 = +, −, 0 [see panel (a)] vs Δ for various dielectric constants 𝜖. Vertical lines: Δ = Δ∗ (𝜖). Note 𝐹+ → 0, 𝐹− → 1 as Δ → Δ∗ , resulting in quantized current. with 𝐹¤𝑘𝛼 (𝑡) = 0. The FBE is valid when 𝐻ˆ ee and 𝐻ˆ b weakly scatter electrons between single-particle Floquet eigenstates. Here, we use the Fermi golden rule modified for transitions between Floquet states [257, 258] to calculate the electron-phonon and b [{𝐹 (𝑡)}] and 𝐼 ee [{𝐹 (𝑡)}], respectively electron-electron collision integrals 𝐼 𝑘𝛼 𝑘𝛼 𝑘𝛼 𝑘𝛼 (see Appendix C). We note that the analysis in terms of the occupations, 𝐹𝑘𝛼 (𝑡), † diagonal in the Floquet basis neglects the coherences ⟨ 𝑓ˆ𝑘𝛼 (𝑡) 𝑓ˆ𝑘𝛼′ (𝑡)⟩ for 𝛼 ≠ 𝛼′. ph el These coherences are suppressed when 1/𝜏𝑘𝛼𝛼′ + 1/𝜏𝑘𝛼𝛼 ′ ≪ |𝜀 𝑘𝛼 − 𝜀 𝑘𝛼 ′ |/ℏ, where ph el 1/𝜏𝑘𝛼𝛼′ and 1/𝜏𝑘𝛼𝛼 ′ are respectively the electron-phonon and electron-electron interband scattering rates between bands 𝛼 and 𝛼′ ≠ 𝛼 [145, 257, 258]. Interband scattering transitions also broaden the electronic spectral function by an energy of tot , relaxing energy conservation in the FBE. roughly 𝛿𝜀 ≈ ℏ/𝜏𝑘𝛼𝛼 ′ 7.4 Phenomenological Rate Equation Before discussing the numerical solution to the FBE, we present a simplified phenomenological analysis. Our goal is to estimate the conditions on the scattering opt opt rates leading to 𝐹𝑘𝛼 ≈ 𝐹𝑘𝛼 in the steady state. Here, 𝐹𝑘𝛼 is the optimal steady opt opt state distribution, defined as 𝐹𝑘0 = 1, and 𝐹𝑘,𝛼≠0 = 0, in which the current is quan- 127 tized. To this end, we identify the key scattering processes that contribute to the steady state, indicated by arrows in Figure 7.2(a)-(b). These processes connect three patches of Floquet states denoted by S+ , S− , and S0 , with approximately uniform electronic occupation 𝐹+ , 𝐹− , and 𝐹0 respectively. The patch S0 includes Floquet states with momentum 𝑘 ∗ ≤ 𝑘 < 𝑞/2 in the 𝛼 = 0 band, while S+ and S− encloses those with momentum −𝑞/2 ≤ 𝑘 < 𝑘 ∗ in the 𝛼 = 1 and 𝛼 = 0 bands, respectively [see Figure 7.2(a)]. To estimate the electronic occupations of the patches, let us first consider the limit ee → 0 in which scattering is mediated by acoustic phonons only. 𝜖 → ∞ and 𝐼 𝑘𝛼 Averaging the FBE over the patches, we obtain rate equations for their occupations, ∑︁ 𝐹¤𝑖 = [R 𝑗𝑖 𝐹 𝑗 (1 − 𝐹𝑖 ) − R𝑖 𝑗 𝐹𝑖 (1 − 𝐹 𝑗 )], (7.5) 𝑗 where 𝑖, 𝑗 = +, −, 0 and R𝑖 𝑗 denotes the average scattering rate of an electron from patch S𝑖 to S 𝑗 . We begin by assuming that the system is optimally doped, i.e., opt 𝑛𝑒 = 𝑞, where 𝑛𝑒 is the density of the electrons. The optimal steady state 𝐹𝑘𝛼 is obtained when the “Floquet-cooling” processes R +0 , R 0− and R −0 [dashed, light purple arrows numbered 1-3 in Figure 7.2(a)] dominate the scattering rates. The rest of the scattering rates R𝑖 𝑗 create excitations in S+ , leading to deviations from opt 𝐹𝑘𝛼 and therefore are dubbed “Floquet-heating” rates [solid red arrows numbered 4-5 in Figure 7.2(a)]. When the incoherent phonons remain at temperature 𝑇 = 0, all “Floquet-heating” processes mediated by acoustic phonons require a small energy and large momentum transfer. This kinematically constrains the rates R +− and R 0+ at high drive intensities, i.e., when Δ > 𝛿𝜀, disabling all “Floquet-heating” processes. Simultaneously, the processes described by the rates R 0− and R −0 are kinematically allowed. These processes are of the Floquet-Umklapp (FU) type and therefore increase with the drive opt intensity as (Δ/ℏ𝜔) 2 yielding 𝐹𝑘𝛼 → 𝐹𝑘𝛼 [76, 241, 257]. This is a remarkable result that shows that the coupling to acoustic phonons can stabilize a quantized current in the non-adiabatic regime 𝛿𝜀 < Δ < ℏ𝜔. In contrast, in the low intensity limit (Δ → 0) of the drive, the rates R 0− and R −0 vanish, whereas the rates R −+ and R 0+ become kinematically enabled. This situation leads to 𝐹+ = 𝐹0 = 1 and 𝐹− = 0, recovering the equilibrium Fermi-Dirac distribution. A finite incoherent phonon temperature 𝑘 𝐵𝑇 < ℏ𝜔 and electron-electron interactions opt (finite 𝜖) cause deviations of the steady state from 𝐹𝑘𝛼 when Δ > 𝛿𝜀. Absorption of incoherent phonons yields a finite but weak electron-phonon heating rate R 0+ , 128 (a) (b) 1 (c) (d) 0.5 0 (e) Figure 7.3: (a) Steady state occupation of the phonon-driven CNT for a weak phonon drive (Δ < Δ∗ ), dielectric constant 𝜖 = 80, and optimal doping 𝑛𝑒 = 𝑞. Inset shows excitations in the 𝛼 = 1 band near the Floquet gap. (b) Same as (a) but for a strong drive amplitude (Δ > Δ∗ ) where the occupation of the 𝛼 = 1 band is negligible, and the 𝛼 = 0 band is fully occupied. (c)-(d) Same as (b), but with two different electronic densities 𝑛𝑒 away from optimal doping. (e) Steady state current vs 𝑛𝑒 evaluated at Δ = 0.8ℏ𝜔 > Δ∗ for 𝜖 = 80. generating small electron and hole densities in S+ and S− respectively. Heating processes due to electron-electron interactions excite electrons into the patch S+ in the 𝛼 = 1 band [pairs of squiggly red arrows in Figure 7.2(b)]. Other electronelectron scattering processes [pairs of zigzag arrows in Figure 7.2(b)] excite electrons to states elsewhere in the 𝛼 = 1 band, which are relaxed to S+ by electron-phonon cooling [dashed, light purple arrow in Figure 7.2(b)]. These processes result in a net increase in 𝐹+ and reduced current response. Electron-electron heating can be suppressed by increasing the drive intensity, since the phase space for such scattering processes is constrained to small energy and momentum transfers near the Floquet gap [see Figure 7.2(b)] and is reduced as Δ is increased. In Figure 7.2(c)-(e), we show the average occupations of the patches as a function of Δ for various dielectric constants 𝜖. We define Δ∗ as the minimal Floquet gap at which 𝐽 = 0.96𝑒𝜔/𝜋. The opt equilibrium Fermi-Dirac distribution (𝐹+ = 𝐹0 = 1 − 𝐹− = 1) transitions to 𝐹𝑘𝛼 as Δ → Δ∗ . 7.5 Numerical Analysis To test our prediction of the steady state current, we solve the FBE numerically in the steady state. We consider a coherent acoustic phonon mode of angular frequency 𝜔 = 6 meV/ℏ with speed of sound 𝑐 ph = 20 km/s, incoherent phonon modes 129 at temperature 𝑇 = 7 K, and optimal electronic density 𝑛𝑒 = 𝑞. Figure 7.3(a)(b) compares the steady state distributions for weak (𝛿𝜀 < Δ < Δ∗ ) and strong (Δ∗ < Δ < ℏ𝜔) non-adiabatic drives. As predicted using the phenomenological model, the density of excitations in the 𝛼 = 1 band is suppressed in the strong drive opt limit, approaching the optimal distribution 𝐹𝑘𝛼 . Figure 7.1(b) shows the current 𝐽 as a function of Δ/ℏ𝜔 for different dielectric constants 𝜖. The vertical line indicates Δ = ℏ𝜔, the boundary between the nonadiabatic and adiabatic drive regimes. The current approaches quantization in the non-adiabatic regime Δ < ℏ𝜔 when 𝜖 is sufficiently large, verifying the phenomenological model. The inset of Figure 7.1(b) shows that the optimal Floquet gap Δ∗ decreases as a function of 𝜖. Finally, we study the current as a function of the doping. The optimal distribution, opt 𝐹𝑘𝛼 , is obtained at optimal doping 𝑛𝑒 = 𝑞. Figure 7.3(c)-(d) compares the steady state distributions for electronic densities below (𝑛𝑒 < 𝑞) and above (𝑛𝑒 > 𝑞) optimal opt doping, where the steady state deviates significantly from 𝐹𝑘𝛼 . Figure 7.3(e) shows 𝐽 as a function of 𝑛𝑒 for Δ ≈ 0.8ℏ𝜔 > Δ∗ and 𝜖 = 80. The quantized current is reached at optimal doping. Experimental realization.—In Figure 7.1(a), we show a CNT suspended above a material of high relative permittivity 𝜖, such as SrTiO3 with 𝜖 ∼ 103 at THz frequencies [69, 243, 295]. The CNT is enclosed by a conducting anti-reflective material (ARM) with thickness tuned for destructive interference of reflected phonon waves [231]. Gold leads attached to the ARM measure the current. On the right, a segment of the CNT extends beyond the ARM. Phonon generation could be achieved by placing the segment under an intense laser pulse [67, 68, 85, 92, 246, 249, 250, 274, 317] or in contact with a material hosting coherent phonons, such as a phonon laser [22, 77, 151, 182, 213, 327] (see the Appendix C for details of potential coherent phonon sources.) The CNT must also be shorter than the coherent phonon decay length, which we estimate by 𝑐 ph 𝜏 ∼ 1 𝜇m, where 𝜏 ∼ 100 ps is the approximate coherent phonon lifetime, set by FU and phonon-phonon scattering [18, 27, 94] (see Appendix C). Additionally, the phonon drive amplitude 𝑢 0 must be much weaker than that which causes melting, i.e., 𝑢 0 ≪ 0.1𝑞 −1 , predicted by the Lindemann criterion [136, 166]. Indeed, the inset of Figure 7.1(b) shows that nearly quantized current is realized for experimentally-accessible values of 𝜖 when the Floquet gap satisfies Δ∗ /ℏ𝜔 ∼ 0.5, corresponding to sufficiently weak driving amplitudes of 𝑢 0 ∼ 0.001𝑎 ≪ 0.1𝑞 −1 . 130 Conclusion.—Our work shows that coherent phonons can induce Floquet bands with quantized transport in screened quantum wires, where the steady state electronic population is stabilized by interactions with low-temperature incoherent phonons [see Figure 7.1(c)]. Furthermore, our device can serve as a detector of coherent phonons, where the measured current indicates the intensity and degree of coherence of the incoming acoustic waves. We anticipate sensitive detection even to weak fields because adiabaticity is not required [see Figs. 7.1(a)-(b)], in contrast to conventional methods utilizing optical reflectivity [72, 96, 157, 315]. A stronger signal, also sensitive to the direction of phonon propagation, could be obtained in higher-dimensional materials and heterostructures. Exploration of these systems is left for future work. While our work specialized on a phonon-driven CNT, the effect is general to various fermionic models with spatio-temoral modulation and energy-relaxing heat bath coupling (see Appendix C for a general model), including solid state and AMO systems [111, 201, 201]. 7.6 Acknowledgements We thank Yang Peng, Michael Kolodrubetz, and Erez Berg for valuable discussions. C.Y. gratefully acknowledges support from the DOE NNSA Stewardship Science Graduate Fellowship program, which is provided under cooperative agreement number DE-NA0003960. W.H. gratefully acknowledges support from the Caltech Cambridge Scholars Exchange Program. G.R. and I.E. are grateful for support from the Simons Foundation and the Institute of Quantum Information and Matter, as well as support from the NSF DMR grant number 1839271. This work is supported by ARO MURI Grant No. W911NF-16-1-0361, and was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. 131 Appendix C APPENDIX C.1 Microscopic Details of the Carbon Nanotube In this section, we discuss in detail the microscopic model for the phonon-driven carbon nanotube, including the single particle electronic physics and electronic coupling to coherent and incoherent phonons. The Carbon Nanotube Hamiltonian The carbon nanotube (CNT) structure is described by the chirality indices (𝑚, 𝑛) [79, 105, 128, 191, 247, 249]. Along the axis of the CNT, the lattice is periodic under translations by 𝑻 = 𝑡1 𝒂 1 + 𝑡2 𝒂 2 , where 𝑡1 = (2𝑚 + 𝑛)/𝑑 𝑅 , 𝑡2 = −(2𝑛 + 𝑚)/𝑑 𝑅 , and 𝑑 𝑅 = gcd(2𝑛 + 𝑚, 2𝑚 + 𝑛), resulting in enlarged CNT unit cells that each contain 𝑁 = 2(𝑛2 + 𝑛𝑚 + 𝑚 2 )/𝑑 𝑅 graphene unit cells. Here, 𝒂 1 = 𝜹1 − 𝜹3 and 𝒂 2 = 𝜹2 − 𝜹3 are the primitive lattice vectors the graphene layer, where 𝜹 𝑗 = √

𝑎/ 3(sin 2𝜋 𝑗/3 , cos 2𝜋 𝑗/3 ) and 𝑎 = 0.246 nm. Along the circumference of the tube, the electron and phonon momenta acquire discrete values, while the momenta remain approximately continuous along the tube axis for a long CNT. The possible momenta can be expressed as 𝒌 = 𝑠𝒌 ⊥ + 𝑘 𝒌ˆ ∥ , where 𝑠 = 0, 1, . . . , 𝑁 and 𝑘 ∈ [−𝜋/|𝑻|, 𝜋/|𝑻|], with momentum vectors 𝒌 ⊥ = (−𝑡2 𝒃 1 + 𝑡 1 𝒃 2 ) and 𝒌 ∥ = (𝑚𝒃 1 − 𝑛𝒃 2 )/𝑁, and unit vectors 𝒌ˆ ⊥ = 𝒌 ⊥ /| 𝒌 ⊥ | and 𝒌ˆ ∥ = 𝒌 ∥ /| 𝒌 ∥ |. We use 𝒃 1 and 𝒃 2 to denote the reciprocal lattice vectors of the graphene layer (𝒂𝑖 · 𝒃 𝑗 = 2𝜋𝛿𝑖 𝑗 ). The family of possible momenta for each value of 𝑠 represents a linear path, or ‘cut,’ along the Brillouin zone of the monolayer graphene. Now, the Hamiltonian for the CNT is of a block-diagonal form, with 𝑁 blocks each corresponding to the Hamiltonian 𝐻ˆ 𝑠 ( 𝒌) = 𝐻ˆ g (𝑠𝒌 ⊥ + 𝑘 𝒌ˆ ∥ ) along a cut 𝑠 = 0, . . . 𝑁, where the single-particle tightbinding Hamiltonian for a monolayer graphene sheet with nearest-neighbor hopping is given by ! ∑︁ † 0 ℎ∗ ( 𝒌) 𝐻ˆ g = 𝝍ˆ 𝒌 𝝍ˆ 𝒌 . (C.1) ℎ( 𝒌) 0 𝒌   Here, 𝝍ˆ 𝒌 ≡ 𝜓ˆ 𝒌,𝐴 𝜓ˆ 𝒌,𝐵 where 𝜓ˆ †𝒌,𝑥 creates a fermion of crystal momentum 𝒌 on Í sublattice 𝑥 = 𝐴, 𝐵 of the graphene sheet, ℎ( 𝒌) = ℎ 𝑗 𝑒𝑖𝒌·𝜹 𝑗 , and ℎ = 2.8 eV. The lowest-energy conduction and highest-energy valence bands are indexed by 132 𝑠m = argmin𝑠 min 𝑘 |𝑠𝒌 ⊥ + 𝑘 𝒌ˆ ∥ − 𝑲|, where 𝑲 is the momentum of the Dirac 𝐾 point in the monolayer graphene Brillouin zone. The lowest energy electronic state along the cut 𝑠𝑚 has momentum 𝑘 𝑚 = argmin 𝑘 |𝑠𝑚 𝒌 ⊥ + 𝑘 𝒌ˆ ∥ − 𝑲|. The Electron-Phonon Hamiltonian In this section, we detail the CNT Hamiltonian and electron-phonon coupling Hamiltonian used in the main text. The lowest conduction and highest valence bands [see Figure 1(c) in the main text] of the CNT are described by the effective Hamiltonian ∫ † 𝐻ˆ e = 𝑑𝑘/2𝜋 𝝍ˆ 𝑘 𝐻e (𝑘) 𝝍ˆ 𝑘 , where 𝐻e (𝑘) = ℏ𝑣 𝐹 (𝛿𝑘 𝒌ˆ ⊥ + 𝑘 𝒌ˆ ∥ ) · 𝝈, (C.2) 𝑣 𝐹 is the Fermi velocity of graphene, 𝝈 = (𝜎𝑥 , 𝜎𝑦 ) is a vector of Pauli matrices acting in the graphene sublattice basis, the 𝑥 axis is aligned along a bond between carbon atoms, and 𝛿𝑘 = |𝑠𝑚 𝒌 ⊥ + 𝑘 𝑚 𝒌ˆ ∥ − 𝑲|. To describe electron-phonon interactions in the CNT, we consider the Hamiltonian ∫ Í (𝜆) † (𝜆) 𝐻ˆ ep (𝑡) = 𝜆 𝐻ˆ ep (𝑡), where 𝐻ˆ ep (𝑡) = 𝑑 2 𝒓 𝝍ˆ 𝒓 𝑉ˆ (𝜆) (𝒓, 𝑡) 𝝍ˆ 𝒓 and (𝜆) (𝜆) 𝑉ˆ (𝜆) (𝒓, 𝑡) = ℏ𝑣 𝐹  ph (𝒓, 𝑡) · 𝝈 + 𝜙ˆph (𝒓, 𝑡) (C.3) describe electronic coupling to a phonon mode indicated by 𝜆, 𝒓 = (𝑥, 𝑦) is the ∫ † † ˆ ˆ spatial coordinate along the tube, and 𝝍ˆ 𝒓 = 𝑃−1 𝑑𝑘/(2𝜋)𝑒 −𝑖(𝛿𝑘 𝒌 ⊥ +𝑘 𝒌 ∥ )·𝒓 𝝍ˆ 𝑘 . The electrons interact with the phonons through the effective vector potential √ (𝜆) (𝜆) (𝜆) (𝜆) (𝒓, 𝑡) − 𝑢ˆ 𝑦𝑦 (𝒓, 𝑡), 2𝑢ˆ 𝑥𝑦 (𝒓, 𝑡))  ph (𝒓, 𝑡) = 3𝛽/(2𝑎)( 𝑢ˆ 𝑥𝑥 (C.4) and through the local scalar potential (𝜆) (𝜆) (𝜆) 𝜙ˆph (𝒓, 𝑡) = 𝐷 [𝑢ˆ 𝑥𝑥 (𝒓, 𝑡) + 𝑢ˆ 𝑦𝑦 (𝒓, 𝑡)] 𝐼, (C.5) where 𝑎 = 0.246 nm, 𝛽 ≈ 3.14, and deformation potential 𝐷 = 15 eV [77, 296]. Here, 𝒖ˆ (𝜆) (𝒓, 𝑡) is the displacement operator of the phonon mode, 𝑢ˆ 𝑏(𝜆) (𝒓, 𝑡) is its (𝜆) 𝑏-th component, and 𝑢ˆ 𝑏𝑐 (𝒓, 𝑡) = [𝜕𝑏 𝑢ˆ 𝑐(𝜆) (𝒓, 𝑡) + 𝜕𝑐 𝑢ˆ 𝑏(𝜆) (𝒓, 𝑡)]/2. Hamiltonian for Interactions Between Electrons and Incoherent Phonons In this section, we provide the full expression for the microscopic Hamiltonian 𝐻ˆ b (𝑡) accounting for electronic coupling to incoherent bath phonon modes. The low-energy longitudinal acoustic phonons dominate the electron-phonon scattering near the Fermi surface of the lightly-doped CNT. Upon writing the displacement 133 operators 𝒖ˆ (𝜆) (𝒓, 𝑡) for such phonon modes in terms of bath acoustic phonon creation operators 𝑏ˆ †𝑝 , we derive the effective Hamiltonian [246]: ∫ 𝑑𝑘 𝑑𝑞 ˆ (C.6) 𝑀𝑘,𝑝 𝑐ˆ†𝑘+𝑝,+ 𝑐ˆ𝑘,+ ( 𝑏ˆ †𝑝 + 𝑏ˆ −𝑝 ) + h.c. 𝐻b = (2𝜋) 2 √︁ √ √︁ where 𝑀𝑘,𝑝 = 𝐷 ℏ𝑐 ph 𝑝/( 2𝐴𝜌𝑐 ph )W𝑘,𝑝 with unit cell area 𝐴 = 3𝑁𝑎 2 /2, and graphene density 𝜌. C.2 Details of the coherent phonon drive In this section, we analyze the coherent phonon drive in detail and propose possible coherent phonon sources. Analytic Expression for the Floquet Gap In this section, we derive the expression for the Floquet gap √ 𝑦 Δ ≈ 𝑞𝑢 0 [𝐷 + 𝑘ˆ ∥ ℏ𝑣 𝐹 3𝛽/2𝑎] (C.7) used in the main text. We begin by constructing an effective drive Hamiltonian by working in the eigenbasis of the electronic Hamiltonian 𝐻e (𝑘) = ℏ𝑣 𝐹 𝒌 · 𝜎, (C.8) where 𝒌 = 𝛿𝑘 𝒌ˆ ⊥ + 𝑘 𝒌ˆ ∥ [see the full definition in the main text]. Let 𝑈 be the matrix that diagonalizes 𝐻e (𝑘), where ! −ℏ𝑣 | 𝒌 | 0 𝐹 𝑈 −1 𝐻e (𝑘)𝑈 = . (C.9) 0 ℏ𝑣 𝐹 | 𝒌 | In the limit near 𝑘 = 0, one can show that 𝐸 − (𝑘) 0 𝑈 −1 𝐻e (𝑘)𝑈 ≈ 0 𝐸 + (𝑘) ! (C.10) where 𝐸 ± (𝑘) = ℏ2 𝑘 2 /(2𝑚) ± ℏ𝑣 𝐹 𝛿𝑘. The electronic coupling to a coherent phonon mode is described by the Hamiltonian (𝜆 ) (𝜆 0 ) 𝑉 (𝒓, 𝑡) = ℏ𝑣 𝐹 ⟨  ph0 (𝒓, 𝑡)⟩ · 𝝈 + ⟨𝜙ˆph (𝒓, 𝑡)⟩. (C.11) Note that √ (𝜆 ) [⟨  ph0 (𝒓, 𝑡)⟩] 𝑥 =
3𝛽 ˆ 𝑦 2 [( 𝑘 ∥ ) − ( 𝑘ˆ 𝑥∥ ) 2 ]𝑞𝑢 sin 𝑞𝑟 − 𝜔𝑡 2𝑎 (C.12) 134 √
3𝛽 (𝜆 0 ) 𝑦 [⟨ Â ph (𝒓, 𝑡)⟩] 𝑦 = (−2 𝑘ˆ 𝑥∥ 𝑘ˆ ∥ )𝑞𝑢 sin 𝑞𝑟 − 𝜔𝑡 2𝑎 (C.13)
(𝜆 0 ) ⟨𝜙ˆph (𝒓, 𝑡)⟩ = 𝐷𝑞𝑢 sin 𝑞𝑟 − 𝜔𝑡 𝐼. (C.14) and To transform 𝑉 (𝒓, 𝑡) into the diagonal basis of 𝐻e (𝑘), first note that (𝜆 ) 𝑈 −1 [ℏ𝑣 𝐹 ⟨ Â ph0 (𝑟, 𝑡)⟩ · 𝝈]𝑈 (𝜆 ) (𝜆 ) 0 0 ©−⟨ Â ph (𝒓, 𝑡)⟩ · 𝒌/| 𝒌 | −𝑖⟨ Â ph (𝒓, 𝑡)⟩ × 𝒌/| 𝒌 | ª =­ ®. (𝜆 ) (𝜆 ) 𝑖⟨ Â ph0 (𝒓, 𝑡)⟩ × 𝒌/| 𝒌 | ⟨ Â ph0 (𝒓, 𝑡)⟩ · 𝒌/| 𝒌 | « ¬ (C.15) In a semiconducting nanotube, near the band extrema, 𝑘 ≪ 𝛿𝑘, so 𝒌 and 𝒌ˆ ⊥ are roughly parallel and √
3𝛽 (𝜆0 ) 𝑦 ⟨ Â ph (𝑟, 𝑡)⟩ · 𝒌/| 𝒌 | ≈ 𝑘ˆ ∥ ℏ𝑣 𝐹 𝑞𝑢 sin 𝑞𝑟 − 𝜔𝑡 . (C.16) 2𝑎 When ℏ𝑣 𝐹 𝛽𝑎 −1 ≪ ℏ𝑣 𝐹 𝛿𝑘, the off-diagonal components of Eq. (C.15) are negligible. In this limit, we finally obtain an effective Hamiltonian for the lowest conduction band of the CNT, [𝑈 −1 𝐻0 (𝑘, 𝑡)𝑈] 00 # " √
3𝛽 𝑦 + 𝐷 𝑞𝑢 sin 𝑞𝑟 − 𝜔𝑡 . ≈ 𝐸 + (𝑘) + 𝑘ˆ ∥ ℏ𝑣 𝐹 2𝑎 (C.17) √ 𝑦 From Eq. (C.17), we read the Floquet gap Δ ≈ 𝑞𝑢 0 [𝐷 + 𝑘ˆ ∥ ℏ𝑣 𝐹 3𝛽/2𝑎] from the amplitude of the spatially- and time-periodic component of the Hamiltonian. Coherent Phonon Occupancy The Floquet theory used in the main text assumes that the coherent phonon occupancy 𝑛 is large. In this section, we show that the assumption is true for the phonon displacement amplitudes required to observe quantized acoustelectric Floquet effect. The coherent phonon amplitude, expressed in terms of the phonon occupation 𝑛, is given by √︄ 𝑢0 = 2 (𝑛 + 1/2)ℏ , 2𝑁 𝑀𝐶 𝜔 (C.18) where 𝑀𝐶 is the mass of a carbon atom and 𝑁 is the total number of graphene unit cells in the carbon nanotube. For a (10, 0) carbon nanotube of length on the order of 1 𝜇m, we estimate 𝑁 ≈ 105 . To estimate the minimum phonon occupancy necessary to achieve nearly-quantized currents, we refer to the inset of Figure 1(b) 135 C Figure C.1: Phonon transfer mechanism when a phonon source (grey block) is placed under the right end of the carbon nanotube [see Figure 1(a) in the main text for the complete, proposed experimental setup]. A carbon atom of mass 𝑀𝑐 (grey circle in magnified picture) experiences a Van der Waals force across the interface, with effective spring constant 𝑘. in the main text, which indicates that the Floquet gap at current saturation is given by Δ∗ ≈ 0.5ℏ𝜔, corresponding to the phonon amplitude [141] √ 𝑢 ∗0 ≈ 0.5ℏ𝑐 ph /[𝐷 + 0.866ℏ𝑣 𝐹 3𝜋/(2𝑎)]. (C.19) (See Section C.2 for a derivation.) Here, 𝐷 is the deformation potential, 𝑐 ph is the speed of sound, 𝑣 𝐹 = 106 m/s is the Fermi velocity, and 𝑎 = 0.246 nm is the graphene lattice constant. As an example, assuming 𝐷 ≈ 10 eV, ℏ𝜔 ≈ 10 meV, and 𝑐 ph = 25 km/s, we arrive at the phonon occupancy 𝑛 ∼ 36. Given 𝑛 ≫ 1, this occupation can provide multi-phonon absorption processes with an approximately constant amplitude. Coherent Phonon Lifetime To estimate the coherent phonon lifetime 𝜏, which limits the total length of the CNT, we note that the Floquet scattering rates shown in Figure C.2 saturate on the order tot ∼ 10 ps for relative permittivities 𝜖 > 500. The Floquet-Umklapp scattering 𝜏𝑘𝛼𝛼 ′ rates contribute to coherent phonon absorption and emission and are suppressed by a factor of (Δ/ℏ𝜔) 2 , occuring on timescales of ∼ 100 ps. Phonon-phonon and phonon-impurity interactions in graphene materials are also estimated to occur on timescales of ∼ 100 ps [18, 27, 94]. Thus, we estimate 𝜏 ∼ 100 ps. Proposals for Experimental Realizations of the Quantized Acoustoelectric Floquet Effect In this section, we investigate possible coherent phonon sources and their coupling to the CNT. There are several potential sources of coherent THz-frequency phonons in materials. Conventionally, THz-frequency phonons have been generated optically, 136 by a direct driving of an IR-active phonon mode [68], or by excitation of electrons that generate phonons through electron-phonon coupling [22, 67, 85, 109, 182, 274, 317]. Alternatively, acoustic phonons can be generated in strongly biased metals through the acoustic Cerenkov [10, 151, 213, 327], Klein-Zehner effects [5, 103], and through the phaser effect [35], which has been theoretically demonstrated in twisted bilayer graphene-based devices. These methods have been demonstrated to induce intense phonon waves. In particular, a medium with a positive phonon gain in an acoustic resonator is predicted to induce acoustic waves with amplitudes of the order of ∼ 0.01 nm, when reaching saturation [77]. In Figure 1(a) of the main text, we show a potential experimental setup, where a segment of the carbon nanotube extends to the right of the anti-reflective material. The coherent phonons can be directly generated by an optical illumination of the segment of CNT [92, 249, 250]. Alternatively, they can be generated in a different system, outside the CNT, and then coupled to the CNT. The latter is a more versatile approach that allows for on-demand coherent phonon generation by various methods. We estimate that the coherent phonon amplitudes in the CNT can reach the order of ∼ 0.01 nm when an acoustic resonator like the phaser is placed in contact with the CNT, provided that the phonon angular frequency does not exceed the natural resonance frequency set by the coupling between the source and the CNT. To estimate the resonance frequency, we consider the Van der Waals forces between the source and CNT, which can be modeled by a forced oscillation problem (see Figure C.1). For efficient transmission of the phonon wave into the CNT, the angular frequency 𝜔 of the coherent phonon should satisfy 𝜔 < 𝜔𝑟 , where the resonance frequency √︁ 𝜔𝑟 = 𝑘/𝑀𝐶 is set by the effective spring constant 𝑘 and the mass 𝑀𝐶 of the carbon atom. We estimate 𝑘 ≈ 2 N/m and ℏ𝜔𝑟 = 7 meV by utilizing the Lennard-Jones potential function calculated from first principles for Van der Waals forces between graphene layers [56]. The angular frequency presented in the main text satisfies 𝜔 < 𝜔𝑟 . Let us finally discuss the necessary conditions on the placement of the dielectric material. Because the characteristic separation between electrons in the tube, 2𝜋/𝑞 ∼ 100 , is much larger than the diameter of the CNT, the dielectric screens the entire CNT. To avoid interference with the coherent phonon wave, one can minimize phonon transfer between the dielectric and CNT by suspending the CNT above the dielectric a distance much larger than the equilibrium interatomic distance ∼ 3 set 137 by Van der Waals (VdW) potential between the materials [56]. C.3 Details on the Steady State Calculation We start by presenting the full expression for the Floquet-Boltzmann equation (FBE) discussed in main text and show that the FBE is valid under the conditions considered in our work. Floquet Boltzmann Equation In this section, we present the full expressions for the electron-phonon and electronelectron collision integrals, discretized on a 1D momentum grid of 𝑁 points. The electron-phonon collision integral is given by b 𝐼 𝑘𝛼 [{𝐹𝑘𝛼 }] = 1 2𝜋 1 ∑︁ ∑︁ ∑︁ ∑︁ 𝑘 ′ 𝛼′ |G𝑘𝛼 (𝑛)| 2 × ℏ 𝑁 𝑘 ′ ∈BZ 𝛼′ 𝑠 𝑛 ℏ𝑐 ph × ({−N (𝜀 𝑘 ′ 𝛼′ − 𝜀 𝑘𝛼 )𝐹𝑘𝛼 (1 − 𝐹𝑘 ′ 𝛼′ )𝛿(𝑘 ′ − 𝑘 + 𝑛𝑞 + 𝑘 𝑠 ) + [1 + N (𝜀 𝑘 ′ 𝛼′ − 𝜀 𝑘𝛼 )] (1 − 𝐹𝑘𝛼 )𝐹𝑘 ′ 𝛼′ 𝛿(𝑘 ′ − 𝑘 + 𝑛𝑞 − 𝑘 𝑠 )}𝜃 (𝜀 𝑘 ′ 𝛼′ − 𝜀 𝑘𝛼 ) + {−[1 + N (𝜀 𝑘 ′ 𝛼′ − 𝜀 𝑘𝛼 )]𝐹𝑘𝛼 (1 − 𝐹𝑘 ′ 𝛼′ )𝛿(𝑘 ′ − 𝑘 + 𝑛𝑞 − 𝑘 𝑠 ) + N (𝜀 𝑘 ′ 𝛼′ − 𝜀 𝑘𝛼 )(1 − 𝐹𝑘𝛼 )𝐹𝑘 ′ 𝛼′ 𝛿(𝑘 ′ − 𝑘 + 𝑛𝑞 + 𝑘 𝑠 )}𝜃 (𝜀 𝑘 ′ 𝛼′ − 𝜀 𝑘𝛼 ) (C.20) ∑︁ 1 𝐷ℏ𝑘 𝑠 𝑘 ′ 𝛼′ ′ 𝑚 ′ ⟨𝜙𝑛+𝑚 G𝑘𝛼 (𝑛) = √ √︁ 𝑘 ′ 𝛼′ |+, 𝑘 ⟩W𝑘,𝑘 −𝑘 ⟨+, 𝑘 |𝜙 𝑘𝛼 ⟩ 𝐴 2𝜌ℏ|𝜀 𝑘 ′ 𝛼′ − 𝜀 𝑘𝛼 | 𝑚 (C.21) where 𝜌 = 1.52 × 10−6 kg/m2 is the 2D density of the graphene layers, 𝐷 is the deformation potential, 𝑘 𝑠 satisfies ℏ𝑐 ph |𝑠𝒌 ⊥ + 𝑘 𝑠 𝒌ˆ ∥ | = 𝜀 𝑘 ′ 𝛼′ − 𝜀 𝑘𝛼 , and N (𝜀) = 1/(𝑒 𝜀/𝑘 𝐵𝑇 − 1) is the Bose-Einstein distribution for incoherent phonons maintained in thermal equilibrium at temperature 𝑇. The electron-electron collision integral is given by ee 𝐼 𝑘𝛼 [{𝐹𝑘𝛼 }] = 4𝜋 1 ∑︁ ∑︁ ∑︁ ∑︁ (𝑘 3 ,𝛼3 ),(𝑘 1 +𝑘 2 −𝑘 3 ,𝛼4 ) (𝑛)| 2 × |V(𝑘,𝛼),(𝑘 ,𝛼 ) 2 2 ℏ 𝑁 2 𝑘 ∈BZ 𝑘 ∈BZ 𝛼 ,𝛼 ,𝛼 𝑛 2 3 2 3 4 × 𝛿(𝜀 𝑘𝛼 + 𝜀 𝑘 2 𝛼2 − 𝜀 𝑘 3 𝛼3 − 𝜀 𝑘+𝑘 2 −𝑘 3 ,𝛼4 + 𝑛ℏΩ)×   × (1 − 𝐹𝑘𝛼 )(1 − 𝐹𝑘 2 𝛼2 )𝐹𝑘 3 𝛼3 𝐹𝑘 1 +𝑘 2 −𝑘 3 ,𝛼4 − 𝐹𝑘𝛼 𝐹𝑘 2 𝛼2 (1 − 𝐹𝑘 3 𝛼3 )(1 − 𝐹𝑘 1 +𝑘 2 −𝑘 3 ,𝛼4 ) (C.22) 138 (𝑘 3 ,𝛼3 ),(𝑘 1 +𝑘 2 −𝑘 3 ,𝛼4 ) (𝑛) = V(𝑘,𝛼),(𝑘 ,𝛼 ) ∑︁ 2 2 2 +𝑛3 +𝑛4 𝑉 (𝑘 2 − 𝑘 3 )W𝑘 1 ,𝑘 3 −𝑘 2 W𝑘 2 ,−(𝑘 3 −𝑘 2 ) ⟨𝜙𝑛−𝑛 |+, 𝑘⟩⟨𝜙𝑛𝑘22 𝛼2 |+, 𝑘 2 ⟩× (C.23) 𝑘𝛼 𝑛2 ,𝑛3 ,𝑛4 4 × ⟨+, 𝑘 3 |𝜙𝑛𝑘33 𝛼3 ⟩⟨+, 𝑘 4 |𝜙𝑛𝑘+𝑘 ⟩. 2 −𝑘 3 ,𝛼4 To solve for the steady-state, we use the Newton-Raphson algorithm to find the roots 𝜕𝑡 𝐹𝑘𝛼 = 0 of the FBE. We set the doping of the system by adding the Lagrange Í multiplier term 𝜆( 𝑘𝛼 𝐹𝑘𝛼 − 𝑁 𝑘 𝐹 /𝑞) with large constant 𝜆 to the FBE. Validity of Floquet Boltzmann Equation and Steady State Scattering Times ph el The interband scattering rates 1/𝜏𝑘𝛼𝛼′ and 1/𝜏𝑘𝛼𝛼 ′ are given by 1 ph = 𝜏𝑘𝛼𝛼′ 2𝜋 1 ∑︁ ∑︁ ∑︁ 𝑘 ′ 𝛼′ 1 × |G𝑘𝛼 (𝑛)| 2 ℏ 𝑁 𝑘 ′ ∈BZ 𝑠 𝑛 ℏ𝑐 ph × {N (𝜀 𝑘 ′ 𝛼′ − 𝜀 𝑘𝛼 )(1 − 𝐹𝑘 ′ 𝛼′ )𝛿(𝑘 ′ − 𝑘 + 𝑛𝑞 + 𝑘 𝑠 )+ (C.24) + [1 + N (𝜀 𝑘 ′ 𝛼′ − 𝜀 𝑘𝛼 )] (1 − 𝐹𝑘 ′ 𝛼′ )𝛿(𝑘 ′ − 𝑘 + 𝑛𝑞 − 𝑘 𝑠 )} and 1 el 𝜏𝑘𝛼𝛼 ′ = 4𝜋 1 ∑︁ ∑︁ ℏ 𝑁 2 𝑘 ∈BZ 𝑘 ∈BZ 2 3 ∑︁ ∑︁ ∑︁ 𝛼2 ,𝛼3 ,𝛼4 𝑛 𝛼3 =𝛼′ or 𝛼4 =𝛼′ 𝐺 (𝑘 3 ,𝛼3 ),(𝑘 1 +𝑘 2 −𝑘 3 ,𝛼4 ) |V(𝑘,𝛼),(𝑘 (𝑛, 𝐺)| 2 × ,𝛼 ) 2 2 (C.25) × 𝛿(𝜀 𝑘𝛼 + 𝜀 𝑘 2 𝛼2 − 𝜀 𝑘 3 𝛼3 − 𝜀 𝑘+𝑘 2 −𝑘 3 ,𝛼4 + 𝑛ℏΩ)× × 𝐹𝑘 2 𝛼2 (1 − 𝐹𝑘 3 𝛼3 )(1 − 𝐹𝑘 1 +𝑘 2 −𝑘 3 ,𝛼4 ). To verify that the steady state coherences are suppressed and that the Floquet Boltztot |𝜀 mann equation is valid, we check that 𝜁 ≪ 1, where 𝜁 ≡ max 𝑘,𝛼,𝛼′ ℏ/(𝜏𝑘𝛼𝛼 ′ 𝑘𝛼 − ph tot el 𝜀 𝑘𝛼′ |), where 1/𝜏𝑘𝛼𝛼′ ≡ 1/𝜏𝑘𝛼𝛼′ + 1/𝜏𝑘𝛼𝛼′ [see full definition in the main text]. Figure C.2 verifies that 𝜁 ≪ 1 as a function of drive amplitude. C.4 Effective Description Using Lindblad Master Equation In the present and following sections, we analyze the quantized acoustoelectric Floquet effect using a toy model that can be analytically solved. The purpose of the toy model is to provide an intuitive and simple picture of the key conditions required to observe the quantized acoustoelectric Floquet effect. We demonstrate that the effect is analogous to a Thouless pump driven non-adiabatically, i.e. Δ < ℏ𝜔, with quantized transport stabilized by an incoherent energy relaxation mechanism, which in the microscopic model arises from phonon cooling [see Figure C.3]. In 139 100 500 2000 Figure C.2: The ratio 𝜁 of the maximum interband scattering rate to the Floquet band energy separation vs. Floquet gap Δ for various dielectric constants 𝜖. Note that 𝜁 ≪ 1, so steady state coherences are suppressed. particular, incoherent phonons relax electrons into the troughs of the moving periodic potential 𝑉 (𝑥, 𝑡), which is induced by the coherent phonon wave. This process is dominant when the relaxation time of incoherent phonon scattering, 𝜏ep , is much faster than that of heating processes, 𝜏ℎ . Importantly, we stress that the Lindblad master equation analysis used in this section is distinct from the full Floquet theory and Floquet-Boltzmann equation approach presented in the main text and previous supplementary materials sections. Effective Hamiltonian Let us first derive an effective Hamiltonian for the driven carbon nanotube. In the limit where coupling strength of the coherent electrons via the vector potential (𝜆) Â ph (𝒓, 𝑡) is much smaller than the separation between the lowest conduction and highest valence bands, i.e., ℏ𝑣 𝐹 𝛽𝑎 −1 ≪ ℏ𝑣 𝐹 𝛿𝑘 (see definitions of 𝛿𝑘 and 𝛽 in Section
C.1) we can approximate 𝑉 ( 𝒓ˆ , 𝑡) ≈ Δ cos 𝑞 𝑥ˆ − 𝜔𝑡 , where 𝑉 ( 𝒓ˆ , 𝑡) ≡ ⟨𝑉ˆ (𝜆0 ) ( 𝒓ˆ , 𝑡)⟩, and 𝒓ˆ = 𝑥ˆ 𝒌ˆ ∥ is the spatial position along the tube axis of the CNT (see Section C.2 for the derivation). We also approximate 𝐸 + (𝑘) ≈ ℏ2 𝑘 2 /(2𝑚) + ℏ𝑣 𝐹 𝛿𝑘 where 𝑚 is the effective mass at the band bottom [see Figure 1(c) in the main text]. Up to a constant energy offset, the resulting effective Hamiltonian is given by
𝐻ˆ (𝑡) = 𝐻ˆ 0 + Δ cos 𝑞 𝑥ˆ − 𝜔𝑡 , (C.26) where 𝐻ˆ 0 = ℏ2 𝑘ˆ 2 /2𝑚, 𝑥ˆ = −𝑖𝜕𝑘ˆ is the position operator, and 𝑘ˆ is the momentum operator. To analyze the problem analytically, we work in the eigenbasis |𝑘⟩ of the free-propagation Hamiltonian 𝐻ˆ 0 . Using the relation 𝑒𝑖𝑞𝑥ˆ |𝑘⟩ = |𝑘 + 𝑞⟩, and considering the limit ℏ2 𝑞 2 /(2𝑚) ≫ Δ, we obtain the approximate matrix form of 140 x Figure C.3: Simplified model of the quantized acoustoelectric Floquet effect. The black curve represents the moving coherent phonon potential 𝑉 (𝑥, 𝑡), and the grey dashed curve sketches the potential at a slightly later time. Electrons (green dots) generate quantized current when confined to the troughs of the potential, which is achieved when the incoherent phonon relaxation time 𝜏ep is much faster than the heating time 𝜏h . 𝐻ˆ (𝑡), ! Δ𝑒𝑖𝜔𝑡 /2 𝐻 (𝑡) ≈ 𝜀¯ 𝑘 + , Δ𝑒 −𝑖𝜔𝑡 /2 −𝛿 𝑘 𝛿𝑘 (C.27) where 𝜀¯ 𝑘 = (𝜀 𝑘 + 𝜀 𝑘−𝑞 )/2, 𝛿 𝑘 = (𝜀 𝑘 − 𝜀 𝑘−𝑞 )/2, and 𝜀 𝑘 = ℏ2 𝑘 2 /(2𝑚). Lindblad Master Equation Our goal is to compute the time-averaged current, given by   ∫ ∫ 𝑞 𝜕𝐻 (𝑡) 2𝑒 1 𝑇 𝑑𝑘 Tr 𝜌(𝑘, 𝑡) 𝑑𝑡 𝐽= ℏ 𝑇 0 𝜕𝑘 0 2𝜋 (C.28) where the factor of two accounts for spin degeneracy, 𝑇 = 2𝜋/𝜔, and 𝜌(𝑘, 𝑡) is the density matrix for an electron with crystal momentum 𝑘. The actual value of 𝜌(𝑘, 𝑡) is controlled by the Hamiltonian 𝐻 (𝑡), collisions with incoherent phonon modes, and electron-electron interactions. We approximate the relaxation dynamics of the electrons in the CNT due to scattering processes by the Lindbladian L{𝜌(𝑘, 𝑡)} ≡ − 𝜌(𝑘, 𝑡) − 𝜌 eq (𝑘, 𝑡) 𝜌(𝑘, 𝑡) − 𝜌 𝐹 (𝑘, 𝑡) − 𝜏h 𝜏ep (C.29) with two characteristic relaxation times 𝜏h and 𝜏ep . Here, heating processes, such as electron-electron interactions, relax electrons into a thermal equilibrium state in the instantaneous basis of 𝐻 (𝑡), denoted by 𝜌 eq (𝑘, 𝑡). Separately, incoherent phonon scattering processes relax electrons into the Floquet ground state 𝜌 𝐹 (𝑘, 𝑡), representing a fully-occupied lowest-energy band of the Floquet Hamiltonian 𝐻 𝐹 ≡ 𝐻 (𝑡) − 𝑖ℏ𝜕𝑡 . Eq. (C.29) captures the key microscopic scattering processes analyzed in the main text. Under these approximations, the time evolution of the density 141 matrix can be described by the master equation [186, 253] 𝑖 𝜌(𝑘, ¤ 𝑡) = [𝜌(𝑘, 𝑡), 𝐻 (𝑡)] + L{𝜌(𝑘, 𝑡)}. ℏ (C.30) To solve the Lindblad master equation analytically, we utilize the time-independent basis, where the Hamiltonian is given by ! 𝛿 Δ/2 𝑘 𝐻˜ = 𝑅(𝑡)𝐻 (𝑡)𝑅(𝑡) † = 𝜀¯ 𝑘 + . (C.31) Δ/2 −𝛿 𝑘 Here, 𝑒 −𝑖𝜔𝑡/2 0 𝑅(𝑡) = 0 𝑒𝑖𝜔𝑡/2 ! (C.32) is the transformation matrix. In the time-independent basis, the master equation is given by ¤̃ 𝑡) = 𝑖 [ 𝜌(𝑘, ˜ 𝑡), 𝐻˜ + 𝐵] + L̃{ 𝜌(𝑘, ˜ 𝑡)} (C.33) 𝜌(𝑘, ℏ † = (ℏ𝜔/2)𝜎 , 𝜌(𝑘, ¤ where 𝐵 = 𝑖 𝑅(𝑡)𝑅(𝑡) 𝑡) = 𝑅(𝑡) 𝜌(𝑘, 𝑡)𝑅(𝑡) † , and L̃{ 𝜌(𝑘, ˜ 𝑡)} = 𝑧 ˜ † 𝑅(𝑡)L{𝜌(𝑘, 𝑡)}𝑅(𝑡) . In the instantaneous basis, we also define 𝜌˜ eq (𝑘) ≡ 𝑅(𝑡) 𝜌 eq (𝑘, 𝑡)𝑅(𝑡) † , (C.34) 𝜌˜ 𝐹 (𝑘) ≡ 𝑅(𝑡) 𝜌 𝐹 (𝑘, 𝑡)𝑅(𝑡) † . (C.35) and Thermal Distribution in the Instantaneous Eigenbasis The instantaneous thermal distribution 𝜌 eq (𝑘, 𝑡) is diagonal in the instantaneous eigenbasis. In this section, we transfer 𝜌 eq (𝑘, 𝑡) into the time-independent basis to determine 𝜌˜ eq (𝑘). We use the operator 𝑈 to transform into the instantaneous eigenbasis of 𝐻 (𝑡), where ! 𝐼 (𝑘) 𝜀 0 ˜ †= − H = 𝑈 𝐻𝑈 (C.36) 0 𝜀+𝐼 (𝑘) √︃ and 𝜀±𝐼 (𝑘) ≡ 𝜀¯ 𝑘 ± 𝛿2𝑘 + (Δ/2) 2 are the instantaneous eigenenergies. It follows that the density matrix corresponding to the thermal distribution in the instantaneous basis is given by ! 𝐼 1 − 𝑓 0 𝑘 𝜌˜ eq (𝑘) = 𝑈 † 𝑈 0 𝑓 𝑘𝐼 (C.37)   1 − 2 𝑓 𝑘𝐼 1 Δ = + √︁ 𝜎𝑥 + 𝛿 𝑘 𝜎𝑧 . 2 2 𝛿 𝑘 + (Δ/2) 2 2 where 𝑓 𝑘𝐼 is the Fermi-Dirac occupation of the electronic state with energy 𝜀−𝐼 (𝑘). 142 Floquet Ground State The Floquet ground state distribution 𝜌 𝐹 (𝑘, 𝑡) is diagonal in the Floquet basis. In this section, we transfer 𝜌 𝐹 (𝑘, 𝑡) into the time-independent basis by determining 𝜌˜ 𝐹 (𝑘). To find the Floquet eigenenergies, we consider the diagonal basis of the Floquet Hamiltonian 𝐻˜ 𝐹 = 𝑅(𝑡)𝐻 𝐹 𝑅(𝑡) † = 𝐻˜ + 𝐵, which can be obtained using the transformation operator 𝑈𝐹 , where ! 𝐹 (𝑘) 𝜀 0 H𝐹 = 𝑈𝐹 𝐻˜ 𝐹 𝑈𝐹† = − (C.38) 0 𝜀+𝐹 (𝑘) √︁ and 𝜀±𝐹 (𝑘) = 𝜀¯ 𝑘 ± (𝛿 𝑘 + ℏ𝜔/2) 2 + (Δ/2) 2 are the quasienergies. The density matrix corresponding to a fully-occupied Floquet ground state is given by ! 𝐹 1 − 𝑓 0 𝑘 𝜌˜ 𝐹 (𝑘) = 𝑈𝐹† 𝑈 0 𝑓 𝑘𝐹 " # (C.39)  𝐹 1 − 2 𝑓 1 Δ + ℏ𝜔 𝑘 = + √︁ 𝜎𝑥 + 𝛿 𝑘 𝜎𝑧 . 2 2 (𝛿 𝑘 + ℏ𝜔/2) 2 + (Δ/2) 2 2 where 𝑓 𝑘𝐹 is the occupation function of the electronic state with quasienergy 𝜀−𝐹 (𝑘). Steady State Solution To calculate the current in the steady state, we solve for the steady state density matrix in the time-independent basis, denoted 𝜌(𝑘), ˜ which satisfies the steady state condition 𝑖 [ 𝜌(𝑘), ˜ 𝐻˜ + 𝐵] + L̃{ 𝜌(𝑘)} ˜ =0 (C.40) ℏ [see Eq. (C.33)]. It can be shown that the steady state solution is given by 1 1 𝐼 + ( 𝜌˜ 𝑥 𝜎𝑥 + 𝜌˜ 𝑦 𝜎𝑦 + 𝜌˜ 𝑧 𝜎𝑧 ), 2 2 (C.41) 𝜂 + 4𝜂𝛽2 − 4𝜁 𝛽2 [𝛼𝛾(𝑥 − 1/2) + 1/2] , 1 + 4𝛽2 {1 + [𝛾(𝑥 − 1/2) + 𝛼−1 /2]} (C.42) 2𝛽{𝜂[𝛾(𝑥 − 1/2) + 𝛼−1 /2] − 𝜁 } , 1 + 4𝛽2 {1 + [𝛾(𝑥 − 1/2) − 𝛼−1 /2]} (C.43) 𝜌(𝑘) ˜ = where 𝜌𝑥 = 𝜌𝑦 = 𝜌𝑧 = 4𝜂2 [𝛾(𝑥 − 1/2) + 𝛼−1 /2] + 𝜁 {1 + 4𝛽2 [𝛾(𝑥 − 1/2) + 𝛼−1 /2] 2 } , 1 + 4𝛽2 {1 + [𝛾(𝑥 − 1/2) + 𝛼−1 /2] 2 } (C.44) 143 (a) (b) 0 0.2 0.4 0.6 0.8 1 Figure C.4: Steady state current 𝐽 as calculated from the Lindblad master equation [Eq. (C.33)]. (a) Steady state current in the regime ℏ/( 𝜏Δ) ¯ ≪ 1 analyzed in the main text using the Floquet Boltzmann equation. The current is quantized when
the system
is driven adiabatically, i.e., log ℏ𝜔/Δ < 0. In the non-adiabatic regime log ℏ𝜔/Δ > 0, the current attains a quantized value in the regime where incoherent phonon relaxation dominates 𝜏ep ≪ 𝜏h . (b) Steady state current for a weak driving potential amplitude Δ satisfying ( 𝜏Δ)/ℏ ¯ = 0.3. In contrast to
panel (a), the current deviates from quantization in the adiabatic regime log ℏ𝜔/Δ < 0, and instead only attains a quantized value when 𝜏ep ≪ 𝜏h . (1 − 2 𝑓 𝑘𝐹 ) [𝛾(𝑥 − 1/2) + 1/2] sign(𝑥 − 1/2) + 𝜈 𝜒 √︁ , 𝜂 = 𝜈 √︁ 1 + 𝛾 −2 (𝑥 − 1/2) −2 [𝛾(𝑥 − 1/2)𝛼 + 1/2] 2 + 𝛼2 (C.45) and (1 − 2 𝑓 𝑘𝐼 ) 𝜁 = 𝜈 √︁ (1 − 2 𝑓 𝑘𝐹 ) [𝛾(𝑥 − 1/2) + 1/2] + 𝜈 𝜒 √︁ . 1 + 𝛾 2 (𝑥 − 1/2) 2 [𝛾(𝑥 − 1/2) + 𝛼−1 /2] 2 + 1 (1 − 2 𝑓 𝑘𝐼 ) (C.46) Here, 𝑥 = 𝑘/𝑞, 𝛼 = Δ/(2ℏ𝜔), 𝛽 = 𝜏Δ/(2ℏ), ¯ 𝛾 = 2[𝑞 2 /(2𝑚)]/Δ, 𝜈 = 𝜏/𝜏 ¯ h, −1 𝜒 = 𝜏h /𝜏ep , and 𝜏¯ = (1/𝜏ep + 1/𝜏h ) . Using Eq. (C.28), we find that the time averaged current is given by ∫ 𝑞 𝐽= 0 𝑑𝑘 𝑒𝜔 (𝜕𝑘 𝛿 𝑘 ) 𝜌 𝑧 = 𝛾𝛼 𝜋 𝜋 ∫ 1 𝑑𝑥 𝜌 𝑧 . (C.47) 0 The integral in Eq. (C.47) can be calculated numerically. Figure C.4 shows the steady state current at several representative values of ℏ/( 𝜏Δ). ¯ We choose a large 𝛾 = 104 to ensure the approximation used in Eq. (C.27) is accurate, 144 and we assume that the system relaxes to zero-temperature thermal distributions in the Floquet and instantaneous bases, i.e., 𝑓 𝑘𝐼 = 𝑓 𝑘𝐹 = 1. In Figure C.4(a), we analyze the regime ℏ/( 𝜏Δ) ¯ ≪ 1 considered in the main text. Consistent with the analysis of the Floquet-Boltzmann equation, the current is quantized in the non
adiabatic regime log ℏ𝜔/Δ > 0 when incoherent phonon relaxation dominates
𝜏ep ≪ 𝜏h . When the system is driven adiabatically, i.e., log ℏ𝜔/Δ < 0, the current is quantized regardless of the strength of incoherent phonon coupling, consistent with the picture of a Thouless pump. In Figure C.4(b), we show the current for a weak driving potential Δ satisfying ( 𝜏Δ)/ℏ ¯ = 0.3. In contrast to panel (a), the system only exhibits quantized quantized current in the adiabatic regime when incoherent phonon relaxation dominates 𝜏ep ≪ 𝜏h . In conclusion, our effective description reproduces both the quantized acoustoelectric Floquet effect for non-adiabatic phonon drives and the Thouless pumping regime for adiabatic phonon drives. In the the adiabatic regime, given by ℏ𝜔 ≪ Δ and ℏ/𝜏 ≪ Δ, the current is quantized regardless of the rate of incoherent phonon relaxation [see Figure C.4(a)]. Such a regime has been realized experimentally using GHz-frequency surface acoustic phonon waves, which have been observed to host a quantized acousto-electric current response [7, 29, 71, 82, 159, 278]. In contrast, the quantized acoustoelectric Floquet effect relies on incoherent phonon relaxation to realize quantized transport in the non-adiabatic regime where ℏ𝜔 > Δ or ℏ/𝜏 ≳ Δ [see Figs. C.4(a-b)]. 145 Chapter 8 CHIRAL PLASMONS AND TOPOLOGICAL PHOTOCURRENT IN WEYL SEMIMETALS We analyze a realizaton of Thouless’ adiabatic charge pump in a Weyl semimetal (WSM) irradiated by circularly polarized light, first reported in Ref. [55]. We show the effect can lead to a strong and highly nonlinear topological photocurrent along the light propagation axis, which is insensitive to anisotropy, lattice orientation, and interactions. The effect can emerge in non-centrosymmetric WSMs where Weyl nodes of opposite chiralities are isolated and offset in energy, such as, potentially, SrSi2 . We estimate the current can reach 10 A/mm2 for THz frequencies at 106 V/m amplitudes. We discuss the possibility of further amplification due to coinciding plasmonic resonances in the THz-low infrared regime. We also report that WSMs with broken time reversal and inversion symmetry may support two nontrivial response phenomena: namely, a helicity-independent photocurrent, and a mechanism for chiral plasmon modes, whose frequencies depend on their helicities relative to an anisotropy axis of the material. The phenomena we uncover may find uses in sensing, photovoltaics, and electronics. 8.1 Introduction The linear dispersion and unique band topology of Weyl semimetals (WSMs) translates to exotic electronic responses [12, 14, 31, 38, 43, 58, 102, 121, 121, 150, 170– 172, 197, 199, 205, 207, 218, 230, 237, 285, 310, 319, 323, 328], with promising applications in optics [9, 44, 122, 150, 156, 194, 198, 219, 283, 305, 311], magnetism [11, 66, 161, 202, 207, 208, 252, 267, 268, 276, 299, 324], and electronics [13, 30, 98, 107, 112, 168, 190, 217, 227, 228, 265, 269, 272, 288, 300, 309, 318, 321]. These effects can arise both from interband photoabsorption, such as the circular photogalvanic effect [58, 102, 121, 121, 172, 197, 218, 230, 237, 323], and from intraband dynamics, such as topological frequency conversion [205], quantum nonlinear Hall effects [264], photovoltaic chiral magnetic effects [51, 132, 277], and shift current-induced bulk photovoltaic effects [218]. Here we study the intra-band dynamics of WSMs driven by circularly-polarized light in the near-adiabatic regime, with frequency in the THz range, and less than the momentum-dependent gap [see Figure 8.1(a)]. In noncentrosymmetric WSMs, 146 (a) (b) (c) z x (d) y + - + + - + + - + (e) + - + - Figure 8.1: Light-induced charge pumping and chiral plasmons. (a) We consider the optical response of a noncentrosymmetric WSMs, where opposite-chirality Weyl nodes (WNs) can have distinct effective chemical potentials (𝜇± ) and tilts (𝜒± ). We show that circularly polarized driving in the near-adiabiatic regime can induce a topological charge pumping effect leading to a strong photocurrent 𝑗¯𝜂𝑧 along the beam axis, where 𝜂 =⟲, ⟳ denotes the helicity of the laser. (b) Numerically 𝑧 computed photocurrent, 𝑗¯⟲ , for 𝜇− = 15 m𝑒𝑉, 𝜇+ = 30 m𝑒𝑉, and ℎ 𝑓 = 5 meV, 𝑧 with 𝜒± = ±𝜒. (c) Maximal 𝑗¯𝜂𝑧 , denoted 𝑗 max , attained numerically (scatter points) and predicted semiclassically (dashed line) [Eq. (8.7)]. Discrepancies arise as ℎ 𝑓 is raised beyond the adiabatic limit. (d) Plasmon enhancement factor 𝐺 𝜂 ≡ 𝐸/𝐸 ext vs ℎ 𝑓 and 𝐸, where 𝐸 ext (𝐸) denotes the drive-field amplitude outside (inside) the WSM. Dashed curve indicates plasmon resonance predicted semiclassically. (e) Same as (d), but for a WSM with broken TRS and a pair of WNs with opposite chirality and tilt. Here, the chiral plasmon resonance differs for left (left panel) and right (right panel) circular polarizations. we show these conditions induce Thouless’ topological charge pumping that generates a strong topological photocurrent along the propagation of light [47, 279]. The photocurrent is insensitive to lattice orientation or anisotropy in the WSM [see Figure 8.1(b)] and can therefore arise even in systems with many domains of crystal orientation. It is pronounced in WSMs where Weyl nodes (WNs) of opposite chiralities are isolated and well-separated in energy [see Eq. (8.7)], such as, potentially, SrSi2 [113]. When the WNs are tilted and time-reversal symmetry (TRS) is broken, the WSM can further support a photocurrent component independent of laser helicity. Interestingly, the THz driving coincides with the plasma resonance frequency range of WSMs [43, 322], enabling significant amplification of the photocurrents at moderate driving intensities [Figure 8.1(c)]. As a second result, when TRS is bro- 147 ken, we show the plasmon frequency becomes chiral, dependent on the alignment of the laser chirality with the anisotropy axis. 8.2 System Let us start with the photoresponse of electrons near a single Weyl node (WN). Focusing on the conduction and valence band, the Hamiltonian near a WN reads 𝐻 ( 𝒌) = 𝜉ℏ𝑣 𝐹 𝒌 · 𝝈 + ℏ𝑽 · 𝒌 [12]. Here, 𝝈 = (𝜎𝑥 , 𝜎𝑦 , 𝜎𝑧 ) denotes the vector of Pauli matrices acting in the orbital basis, and the crystal momentum 𝒌 = (𝑘 𝑥 , 𝑘 𝑦 , 𝑘 𝑧 ) is measured from the WN located at 𝒌 = 0. The Fermi velocity 𝑣 𝐹 controls the gap size of the energy dispersion away from the WN, while 𝑽 parameterizes a tilt in the WN dispersion. Finally, 𝜉 = +, − defines the chirality of the WN. Below, 𝜀 𝒌𝜈 and |𝜓 𝒌𝜈 ⟩ respectively denote the single-particle energies and eigenstates of 𝐻 ( 𝒌), where 𝜈 = 0 (𝜈 = 1) indicate the conduction (valence) band. The WN provides a point source of the Berry curvature of the bands, defined as \Ω𝜈 ( 𝒌) ≡ (2𝜋) −1 ∇ 𝒌 × ⟨𝜓 𝒌𝜈 |𝑖∇ 𝒌 |𝜓 𝒌𝜈 ⟩, where ∇ 𝒌 indicates the gradient with respect to 𝒌. Specifically, ∇ · \Ω𝜈 ( 𝒌) = 2𝜋𝜉 (−1) 𝜈 𝛿( 𝒌), where 𝛿( 𝒌) denotes the Dirac delta function. Nonzero Berry-curvature divergence is the hallmark of WSMs. The system is illuminated by coherent circularly-polarized light traveling along the 𝑧-direction with frequency 𝑓 and polarization 𝜂, which can take value ⟲ or ⟳, corresponding to left or right polarization, respectively. The time-dependent

vector potential inside the material is then 𝑨(𝑡) = 𝐴(cos 2𝜋 𝑓 𝑡 , 𝑐 𝜂 sin 2𝜋 𝑓 𝑡 , 0), where 𝑐 ⟲ = +1 and 𝑐 ⟳ = −1, with the corresponding internal electric field given by 𝑬 (𝑡) = 𝜕𝑡 𝑨(𝑡). The electrons couple to the light via 𝒌 → 𝒌 + 𝑒 𝑨(𝑡)/ℏ, where 𝑒 is the electron charge, resulting in the time-periodic Hamiltonian 𝐻 ( 𝒌, 𝑡) = 𝐻 ( 𝒌 +𝑒 𝑨(𝑡)/ℏ). We are interested in the photocurrent density in the system, denoted ∫𝑡 𝒋 𝜂 (𝑡), and its time-averaged value ¯𝒋 𝜂 ≡ lim𝑡→∞ 1𝑡 0 𝑑𝑡 ′ 𝒋 𝜂 (𝑡 ′) in the adiabatic regime given by ℎ 𝑓 ≪ 𝑣 𝐹 𝑒 𝐴/ℏ. This expression is derived from the adiabatic conditions ℏ||𝜕𝑡 𝐻ˆ ( 𝒌, 𝑡)|| ≪ 𝛿𝜀 2 ( 𝒌, 𝑡) and ℏ𝜔 ≪ 𝛿𝜀 2 ( 𝒌, 𝑡), where ||·|| denotes the operator spectral norm and 𝛿𝜀( 𝒌, 𝑡) ≡ 𝜀1 ( 𝒌, 𝑡) − 𝜀0 ( 𝒌, 𝑡) denotes the spectral gap. 8.3 Origin of Topological Response To elucidate the photoresponse of an isolated WN in the adiabatic regime [55], we concentrate on the subset of electron modes with 𝑥𝑦-plane momentum 𝒌 ⊥ = (𝑘 𝑥 , 𝑘 𝑦 ), represented by the vertical yellow line in Figure 8.2(a). These modes can be viewed as a 1D Fermionic chain aligned in the 𝑧-direction, whose Bloch Hamiltonian 𝐻 ( 𝒌, 𝑡) is controlled by 𝒌 ⊥ . For now, we fix 𝜇 = 0, and assume that the bands of this chain 148 are fully gapped over a full driving cycle, such that they do not intersect the WN. The light’s vector potential drives the Fermionic chain along a closed loop, indicated by the yellow arrow in Figure 8.2(a). This adiabatic trajectory produces a Thouless charge pump [201], which transports one electron per cycle in the 𝑧-direction: ∫ ∫ I𝜂𝑧 ( 𝒌 ⊥ ) = −𝑐 𝜂 𝜉C𝒌 ⊥ 𝑒 𝑓 . Here C𝒌 ⊥ ≡ (2𝜋) −1 S 𝑑𝑺 · 𝛀0 ( 𝒌), where S 𝑑𝑺 denotes the oriented surface integral on the closed cylinder S defined by 𝒌 ′ (𝑡) ≡ 𝒌 + 𝑒 𝑨(𝑡)/ℏ for 𝑡 ∈ [0, 𝑇] and 𝒌 ⊥ fixed [see Figure 8.2(a)]. Stoke’s theorem implies that C𝒌 ⊥ is given by the net charge of WNs inside S, leading to C𝒌 ⊥ = 𝜉𝜃 (|𝑒 𝐴/ℏ| − | 𝒌 ⊥ |), with 𝜃 the step function. We verify this quantization in simulations, see Figure 8.2(b). Summing all cylindrical trajectories that enclose the WN, the average current density is the quantized current −𝑒 𝑓 times the reciprocal space area formed by the disk of radius 𝑒 𝐴/ℏ. This results in 𝑗¯𝜂𝑧 = 𝑐 𝜂 𝜉 𝑗 t𝑜 𝑝 , where  2 𝑒 𝑓 𝑒𝐴 . 𝑗t𝑜 𝑝 = − 4𝜋 ℏ (8.1) This topological photoresponse is unique to WSMs. 8.4 Nonzero Filling Realistically, we must consider a nonzero chemical potential. To this end, we consider the reduced density operator of the electronic many-body state, 𝜌( ˆ 𝒌, 𝑡), obtained by tracing over all electronic modes with crystal momenta distinct from 𝒌. This operator acts on a 4-dimensional Fock space of the 2 Bloch space orbitals 𝑠 =↓, ↑, spanned by the states |𝑛↑, 𝑛↓⟩ where 𝑛 𝑠 = 0, 1 gives the occupation number in Bloch state |𝑠⟩. The photocurrent is found from 𝜌( ˆ 𝒌, 𝑡) via ∫ 𝑑3 𝒌 Tr[ 𝜌( ˆ 𝒌, 𝑡)ℏ−1 ∇ 𝒌 𝐻ˆ ( 𝒌, 𝑡)], (8.2) 𝒋 𝜂 (𝑡) = 𝑒 (2𝜋) 3 where 𝐻ˆ ( 𝒌, 𝑡) = ( 𝑐ˆ↑† , 𝑐ˆ↓† )𝐻 ( 𝒌, 𝑡)( 𝑐ˆ↑, 𝑐ˆ↓)𝑇 denotes the Hamiltonian acting on the Fock space, with 𝑐ˆ†𝑠 creating a fermion in the orbital 𝑠. In addition to the coherent evolution generated by 𝐻ˆ ( 𝒌, 𝑡), 𝜌( ˆ 𝒌, 𝑡) is affected by incoherent processes such as electron-electron and electron-phonon collisions. We model these phenomenologically, through the master equation 𝜕𝑡 𝜌( ˆ 𝒌, 𝑡) = 𝑖 1 [ 𝜌( ˆ 𝒌, 𝑡), 𝐻ˆ ( 𝒌, 𝑡)] − [ 𝜌( ˆ 𝒌, 𝑡) − 𝜌ˆ eq ( 𝒌, 𝑡)]. ℏ 𝜏 (8.3) Here, we assume that electrons relax to a thermal state in the instantaneous eigenbasis of 𝐻ˆ ( 𝒌, 𝑡) (this is physically justified when the correlation time of the environment 149 is fast relative to the driving frequency [203]), ˆ 𝜌ˆ eq ( 𝒌, 𝑡) = 𝑒 −[ 𝐻 ( 𝒌,𝑡)−𝜇 ˆ Í Í † 𝑠 𝑐ˆ𝑠 𝑐ˆ𝑠 ]/𝑘 𝐵 𝑇 /N (8.4) † where N = Tr[𝑒 −[ 𝐻 ( 𝒌,𝑡)−𝜇 𝑠 𝑐ˆ𝑠 𝑐ˆ𝑠 ]/𝑘 𝐵𝑇 ], 𝜇 is the chemical potential, and 𝑇 the lattice temperature. We assume a uniform phenomenonlogical relaxation time 𝜏. The explicit dissipator form is not essential, and chosen for simplicity. In the adiabatic regime, ℎ 𝑓 ≪ 𝑣 𝐹 𝑒 𝐴/ℏ 1 , 𝒋 𝜂 (𝑡) can be approximated semiclassiÍ cally [205] via Tr[ 𝜌( ˆ 𝒌, 𝑡)ℏ−1 ∇ 𝒌 𝐻ˆ ( 𝒌, 𝑡)] ≈ 𝜈 𝑔𝜈 ( 𝒌, 𝑡)𝒗 𝜈 ( 𝒌, 𝑡), where   ∇𝒌 𝑒 𝑬 (𝑡) 𝒗 𝜈 ( 𝒌, 𝑡) ≡ 𝜀 𝒌𝜈 (𝑡) + 𝑒𝛀𝜈 𝒌 + 𝑨(𝑡) × , (8.5) ℏ ℏ ℏ 𝑔𝜈 ( 𝒌, 𝑡) = Tr[ 𝜌( ˆ 𝒌, 𝑡) 𝜓ˆ †𝒌𝜈 (𝑡) 𝜓ˆ 𝒌𝜈 (𝑡)] denotes the instantaneous occupation of band Í 𝜈, and 𝜓ˆ †𝒌𝜈 (𝑡) = 𝑠 ⟨𝑠|𝜓 [𝒌+𝑒 𝑨(𝑡)/ℏ]𝜈 ⟩𝑐†𝑠 creates an electron in the instantaneous eigenstate in band 𝜈. The first and second term in Eq. (8.5) are the group velocity and the anomalous velocity generated by the Berry curvature 𝛀𝜈 , respectively. Near the WN, 𝛀𝜈 ( 𝒌) = 𝜉 (−1) 𝜈 𝒌/2| 𝒌 | 3 . For a circularly polarized field 𝑬 (𝑡), the 𝑧-component of the anomalous velocity is oriented in the same direction inside a radius 𝑒 𝐴/ℏ from the origin (i.e., where C𝒌 ⊥ ≠ 0). This generates a topological photocurrent which is always normal to the electric field plane. 8.5 Photoresponse of a Single WN Let us now consider the photoresponse of a WN with finite 𝜇. We focus on the experimentally-relevant regime of slow relaxation 𝜏 𝑓 ≫ 1, realized for THz or above in typical WSMs, where 𝜏 ∼ 1 − 100 ps [38, 320]. In this limit and at 𝑘 𝐵𝑇 ≪ 𝜇, the photocurrent in Eq. (8.2) can be approximated by  𝜑(𝜇) !  𝑒𝜇(𝑒 𝐴/ℏ) 2 |𝜇| 𝑗¯𝜂𝑧 ≈ 𝑐 𝜂 𝜉 𝑗t𝑜 𝑝 1 − 𝛾(𝜇) +𝜒 , (8.6) 𝑣𝐹 𝑒 𝐴 12𝜋 2 ℏ [see Supplementary Material (SM) for details], where we assumed anisotropy along the 𝑧-axis, 𝑽 𝑖 = (0, 0, ±𝑣 𝐹 𝜒), with 𝜒 parameterizing the tilt of the WN. The tilt gives rise to the second term in Eq. (8.6), which describes a helicity-independent photocurrent originating from the group velocity term in Eq. (8.5). The first term 1 This expression is derived from the adiabatic conditions ℏ||𝜕 𝐻 2 𝑡 ˆ ( 𝒌, 𝑡)|| ≪ 𝛿𝜀 ( 𝒌, 𝑡) and ℏ𝜔 ≪ 2 𝛿𝜀 ( 𝒌, 𝑡), where ||·|| denotes the operator spectral norm and 𝛿𝜀( 𝒌, 𝑡) ≡ 𝜀 1 ( 𝒌, 𝑡) − 𝜀 0 ( 𝒌, 𝑡) denotes the spectral gap. 150 (a) + (b) + WN Figure 8.2: Topological photocurrent as quantized charge pumping. (a) Brillouin zone region (purple) surrounding a single WN (bright yellow) at 𝜇 = 0. Under illumination by circularly polarized light, all electronic states with a given 𝑘 𝑥 and 𝑘 𝑦 (yellow vertical line) traces out a cylinder (pink) in momentum space during a period of the drive. This generates a quantized charge transfer per cycle along the 𝑧 direction, given by −𝑒 times the total flux of Berry curvature through the surface of the cylinder (differential area element indicated by dark purple). When the cylinder encloses the WN, this integral is nonzero and given by ±1. (b) Total time-averaged current I⟳𝑧 ( 𝒌 ⊥ ) in the system generated by electrons with a fixed in-plane momentum 𝒌 ⊥ = (𝑘 𝑥 , 𝑘 𝑦 ). Left and right panels depict 𝑣 𝐹 𝑒 𝐴/ℏ ≈ 5ℎ 𝑓 and 𝑣 𝐹 𝑒 𝐴/ℏ ≈ 0.2ℎ 𝑓 , respectively, corresponding to adiabatic and nonadiabatic driving. The dashed curve indicates the trajectory of the WN. is the topological photocurrent originating from the anomalous velocity term in Eq. (8.5), and generates the charge pumping effect described above. This term is insensitive to the tilt, but controlled by the drive helicity. The topological current is set by 𝛾(𝜇) and 𝜑(𝜇), which cross over from 𝛾 ≈ 0.4 and 𝜑 = 3 for |𝜇| ≪ 𝑒 𝐴𝑣 F to 𝛾 ≈ 0.2 and 𝜑 = 2 for 𝑒 𝐴𝑣 𝐹 < |𝜇| < 2𝑒 𝐴𝑣 𝐹 . The topological current scales with the number of carriers (∝ 𝜇3 ) near the WN in the light-doping regime |𝜇|/(ℏ𝑣 𝐹 ) ≪ 𝑒 𝐴/ℏ, and with the area of charge-pumping fermionic chains intersecting the Fermi sea (∝ 𝜇2 ) at higher 𝜇. It decreases monotonically with |𝜇|, and plateaus in the regime |𝜇|/(ℏ𝑣 𝐹 ) ≫ 2𝑒 𝐴/ℏ, at a value given by 𝐵𝑐 𝜂 𝜉 𝑗 t𝑜 𝑝 , where 𝐵 depends on the Fermi volume geometry. The residual topological current in this regime originates from the finite height of the cylinder S depicted in Figure 8.2(a), which terminates at the Fermi surface. The Berry flux through the top and bottom ends of S leads to a reduction of I𝜂𝑧 ( 𝒌 ⊥ ) from the quantized value 𝑐 𝜂 𝜉𝑒 𝑓 derived above Eq. (8.1). A semiclassical analysis (see SM) yields 𝐵 = 1/3 for spherical Fermi volumes in the |𝜇|/(ℏ𝑣 𝐹 ) ≫ 2𝑒 𝐴/ℏ limit. When the Fermi volume extends beyond the periodic Brillouin zone boundaries, S is no longer closed in the 𝑘 𝑧 -direction, resulting in 𝐵 = 0. (d) + jGz [A/mm2] (a) 151 50 + - 0 0.1 0.2 0.3 E [MV/m] (b) jGz [A/mm2] (c) + 50 + - - 0 0.1 0.2 0.3 E [MV/m] 0.6 0.4 0.2 0.0 (e) 0.6 0.4 0.2 0.0 Figure 8.3: Photoresponse from doped and tilted WNs, with parameters 𝐸 = 1.5 MV/m, ℎ 𝑓 = 5 meV and 𝜏 = 5 ps, corresponding to 𝜏 𝑓 ≈ 6. (a) Photocurrent 𝑗¯𝜂𝑧 vs. 𝜇 for an isolated, untilted WN. The orange curve indicates semiclassical prediction [see discussion below Eq. (8.6)]. (b)-(c) Photocurrent 𝑗¯𝜂𝑧 from a pair of opposite-chirality WNs with opposite tilts, 𝜒 ≡ 𝜒+ = −𝜒− under left (b) and right (c) laser polarization. In the limit 𝜒 → 0, the photoresponse is dominated by the chirality-dependent component generated by the anomalous velocity. (d) 𝑧 𝑧 + 𝑗¯⟳ )/2 of 𝑗¯𝜂𝑧 . Dashed line indicates Chirality-independent component 𝑗¯𝐺𝑧 = ( 𝑗¯⟲ prediction from semiclassical analysis [Eq. (8.6)]. (e) Steady state occupation function 𝑔1 ( 𝒌, 0) at time 𝑡 = 0 in the 𝜈 = 1 band, for the same parameters as panel (a). Black dot indicates the position of the WN, and vectors indicate the oscillating current components generated parallel 𝑗 ∥ and perpendicular 𝑗⊥ to the internal electric field 𝑬 (𝑡) and the resulting polarization 𝑷𝜂 (𝑡), along with induced field 𝑬 ind (𝑡). 8.6 Symmetry Constraints The total photocurrent 𝑗¯𝜂𝑧 of a WSM is obtained by adding contributions from all WNs, whose chemical potentials, tilts, and chiralities are related by the symmetries of the system. WSMs must break either TRS, inversion symmetry (IS), or both. Noncentrosymmetric WSMs with TRS are characterized by chemical potential imbalance between nodes of opposite chiralities 𝜉, allowing for a nonzero net topological photocurrent. TRS relates each node to another at opposite crystal momentum with identical chirality and opposite tilt [see Figure 8.1(a)], implying a minimum of four WNs, two 𝜉 = +1 nodes with chemical potential 𝜇+ and two 𝜉 = −1 nodes with chemical potential 𝜇− [208]. TRS also implies that each pair of equal-chirality WNs have opposite tilt, resulting in no net helicity-independent photocurrent [see Eq. (8.6)]. Without loss of generality, we let 𝜇− ≤ 𝜇+ . The photocurrent 𝑗¯𝜂𝑧 sharply changes as the electric field amplitude 𝐸 approaches two critical values set by the chemical potentials, 𝐸 ± ≡ ℎ 𝑓 𝜇± /(2ℏ𝑣 𝐹 𝑒) [Figure 8.1(b)]. While for 152 𝐸 < 𝐸 − none of the WNs exhibit photocurrents [see discussion below Eq. (8.6)], for 𝐸 − < 𝐸 < 𝐸 + , the 𝜉 = − WNs contribute a nonzero topological current and the 𝜉 = + WNs remain inactive. Finally, for 𝐸 > 𝐸 + both 𝜉 = +, − WNs contribute. The sharp changes of 𝑗¯𝜂𝑧 imply a highly nonlinear nature of the topological photocurrent, and can not be captured perturbing about 𝐸 = 0 [205]. Eq. (8.6) implies that 𝑗¯𝜂𝑧 attains its maximum when 𝐸 > 𝐸 + , 𝐸 − , 𝑧 𝑗max ≈ 0.016𝑒 𝑓 [(𝜇+ ) 2 − (𝜇− ) 2 ]/(ℏ𝑣 𝐹 ) 2 . (8.7) Hence the topological photocurrent is most pronounced when 𝜇+ differs significantly from 𝜇− , such as, potentially, in SrSi2 [113]. If inversion and TRS are broken, the WSM hosts a minimum of two WNs with opposite 𝜉 and different chemical potentials, but generically different tilts. The lack of symmetry constraints on the tilts enables a photocurrent with both helicitydependent and independent components. Finally, WSMs with broken TRS but preserved inversion symmetry do not support a photocurrent. 8.7 Numerical Verification To numerically verify our results, we first use Eq. (D.1) to compute 𝑗¯𝜂𝑧 from an isolated WN with no tilt, using parameters 𝑣 𝐹 = 5 × 105 m/s, 𝜏 = 5 ps, 𝑇 = 20 K, ℎ 𝑓 = 5 meV, and 𝐸 = 5 × 105 V/m, reflecting experimentally achievable conditions [38]. Note that our simulation is agnostic to the estimates in Eqs. (8.5)-(8.6). Figure 8.2(b) shows the in-plane momentum-resolved photocurrent, I⟲𝑧 ( 𝒌 ⊥ ) for 𝜇 = 0. For adiabatic driving (left panel), 𝑒 𝐴/ℏ ≈ 5ℎ 𝑓 , we see a clear quantized plateau with I⟲𝑧 ( 𝒌 ⊥ ) ≈ 𝑒 𝑓 , of disk shape and radius 𝑒 𝐴/ℏ, as the semiclassical analysis above Eq. (8.1) predicts. Illustrating the role of adiabaticity, the right half of Figure 8.2(b) depicts the same data for non-adiabatic driving, 𝑒 𝐴 ≈ 0.2ℎ 𝑓 . Here, non-adiabatic heating reduces the photocurrent of electronic states with momentum close to the trajectory (black dashed curve) of the WN [205]. Figure 8.3(a) plots 𝑗¯𝜂𝑧 vs. 𝜇. The dashed line is the analytical result, Eq. (8.6). For 𝜇/(𝑣 𝐹 𝑒 𝐴) ≫ 2, the current 𝑧 plateaus near 𝑗¯𝜂𝑧 = 𝑗top /3, as semiclassically predicted. To calculate 𝑗¯𝜂𝑧 in a noncentrosymmetric WSM with TRS, we take 𝜇− = 15 meV and 𝜇+ = 30 meV. Time-reversal conjugate nodes have opposite tilt, and, for simplicity, we assume that non-conjugate pairs of opposite-chirality WNs have opposite tilts, 𝑧 𝑧 ±𝜒. Figure 8.1(b) plots 𝑗¯⟲ for various values of 𝜒. As we expect, 𝑗¯⟲ exhibits sharp changes at the predicted field amplitudes 𝐸 ± [defined above Eq. (8.7)], and 𝑧 is moreover insensitive to 𝜒. Figure 8.1(c), shows 𝑗m𝑎𝑥 calculated numerically and 153 as predicted by Eq. (8.7). We expect the discrepancy for ℎ 𝑓 ≳ 𝑒 𝐴/ℏ are due to non-adiabatic effects. 𝑧 𝑧 Lastly, Figure 8.3(b)-(c) compares 𝑗¯⟲ and 𝑗¯⟳ for a WSM with broken TRS. When 𝑧 𝜒 → 0, 𝑗¯𝜂 exhibits a sign flip upon reversing the laser helicity 𝜂, as expected. The sign flip does not persist for finite tilts 𝜒 > 0, where the helicity-independent photocurrent emerges. Figure 8.3(d) shows that the helicity-independent component 𝑧 𝑧 𝑗¯𝐺𝑧 = ( 𝑗¯⟲ + 𝑗¯⟳ )/2 scales quadratically with the field amplitude in agreement with Eq. (8.6) (dashed lines) for 𝜒 ≪ 1. 8.8 Plasmon Enhancement We now discuss how the plasmonic response of WSMs may enhance the topological photocurrent. We assume that the size of the WSM sample is smaller than the skin depth, estimated to be 0.1-1 µm for THz frequencies [183]. The light-induced plasma oscillations of the WSMs induce a surface charge oscillation on the sample. This in turn induces an electric field 𝑬 ind (𝑡) that modifies the internal electric field 𝑬 (𝑡) relative to the external field provided by the light outside the WSM, 𝑬 ext (𝑡), via 𝑬 (𝑡) = 𝑬 ind (𝑡) + 𝑬 ext (𝑡). The modification can induce either damping or amplification, depending on whether the gain ratio 𝐺 𝜂 ≡ |𝑬 (𝑡)|/|𝑬 ext (𝑡)| is smaller or larger than 1. The induced field 𝑬 ind (𝑡) is appreciable in 3D geometries, but may be negligble in quasi-2D settings such as thin films. To estimate 𝑬 ind (𝑡) in a 3D configuration, we consider a spherical geometry. The drive-induced in-plane current, 𝒋 𝜂 (𝑡), creates surface charge corresponding to a uniform internal polarization density 𝑷𝜂 (𝑡) such that 𝜕𝑡 𝑷𝜂 (𝑡) = 𝒋 𝜂 (𝑡). With spherical 𝜂 𝜂 𝜂 geometry, 𝑬 ind (𝑡) = 𝑷𝜂 (𝑡)/(3𝜖0 ). Writing 𝒋 𝜂 (𝑡) = 𝑗 𝑧 (𝑡) 𝑧ˆ + 𝑗 𝑃 𝒆ˆ (𝑡) + 𝑗 𝑁 𝑧ˆ × 𝒆ˆ (𝑡), where 𝒆ˆ (𝑡) ≡ 𝑬 (𝑡)/𝐸 and 𝐸 = |𝑬 (𝑡)|, we find 𝐺 𝜂 = 𝐸 {[𝐸 + 𝑐 𝜂 𝑗 𝑁 /(6𝜋𝜖 0 𝑓 )] 2 + [ 𝑗 𝑃 /(6𝜋𝜖 0 𝑓 )] 2 }−1/2 . 𝜂 𝜂 (8.8) In noncentrosymmetric WSMs with TRS, the instantaneous population distribution for 𝜏 𝑓 ≫ 1 [see color plot in Figure 8.3(e)] is anisotropic about the WN, yielding Í 𝜂 𝜂 𝑗 𝑃 ≈ 0, and 𝑗 𝑁 ≈ 𝑐 𝜂 𝑒𝑣 𝐹 𝜉=± [𝜇𝜉 /(ℏ𝑣 𝐹 )] 3 /(3𝜋 3 ) in the limit 𝜇𝜉 ≪ ℏ𝑣 𝐹 𝑒 𝐴/ℏ. This result is derived from Eq. (8.5) [see SM], and implies a plasma resonance (divergent 𝐺 𝜂 ) which takes the internal field amplitude inside system towards the critical value 𝐸 𝑝 ( 𝑓 ) ≈ 𝑒𝑣 𝐹 [(𝜇+ ) 3 + (𝜇− ) 3 ]/[(ℏ𝑣 𝐹 ) 3 (18𝜋 4 𝜖0 𝑓 )]. (8.9) The existence of a characteristic internal field amplitude is a consequence of the linear dispersion of the WSM, which gives rise to an inherently nonlinear response. 154 Figure 8.1(d) plots the numerically obtained value of 𝐺 𝜂 , along with 𝐸 𝑝 ( 𝑓 ) as predicted above. 8.9 Chiral Plasmons in WSMs with broken TRS When TRS is broken, noncentrosymmetric WSMs support a chiral plasmonic resonance, where plasma modes with opposite helicities have different frequencies. To illustrate this, we again consider the minimal example of a pair of WNs of opposite chirality, with unequal chemical potentials 𝜇± and tilts 𝜒± . The tilt reshapes the Fermi surface in the 𝜈 = 1 band of each WN, creating an imbalance of electronic carriers with positive and negative 𝑘 𝑧 . In this regime, an additional, in-plane topological Í 𝜂 current emerges, given by 𝛿 𝒋 𝑁 (𝑡) ≈ − 𝜉=± 𝜉 𝜒𝜉 𝑒 𝑓 (𝜇𝜉 /ℏ𝑣 𝐹 ) 2 𝑧ˆ × 𝒆ˆ (𝑡)/(4𝜋 2 ) in the slow relaxation limit 𝜏 𝑓 ≫ 1. The topological current arises from the in-plane, oscillating component of the anomalous velocity 𝑒𝛀𝜈 × 𝑬 (𝑡)/ℏ which is nonzero after integration over 𝑘 𝑧 for a tilted WN. The resonant electric field amplitude is modified as a result, and, in the limit 𝜒± ≪ 1, is now given by   3 𝜇2𝜉 𝑒 𝑓  1 ∑︁  𝑒𝑣 𝐹 𝜇𝜉  𝐸𝑝 ( 𝑓 ) ≈  3𝜋 3 (ℏ𝑣 ) 3 − 𝜉𝑐 𝜂 𝜒𝜉 (ℏ𝑣 ) 2 4𝜋 2  . 6𝜋𝜖 0 𝑓 𝐹 𝐹   {𝜉}   (8.10) This chirality-dependent plasma resonance is verified numerically in Figure 8.1(e) for the case where the WNs of positive and negative chirality exhibit opposite tilts, 𝜒± = ±𝜒. Notably, the plasma resonance amplitudes and frequencies depend on the helicity of the laser drive. 8.10 Conclusions Here we analyzed the generation of robust topological photocurrent induced by circularly polarized light in noncentrosymmetric WSMs, first reported in Ref. [55]. We showed that the effect is insensitive to anisotropy and electronic interactions. The effect may be observed in noncentrosymmetric WSMs where nodes of opposite chiralities are well-separated in energy, such as, possibly, SrSi2 [113]. When timereversal symmetry is broken and the WNs are tilted, the transport further exhibits a giant helicity-independent photocurrent and a helicity-dependent plasmon resonance. These exotic topological electronic and plasmonic responses constitute a new class of highly nonlinear photoresponses in Weyl semimetals with low symmetry constraints. Multiple works have investigated the light-induced current responses of WSMs [38, 58, 102, 117, 118, 121, 121, 147, 172, 197, 218, 230, 237, 264, 323]. The pho- 155 tocurrent we analyze here (first reported in Ref. [55]) is different from these works, since it is of intra-band origin and of strongly nonlinear nature, beyond the paradigm of conventional nonlinear response theory [55, 205]. The topological photocurrent moreover does not require any explicit band structure anisotropy, and can be orders of magnitude stronger than in Refs. [117, 118]. 8.11 Acknowledgments We thank Ivar Martin, Cyprian Lewandowski, Elio König, Tobias Hölder, Takahiro Morimoto, Alexander Tyner, and Hiro Ishizuka for insightful discussions. C.Y. gratefully acknowledges support from the DOE NNSA Stewardship Science Graduate Fellowship program, which is provided under cooperative agreement number DE-NA0003960. G.R. is grateful for support from the Simons Foundation and the Institute of Quantum Information and Matter, as well as support from the NSF DMR grant number 1839271. This work is supported by ARO MURI Grant No. W911NF-16-1-0361, and was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. F.N. was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under award DE-SC0019166, the Simons Foundation under award 623768, and the Carlsberg Foundation, grant CF22-0727. 156 Appendix D APPENDIX Here we provide the semiclassical calculation of the photocurrent 𝒋 (𝑡) quoted in the main text. Before we proceed to the technical details of the derivation, which are provided in the following sections, we begin by outlining the structure of the calculation. Our calculation of 𝒋 (𝑡) is based on solving the Lindblad master equation in Eq. (D.1) of the main text, 𝜕𝑡 𝜌( ˆ 𝒌, 𝑡) = 1 𝑖 [ 𝜌( ˆ 𝒌, 𝑡), 𝐻ˆ ( 𝒌, 𝑡)] − [ 𝜌( ˆ 𝒌, 𝑡) − 𝜌ˆ eq ( 𝒌, 𝑡)], ℏ 𝜏 (D.1) with 𝐻ˆ ( 𝒌, 𝑡), 𝜌( ˆ 𝒌, 𝑡), and 𝜌ˆ e𝑞 ( 𝒌, 𝑡) denoting the second-quantized Bloch Hamiltonian, density matrix and equilibrium density matrix, respectively, while 𝜏 is the phenomenological relaxation time of the system (see main text for more details). The photocurrent is obtained from the steady-state solution to Eq. (D.1), 𝜌ˆ s ( 𝒌, 𝑡) ∫ 1 𝑑 3 𝒌Tr[ 𝜌ˆ s ( 𝒌, 𝑡)ℏ−1 ∇ 𝒌 𝐻ˆ ( 𝒌, 𝑡)]. In the quasi-adiabatic limit we via 𝒋 (𝑡) = (2𝜋) 3 consider, this integral can be approximated by [205] ∑︁ 𝒋 (𝑡) ≈ 𝑔𝜈 ( 𝒌, 𝑡)𝒗 𝜈 ( 𝒌, 𝑡). (D.2) 𝜈 Here 𝑔𝜈 ( 𝒌, 𝑡) = Tr[ 𝜌ˆ s ( 𝒌, 𝑡) 𝜓ˆ †𝒌𝜈 (𝑡) 𝜓ˆ 𝒌𝜈 (𝑡)] denotes the steady-state occupation of the instantaneous electronic bands, with 𝜓ˆ 𝒌𝜈 (𝑡) denoting the creation operator of an electron with crystal momentum 𝒌 in the instantaneous eigenstate in band 𝜈 [see full definition below Eq. (8.5) in the main text]. Moreover,   ∇𝒌 𝑒 𝑬 (𝑡) 𝜀 𝒌𝜈 (𝑡) + 𝑒𝛀𝜈 𝒌 + 𝑨(𝑡) × (D.3) 𝒗 𝜈 ( 𝒌, 𝑡) ≡ ℏ ℏ ℏ denotes the phase velocity of the electrons in band 𝜈. Here, 𝛀𝜈 ( 𝒌 + 𝑒 𝑨(𝑡)/ℏ) and 𝜀 𝒌𝜈 (𝑡) are respectively the Berry curvature and eigenenergies of the bands of the Bloch Hamiltonian 𝐻 ( 𝒌, 𝑡), 𝑨(𝑡) is the drive-induced magnetic vector potential, and 𝑬 (𝑡) = 𝜕𝑡 𝑨(𝑡) the corresponding electric field. We provide more details on the quantities above in the main text. The remainder of this supplement is devoted to calculating the photocurrent 𝒋 (𝑡) using the results above. The calculation is structured as follows: in Section D.1, 157 we derive approximate steady state solutions for 𝑔𝜈 ( 𝒌, 𝑡). In Section D.2, we use this solution along with Eqs. (D.2)-(D.3) to compute 𝒋 (𝑡) for an isolated Weyl node with no tilt, focusing on the out-of-plane (Section D.2) and in-plane (Section D.2) components separately. In Section D.3, we extend the calculation to the case of a finite Weyl node tilt to demonstrate the emergence of the helicity-independent photocurrent and chiral plasmons. D.1 Steady State Electronic Distribution We first derive an approximate analytic expression for the steady state occupation function of the electronic bands, 𝑔𝜈 ( 𝒌, 𝑡). By explicitly solving the master equation [see Eq. (D.1)] [205], we obtain the solution ∫ 1 𝑡 eq 𝑔𝜈 ( 𝒌, 𝑡) = 𝑑𝑠 𝑒 −(𝑡−𝑠)/𝜏 𝑓𝜈 ( 𝒌, 𝑠), (D.4) 𝜏 −∞ eq where 𝑓𝜈 ( 𝒌, 𝑡) defines the electronic occupation function of band 𝜈 in thermal equilibrium, and is given by eq 𝑓𝜈 ( 𝒌, 𝑡) = 1 𝑒 [𝜀𝒌 𝜈 (𝑡)−𝜇]/𝑘 𝐵𝑇 + 1 , (D.5) where 𝜇 is the chemical potential, 𝑘 𝐵 is the Boltzmann constant, and 𝑇 is the lattice temperature. As discussed in the main text, we focus on the slow-relaxation limit 𝜏 𝑓 ≫ 1, which is realized for picosecond-scale relaxation times and terahertz or higher frequency drives [38, 320]. In this limit, using Eq. (D.4) and exploiting the eq time-periodicity of 𝑓𝜈 ( 𝒌, 𝑡), we find that ∫ 𝑇dr 1 eq 𝑑𝑢 𝑓𝜈 ( 𝒌, 𝑢), (D.6) 𝑔𝜈 ( 𝒌, 𝑡) ≈ 𝑇dr 0 where 𝑇dr = 1/ 𝑓 is the period of the drive. Thus, the steady-state distribution 𝑔𝜈 ( 𝒌, 𝑡) approaches a distribution obtained from “smearing” the equilibrium distribution 𝑓𝜈 ( 𝒌, 𝑡) along the circular trajectory 𝒌 + 𝑒 𝑨(𝑡)/ℏ defined by the drive-induced vector potential. D.2 Photoresponse of an Isolated, Untilted Weyl Node Here, we use the results above to compute 𝒋 𝜂 (𝑡) for an isolated, untilted Weyl node driven by circularly polarized light in the slow relaxation limit 𝜏 𝑓 ≫ 1. The Hamiltonian we consider is given by 𝐻 ( 𝒌) = 𝜉ℏ𝑣 𝐹 𝒌 · 𝝈, (D.7) where 𝑣 𝐹 is the Fermi velocity. We first consider the photocurrent component in the 𝑧-direction (Section D.2), and finally in the 𝑥-𝑦 plane (Section D.2). 158 Photocurrent along the 𝑧-axis We compute the photocurrent in the 𝑧-direction by analyzing the contribution from Í 𝑧 𝑧 , where 𝑗¯𝜈,𝜂 each electronic band separately. To this end, we define 𝑗¯𝜂𝑧 = 𝜈 𝑗¯𝜈,𝜂 is the time-averaged contribution to the current from the 𝜈-th band. Using the semiclassical equations of motion [Eqs. (D.2)-(D.3)], we find ( )     ∫ 𝑇dr ∫ 3𝒌 1  1 𝑑 𝑒 𝑬 (𝑡) 𝑧  𝜕𝑘 𝜀 𝒌𝜈 (𝑡) + 𝑒𝛀𝜈 𝒌 + 𝑨(𝑡) ×  𝑔𝜈 ( 𝒌, 𝑡). 𝑗¯𝜈,𝜂 = 𝑑𝑡 · 𝑧 ˆ  𝑇dr 0 ℏ ℏ (2𝜋) 3  ℏ 𝑧    (D.8) In untilted Weyl nodes, the occupation function 𝑔𝜈 ( 𝒌, 𝑡) is symmetric upon inversion along 𝑘 𝑧 (𝑘 𝑧 → −𝑘 𝑧 ), and 𝜕𝑘 𝑧 𝜀 𝒌𝜈 (𝑡) is an odd function of 𝑘 𝑧 . Therefore, the group velocity contribution vanishes. The remaining anomalous velocity is the origin of the topological current. For the positive-chirality Weyl node we consider in Eq. (D.7), the Berry curvature is given by 𝛀𝜈 ( 𝒌) = 𝜉 (−1) 𝜈 𝒌/2| 𝒌 | 3 . Moreover using that 𝑬 (𝑡) = −2𝜋 𝑓 𝑐 𝜂 𝑧ˆ × 𝑨(𝑡) for circularly polarized light, we obtain the photocurrent 3 𝒌 ∫ 𝑇dr 2 𝑓 ∫ 𝒌 · 𝑨(𝑡) + 𝑒 𝐴2 /ℏ 𝑑 𝑒 𝑧 𝜈 ¯𝑗 𝜈,𝜂 = −𝑐 𝜂 𝜉 (−1) 𝑑𝑡 𝑔𝜈 ( 𝒌, 𝑡). (D.9) 2ℏ 2𝜋 (2𝜋) 3 0 | 𝒌 + 𝑒 𝑨(𝑡)/ℏ| 3 We now compute the contributions from the 𝜈 = 0 and 𝜈 = 1 bands separately. In our analysis, we focus on the case 𝜇 ≥ 0, noting that the results can be easily generalized to 𝜇 < 0. 𝑧 Computing 𝑗¯0,𝜂 To compute the photocurrent from electronic states in the fully-occupied 𝜈 = 0 valence band, we note that the inner integral over 𝑡 in Eq. (D.9) is symmetric under rotation around the 𝑧-axis. We can therefore set 𝑘 𝑥 = 𝑘 ⊥ and 𝑘 𝑦 = 0 to evaluate it, where 𝒌 ⊥ = (𝑘 𝑥 , 𝑘 𝑦 ) denotes the in-plane momentum and 𝑘 ⊥ = | 𝒌 ⊥ | denotes its magnitude. Introducing the unitless parameters 𝑥 ⊥ = 𝑘 ⊥ /(𝑒 𝐴/ℏ), 𝑥 𝐹 ≡ 𝑘 𝐹 /(𝑒 𝐴/ℏ), 𝑥 𝑧 ≡ 𝑘 𝑧 /(𝑒 𝐴/ℏ), and 𝜙 = 2𝜋𝑡, where 𝑘 𝐹 = 𝜇/(ℏ𝑣 𝐹 ), the net current produced by electrons in the 𝜈 = 0 band is given by   2 ∫ ∞ ∫ 2𝜋 ∫ ∞ 𝑒 𝑓 𝑒 𝐴 𝑥 ⊥ cos 𝜙 + 1 𝑧 𝑗¯0,𝜂 = −𝑐 𝜂 𝜉 2 𝑑𝑥 𝑑𝜙 𝑑𝑥 𝑧 2 . ℏ 8𝜋 (𝑥 ⊥ + 2𝑥⊥ cos 𝜙 + 1 + 𝑥 𝑧2 ) 3/2 0 0 −∞ (D.10) Here, we have used 𝑔0 ( 𝒌, 𝑡) = 1 because the valence band is fully occupied. We first perform integration over 𝑥 𝑧 , which yields   2 ∫ ∞ ∫ 2𝜋 𝑥 ⊥ cos 𝜙 + 1 ¯𝑗 𝑧 = −𝑐 𝜂 𝜉 𝑒 𝑓 𝑒 𝐴 𝑑𝑥 𝑑𝜙 2 . (D.11) 0,𝜂 2 ℏ 8𝜋 𝑥 ⊥ + 1 + 2𝑥 ⊥ cos 𝜙 0 0 159 Using the identity ∫ 2𝜋 𝑑𝜙 0 −1 cos 𝜙 + 𝑥 ⊥ = 4𝜋𝜃 (1 − 𝑥 ⊥ ), −1 + 𝑥 )/2 cos 𝜙 + (𝑥 ⊥ ⊥ (D.12) 𝑧 it follows that 𝑗¯0,𝜂 = 𝑐 𝜂 𝜉 𝑗 top , where  2 𝑒 𝑓 𝑒𝐴 . 𝑗top = − 4𝜋 ℏ (D.13) Thus, we have reproduced the quantized current result in Eq. (1) of the main text, which was obtained using the charge pump picture [see discussion above Eq. (1)]. 𝑧 Computing 𝑗1,𝜂 𝑧 We next calculate the current 𝑗1,𝜂 generated by electronic states in the partiallyfilled 𝜈 = 1 conduction band, which hosts a spherical Fermi volume with radius 𝑘 𝐹 = 𝜇/(ℏ𝑣 𝐹 ). We again exploit that the inner integral in Eq. (D.9) is symmetric under rotation around the 𝑧-axis, and, without loss of generality, set 𝑘 𝑥 = 𝑘 ⊥ and 𝑘 𝑦 = 0. Using the approximate expression for the occupation function in Eq. (D.6), and the same unitless paramters as in Section D.2, we find that   2 ∫ 1+𝑥 𝐹 ∫ √𝑥 2 −(𝑥⊥ −1) 2 𝐹 𝑒 𝐴 𝑒 𝑓 𝑧 𝑑𝑥 𝑧 𝑑𝑥 ⊥ √ 𝑗1,𝜂 = 𝑐𝜂 𝜉 2 ℏ 8𝜋 − 𝑥 2𝐹 −(𝑥⊥ −1) 2 max(0,1−𝑥 𝐹 ) (D.14) ∫ 2𝜋 𝑥⊥ cos 𝜙 + 1 G1 (𝑥 ⊥ , 𝑥 𝑧 , 𝑡). × 𝑑𝜙 2 (𝑥 ⊥ + 2𝑥 ⊥ cos 𝜙 + 1 + 𝑥 𝑧2 ) 3/2 0 Here, G1 (𝑥⊥ , 𝑥 𝑧 , 𝑡) = 𝑔1 [𝑒 𝐴/ℏ(𝑥 ⊥ , 0, 𝑥 𝑧 ), 𝑡]. Eq. (D.14) can be integrated numerically for different values of 𝑥 𝐹 . We show the numerical result in Figure D.1(a). We 𝑧 to the function fit 𝑗1,𝜂 𝜑(𝜇) 𝑧 𝑗1,𝜂 = −𝑐 𝜂 𝜉 𝑗 t𝑜 𝑝 𝛾(𝜇)𝑥 𝐹 , (D.15)
2 𝑧 where 𝑗top = −𝑒 𝑓 𝑒 𝐴/ℏ /(4𝜋). In Figure D.1(b), we plot log[− 𝑗1,𝜂 /(𝑐 𝜂 𝜉 𝑗top )] vs. log(𝑥 𝐹 ) and we perform fits to the numerically-calculated data using 𝜑 = 3 in the regime 0.05 < 𝑥 𝐹 < 0.3 and 𝜑 = 2 in the regime 1.0 < 𝑥 𝐹 < 1.7 to arrive at the values of 𝛾(𝜇) provided in the main text. For 𝑥 𝐹 ≫ 2, the coefficients saturate to 𝛾 = 2/3, and 𝜑 = 0, as we show below. 𝑧 Saturation value of 𝑗1,𝜂 for 𝑘 𝐹 ≫ 2𝑒 𝐴/ℏ. We can analytically compute the 𝑧 saturation value of 𝑗1,𝜂 in the limit 𝑘 𝐹 ≫ 2𝑒 𝐴/ℏ. To this end, we first consider 160 (a) (b) Figure D.1: (a) Topological current produced by conduction band (𝜈 = 1) states in an isolated Weyl node, calculated by numerically integrating the semiclassical equations of motion [see Eq. (D.14)]. Here, 𝑗top = −𝑒 𝑓 (𝑒 𝐴/ℏ) 2 /(4𝜋) denotes the topological current magnitude produced by a fully-occupied valence band [see Eq. (1) in the main text], and 𝑥 𝐹 = ℏ𝑘 𝐹 /(𝑒 𝐴). (b) Log-scale plot of panel (a), with fits (solid lines) to the numerical data. Figure D.2: Illustration of the coordinate system used to calculate the saturation 𝑧 value of 𝑗1,𝜂 for 𝑘 𝐹 ≫ 2𝑒 𝐴/ℏ. Here, 𝒌 ′ = 𝒌 + 𝑒 𝑨(𝑡)/ℏ is the shifted coordinate system, which fixes the WN at the origin. Purple sphere indicates the Fermi volume in the conduction band, which is centered at the WN (center white dot). Under illumination by circularly polarized light, populated electronic states with a given 𝒌 ⊥ = (𝑘 𝑥 , 𝑘 𝑦 ) traces out a cylinder S𝑡 ( 𝒌 ⊥ ) (dark purple) in momentum space during a period of the drive. Due to the finite Fermi momentum 𝑘 𝐹 , finite Berry curvature √︃ 𝛀1 penetrates the “caps” S𝑡 ( 𝒌 ⊥ ) (light green) of the cylinder located at 𝑘 𝑧 = ± 𝑘 2𝐹 − | 𝒌 ⊥ | 2 . The Berry flux through the caps S𝑡 ( 𝒌 ⊥ ) produces a saturation 𝑧 value of 𝑗 1,𝜂 in the limit 𝑘 𝐹 ≫ 2𝑒 𝐴/ℏ. 161 the current produced by a single Fermionic chain of states with fixed in-plane momentum 𝒌 ⊥ , given by )  (    ∫ 𝑇dr ∫ ∞  𝑑𝑘 𝑧  1 𝑬 (𝑡) 1 𝑒 𝑑𝑡 𝜕𝑘 𝑧 𝜀 𝒌1 (𝑡) + 𝑒𝛀1 𝒌 + 𝑨(𝑡) × · 𝑧ˆ 𝑔1 ( 𝒌, 𝑡). I ( 𝒌 ⊥) =  𝑇dr 0 ℏ ℏ −∞ 2𝜋  ℏ    (D.16) 𝑧 This in-plane-momentum-resolved current determines 𝑗1,𝜂 via ∫ 𝑧 𝑗1,𝜂 = 𝑑2 𝒌⊥ I ( 𝒌 ⊥ ). (2𝜋) 2 (D.17) Noting again that the group velocity term in Eq. (D.16) vanishes by symmetry, it follows that ∫ ∞ ∫ 𝑇dr 𝑒 𝑒𝑓 𝑑𝑘 𝑧 𝑑𝑡 [𝛀1 ( 𝒌 + 𝑒 𝑨(𝑡)/ℏ) × 𝑬 (𝑡)] · 𝑧ˆ 𝑔1 ( 𝒌, 𝑡). (D.18) I ( 𝒌 ⊥) = 2𝜋 −∞ ℏ 0 To further simplify the expression, we note that the magnetic vector potential 𝑨(𝑡) is always perpendicular to the electric field 𝑬 (𝑡), and, therefore,   ∫ ∞ ∫ 𝑇dr 𝑒𝑓 𝑒 ˆ I ( 𝒌 ⊥) = 𝑐𝜂 𝛀1 ( 𝒌 + 𝑒 𝑨(𝑡)/ℏ) · 𝐸 𝑨(𝑡)𝑑𝑡𝑑𝑘 𝑧 𝑔1 ( 𝒌, 𝑡), (D.19) 2𝜋 −∞ 0 ℏ ˆ = 𝑨(𝑡)/| 𝑨(𝑡)|. We identify where 𝐸 = |𝑬 (𝑡)| and 𝑨 ˆ 𝛀1 ( 𝒌 + 𝑒 𝑨(𝑡)/ℏ) · (𝑒/ℏ)𝐸 𝑨(𝑡)𝑑𝑡𝑑𝑘 𝑧 (D.20) [see Figure 2(a) in the main text] as the flux of Berry curvature through an infinitesimal surface area element for a cylinder S0 ( 𝒌 ⊥ ) of radius 𝑒 𝐴/ℏ oriented along the 𝑧 axis and with center located at 𝒌 ⊥ = (𝑘 𝑥 , 𝑘 𝑦 ). Next, we note that 𝑔1 ( 𝒌, 𝑡) ≈ 𝜃 (𝑘 𝐹 − | 𝒌 |) for 𝑘 𝐹 ≫ 𝑒 𝐴/ℏ. Thus I ( 𝒌 ⊥ ) is given by the total Berry flux through the open surface (or “ribbon”) defined by the segment of S0 ( 𝒌 ⊥ ) with √︃ |𝑘 𝑧 | < 𝑘 2𝐹 − | 𝒌 ⊥ | 2 : ∫ 𝑒𝑓 I ( 𝒌 ⊥) = 𝑐𝜂 𝛀1 · 𝑑𝑺. (D.21) 2𝜋 S0 ( 𝒌 ⊥ ) To compute this, we consider the closed surface S( 𝒌 ⊥ ) formed by closing the ′ ends of S√︃ 0 ( 𝒌) by adding the disk-shaped “caps” S𝑡 ( 𝒌 ⊥ ), consisting of points 𝒌 = (𝑘 𝑥 , 𝑘 𝑦 , ± 𝑘 2𝐹 − | 𝒌 ⊥ | 2 ), see Figure D.2 for an illustration. The surface integral over S0 ( 𝒌 ⊥ ) is given by the surface integral over S( 𝒌 ⊥ ) minus the oriented surface integral over S𝑡 ( 𝒌 ⊥ ): ! ∫ ∫ 𝑒𝑓 I ( 𝒌 ⊥) = 𝑐𝜂 𝛀1 · 𝑑𝑺 − 𝛀1 · 𝑑𝑺 . (D.22) 2𝜋 S( 𝒌 ⊥ ) S𝑡 ( 𝒌 ⊥ ) 162 The first integral evaluates to 𝑗t𝑜 𝑝 following the analysis presented before Eq. (8.1) in the main text. To evaluate the second integral, we recall that 𝛀1 ( 𝒌) = 𝜉 𝒌/| 𝒌 | 3 . √︃ When 𝑘 𝐹 ≫ 2𝑒 𝐴/ℏ, we can effectively set 𝑘 𝑧 = 𝑘 2𝐹 − | 𝒌 ⊥ | 2 on the surface S𝑡 ( 𝒌√︃ ⊥ ), implying that the Berry flux density through S𝑡 ( 𝒌 ⊥ ) is uniform and given by 𝜉 𝑘 2𝐹 − | 𝒌 | 2 /𝑘 3𝐹 . Noting that S𝑡 ( 𝒌 ⊥ ) has area 2𝜋(𝑒 𝐴/ℏ) 2 , this leads us to √︃   2 𝑘 2 − | 𝒌 |2 ∫ 𝐹 𝑒𝐴 𝛀1 ( 𝒌) · 𝑑𝑺 ≈ 2𝜋𝜉 . (D.23) ℏ 𝑘 3𝐹 S𝑡 ( 𝒌 ⊥ ) Using Eq. (D.17) and Eq. (D.22) to compute the total current produced all Fermionic chains in the conduction band, we find that  2 2 𝑒 𝑓 𝑒𝐴 𝑧 𝑗1,𝜂 = 𝑐 𝜂 𝜉 f𝑜𝑟 𝑘 𝐹 ≫ 2𝑒 𝐴/ℏ. (D.24) 3 4𝜋 ℏ Photocurrent in the 𝑥-𝑦 plane for 𝑘 𝐹 ≪ 𝑒 𝐴/ℏ We next calculate the (oscillating) photocurrent in the 𝑥-𝑦 plane induced by the circularly polarized laser. We focus on the low temperature regime 𝑇 ≪ 𝜇/𝑘 𝐵 and for small Fermi momenta 𝑘 𝐹 ≪ 𝑒 𝐴/ℏ. This current gives rise to the plasma resonances described in the main text [see Eq. (8.8)]. For concreteness, we set 𝑡 = 0, noting that the results for later times are related to this case through trivial rotation. Using the semiclassical equations in Eqs. (D.2) and (D.3), we find that the total current is given by ( )   ∑︁ ∫ 𝑑 3 𝒌  ∇  𝑒 𝑬 (0) 𝒌 𝜂  𝜀 𝒌𝜈 (𝑡) + 𝑒𝛀𝜈 𝒌 + 𝑨(0) ×  𝑔𝜈 ( 𝒌, 𝑡), 𝒋 (𝑡) = 𝑒  3  ℏ ℏ ℏ (2𝜋)   𝜈  ⊥ (D.25) 𝜂 To evaluate the in-plane component of 𝒋 (𝑡), we first note that the anomalous velocity term does not contribute to the in-plane photocurrent for untilted Weyl nodes, because the in-plane component of the anomalous velocity reverses direction upon inversion along the 𝑘 𝑧 axis (𝑘 𝑧 → −𝑘 𝑧 ), and the Fermi volume (and hence also 𝑔𝜈 ( 𝒌, 𝑡)) is symmetric upon the same transformation. We also note that the 𝜈 = 0 band does not contribute, because 𝑔0 ( 𝒌, 𝑡) = 1 and ∇ 𝒌 𝜀 𝒌0 (0) is odd under momentum inversion about the Weyl node (𝒌 + 𝑒 𝑨(0)/ℏ → −𝒌 + 𝑒 𝑨(0)/ℏ) and the Fermi volume is also symmetric about the same transformation. Thus, ∫ 𝑑3 𝒌 𝜂 𝑗𝑥,𝑦 (𝑡) = 𝑒 ∇ 𝑘 ,𝑘 𝜀 𝒌1 (0)𝑔1 ( 𝒌, 𝑡). (D.26) (2𝜋) 3 𝑥 𝑦 𝜂 𝜂 We next decompose the in-plane photocurrent into its components 𝑗 𝑃 and 𝑗 𝑁 parallel and normal to the direction of the instantaneous electric field, 𝒆ˆ (𝑡) ≡ 𝑬 (𝑡)/𝐸, 163 respectively, such that 𝜂 𝜂 𝜂 𝒋 𝜂 (𝑡) = 𝑗 𝑧 (𝑡) 𝑧ˆ + 𝑗 𝑃 𝒆ˆ (𝑡) + 𝑗 𝑁 𝑧ˆ × 𝒆ˆ (𝑡). (D.27) The combination of rotation and time translation symmetry of the problem implies 𝜂 𝜂 𝜂 that 𝑗 𝑧 , 𝑗 𝑁 , and 𝑗 𝑃 are all time-independent in the steady-state. We moreover identify 𝜂 𝜂 𝑗 𝑁 = 𝑐 𝜂 𝑗𝑥 (0), 𝑗 𝑃 = 𝑐 𝜂 𝑗 𝑦 (0). Also using the expression for the energies 𝜀 𝒌1 (𝑡), we find ∫ 𝑑3 𝒌 𝑘 𝑥 − 𝑒 𝐴/ℏ 𝜂 𝑗 𝑁 = 𝑐 𝜂 𝑒𝑣 𝐹 𝑔1 ( 𝒌, 0), (D.28) √︃ 3 (2𝜋) [𝑘 𝑥 + 𝑒 𝐴/ℏ] 2 + 𝑘 2𝑦 + 𝑘 2𝑧 ∫ 𝑘𝑦 𝑑3 𝒌 𝜂 𝑔1 ( 𝒌, 0). (D.29) 𝑗 𝑃 = 𝑐 𝜂 𝑒𝑣 𝐹 √︃ (2𝜋) 3 [𝑘 + 𝑒 𝐴/ℏ] 2 + 𝑘 2 + 𝑘 2 𝑥 𝑦 𝑧 To evaluate the above, we need to find an approximate expression for the occupation ∫𝑇 e𝑞 function 𝑔1 ( 𝒌, 𝑡) = 𝑇d𝑟 −1 0 dr 𝑑𝑡 ′ 𝑓1 ( 𝒌, 𝑡 ′) [See Eq. (D.6)]. We transform into the cylindrical coordinates 𝒌 = (𝑘 ⊥ cos 𝜑, 𝑘 ⊥ sin 𝜑, 𝑘 𝑧 ) and first find an approximate expression for the zero-temperature instantaneous equilibrium occupation function eq 𝑓1 ( 𝒌, 𝑡) = 𝜃 (𝑘 𝐹 − | 𝒌 + 𝑒 𝑨(𝑡)/ℏ|), where 𝜃 is the Heaviside step function. In the regime 𝑘 𝐹 ≪ 𝑒 𝐴/ℏ, it can be approximated by eq 𝑓1 ( 𝒌, 𝑡) ≈ 𝜃 (min |𝜑−2𝜋 𝑓 𝑡 +2𝜋𝑛|−𝛿𝜑(𝑘 ⊥ , 𝑘 𝑧 ))𝜃 (𝑘 2𝐹 −𝑘 2𝑧 − (𝑘 ⊥ −𝑒 𝐴/ℏ) 2 ), (D.30) 𝑛∈Z where 𝛿𝜑(𝑘 ⊥ , 𝑘 𝑧 ) measures the azimuthal angle taken up by the Fermi volume at fixed 𝑘 ⊥ and 𝑘 𝑧 [see Figure D.3(a)]. The second step function in Eq. (D.30) requires that the in-plane momentum magnitude 𝑘 ⊥ of occupied states lies between the two dashed blue circles in Figure D.3(a), while the first step function in Eq. (D.30) requires that the azimuthal angle 𝜑 of the in-plane momentum lies between the two angles indicated by green dashed lines in Figure D.3(a). The angle between the two green dashed lines can be estimated by 𝛿𝜑(𝑘 ⊥ , 𝑘 𝑧 ) = 𝜑+ (𝑘 ⊥ , 𝑘 𝑧 ) − 𝜑− (𝑘 ⊥ , 𝑘 𝑧 ), where 𝜑± (𝑘 ⊥ , 𝑘 𝑧 ) are defined via the relation ℏ𝑣 𝐹 |(𝑘 ⊥ cos[𝜑± (𝑘 ⊥ , 𝑘 𝑧 )] − 𝑒 𝐴/ℏ, 𝑘 ⊥ sin[𝜑± (𝑘 ⊥ , 𝑘 𝑧 )], 𝑘 𝑧 )| = ℏ𝑣 𝐹 𝑘 𝐹 , (D.31) which parameterizes the boundary of the Fermi volume. Expanding cos[𝜑± (𝑘 ⊥ , 𝑘 𝑧 )] ≈ 1 − [𝜑± (𝑘 ⊥ , 𝑘 𝑧 )] 2 /2 in the limit 𝑘 𝐹 ≪ 𝑒 𝐴/ℏ, solving for 𝜑± (𝑘 ⊥ , 𝑘 𝑧 ) yields √︃ 𝛿𝜑(𝑘 ⊥ , 𝑘 𝑧 ) ≈ (ℏ/𝑒 𝐴) 𝑘 2𝐹 − 𝑘 2𝑧 − (𝑘 ⊥ − 𝑒 𝐴/ℏ) 2 . (D.32) 164 (a) (b) Figure D.3: (a) Illustration of the characteristic in-plane momentum dynamics relevant for analyzing the photocurrent. Solid black circle indicates the edge of the instantaneous equilibrium Fermi volume of 𝐻 ( 𝒌, 𝑡) as a function of in-plane momentum, 𝒌 ⊥ = (𝑘 𝑥 , 𝑘 𝑦 ), for an arbitrary, fixed value of 𝑘 𝑧 . Black dot indicates the instantaneous location of the Weyl node, which is located at an azimuthal angle of 2𝜋 𝑓 𝑡 from the 𝑘 𝑥 axis (red arrow). We approximate the instantaneous equilibrium eq occupation 𝑓1 ( 𝒌, 𝑡) [Eq. (D.30)] to be given by 1 if 𝒌 ⊥ is located within the annulus segment (shaded blue) and between the two angles indicated by green lines, and 0 otherwise. (For the choice of 𝒌 ⊥ sketched in the figure, 𝒌 ⊥ is positioned such eq that 𝑓1 ( 𝒌, 𝑡) = 0.) Such an approximation is accurate in the limit where the Fermi momentum 𝑘 𝐹 is much less than the vector potential 𝑒 𝐴/ℏ. (b) Occupation function √︃ 𝑔1 ( 𝒌, 𝑡) as a function of the in-plane momentum magnitude 𝑘 ⊥ = 𝑘 𝑥2 + 𝑘 2𝑦 in the slow relaxation time limit 𝜏 𝑓 ≫ 1, plotted for several values of the 𝑧-momentum 𝑘𝑧. The steady state occupation function [see Eq. (D.6)] in the 𝜏 𝑓 ≫ 1 limit is the time-average of Eq. (D.30), which is time-independent and given by   √︃ 1 ℏ 𝑘 2𝐹 − 𝑘 2𝑧 − (𝑘 ⊥ − 𝑒 𝐴/ℏ) 2 𝜃 (𝑘 2𝐹 − 𝑘 2𝑧 − (𝑘 ⊥ − 𝑒 𝐴/ℏ) 2 ). (D.33) 𝑔1 ( 𝒌, 𝑡) ≈ 𝜋 𝑒𝐴 We plot 𝑔1 ( 𝒌, 𝑡) as a function of 𝑘 ⊥ for several values of 𝑘 𝑧 in Figure D.3(b). We finally exploit the rotational symmetry of 𝑔1 ( 𝒌, 𝑡) to express it in terms of the dimensionless cylindrical coordinates 𝑥 ⊥ , 𝑥 𝑧 , 𝜙, defined via (𝑘 𝑥 , 𝑘 𝑦 , 𝑘 𝑧 ) = 𝑒 𝐴/ℏ(𝑥 ⊥ cos 𝜙, 𝑥⊥ sin 𝜙, 𝑥 𝑧 ), (D.34) G(𝑥 ⊥ , 𝑥 𝑧 , 𝜙) ≡ 𝑔1 [𝑒 𝐴/ℏ(𝑥⊥ cos 𝜙, 𝑥⊥ sin 𝜙, 𝑥 𝑧 ), 0]. (D.35) giving In terms of this function, the in-plane components of the current read, for 𝛼 = 𝑁, 𝑃, 165  3 ∫ ∞ ∫ ∞ ∫ 2𝜋 ℎ𝛼 (𝜙) 𝑑𝑥 𝑧 √︃ G(𝑥 ⊥ , 𝑥 𝑧 , 𝜙), 0 0 −∞ 2 + 𝑥 2 + 1 − 2𝑥 cos 𝜙 𝑥⊥ ⊥ 𝑧 (D.36) where ℎ 𝑁 (𝜙) = 1 − 𝑥 ⊥ cos 𝜙 and ℎ 𝑃 (𝜙) = −𝑥 ⊥ sin 𝜙. Because G(𝑥 ⊥ , 𝑥 𝑧 , 𝜙) is only nonzero when |𝑥 𝑧 | ≤ 𝑘 𝐹 /(𝑒 𝐴/ℏ) and we assume 𝑘 𝐹 ≪ 𝑒 𝐴/ℏ, we have 𝑥 𝑧 ≪ 1. As a result, we can approximate 𝜂 𝑗𝛼 = 𝑐𝜂 𝑒𝑣 𝐹 𝑒 𝐴 (2𝜋) 3 ℏ 𝑑𝑥⊥ 𝑥⊥ 𝑑𝜙  3 ∫ ∞ ∫ 2𝜋 ∫ ∞ 𝑒𝑣 𝐹 𝑒 𝐴 ℎ𝛼 (𝜙) 𝑑𝑥⊥ 𝑥⊥ 𝑑𝜙 √︃ 𝑑𝑥 𝑧 G(𝑥⊥ , 𝑥 𝑧 , 𝜙). (2𝜋) 3 ℏ 0 0 2 + 1 − 2𝑥 cos 𝜙 −∞ 𝑥⊥ ⊥ (D.37) Converting the expression for 𝑔1 ( 𝒌, 0) in Eq. (D.33) to the cylindrical coordinates, we find √︃ 1 2 (D.38) 𝑥 𝐹 − 𝑥 𝑧2 − (𝑥 ⊥ − 1) 2 𝜃 (𝑥 2𝐹 − 𝑥 𝑧2 − (𝑥 ⊥ − 1) 2 ). G(𝑥 ⊥ , 𝑥 𝑧 , 𝜙) ≈ 𝜋 𝜂 𝑗𝛼 ≈ 𝑐𝜂 We next perform the integral over 𝑥 𝑧 , finding ∫ √2 2 𝑥 𝐹 −(𝑥 ⊥ −1) √2 𝑑𝑥 𝑧 G(𝑥⊥ , 𝑥 𝑧 , 𝜙) = − 𝑥 𝐹 −(𝑥 ⊥ −1) 2 1 2 [𝑥 𝐹 − (𝑥 ⊥ − 1) 2 ]. 2 (D.39) Furthermore, in the 𝑥 𝐹 ≪ 1 limit, we approximate !∫   3 ∫ 1+𝑥 𝐹 2𝜋 𝑒𝑣 ℎ𝛼 (𝜙) 1 𝑒 𝐴 𝐹 𝜂 2 2 𝑗𝛼 ≈ 𝑐𝜂 − (𝑥 − 1) ] 𝑑𝜙 𝑑𝑥 𝑥 [𝑥 . √ √︁ ⊥ 𝐹 (2𝜋) 3 2 2 ℏ 0 1−𝑥 𝐹 1 − cos 𝜙 (D.40) Integration over 𝑥 and 𝜙 yields 𝜂 𝑗𝑃 = 0 a𝑛𝑑 𝑒𝑣 𝐹 𝜂 𝑗 𝑁 ≈ 𝑐 𝜂 3 𝑥 3𝐹 3𝜋  𝑒𝐴 ℏ 3 f𝑜𝑟 𝑘 𝐹 ≪ 𝑒 𝐴/ℏ (D.41) for the components parallel and perpendicular to the electric field 𝑬 (𝑡), respectively, where we have used that ℎ 𝑁 (𝜙) ≈ 1 − cos 𝜙 throughout the integration range and ∫ 2𝜋 √︁ √ 𝑑𝜙 1 − cos 𝜙 = 4 2. 0 D.3 Consequences of Finite tilt In this section, we finally consider the photocurrent produced by Weyl node with finite tilt. We focus on an isolated Weyl node with anisotropy along the 𝑧-direction, described by the Hamiltonian 𝐻 ( 𝒌) = ℏ𝑣 𝐹 𝒌 · 𝝈 + ℏ𝑉𝑧 𝜎𝑧 . (D.42) 166 In what follows, we use the unitless tilt parameter 𝜒 ≡ 𝑉𝑧 /𝑣 𝐹 and focus on the slow relaxation limit 𝜏 𝑓 ≫ 1 at temperature 𝑇 ≪ 𝜇/𝑘 𝐵 , 𝑣 𝐹 𝑒 𝐴/𝑘 𝐵 . The anisotropy gives rise to a laser helicity-independent photocurrent in the 𝑧-direction, which we analyze semiclassically in Section D.3. In Section D.3, we show that the anistropy contributes an anomalous velocity component to the oscillating current in the plane of the electric field, giving rise to a laser helicity-dependent plasmon resonance frequency. We semiclassically estimate the additional topological current 𝜂 perpendicular to the electric field, denoted 𝑗 𝑁𝐻 ( 𝜒), and the modifications to 𝑗 𝑁,𝐺 𝜂 and 𝑗 𝑃,𝐺 in leading order of 𝜒. These calculations allow us to calculate the total photocurrent in the presence of a tilt, given by 𝜂 𝜂 𝜂 𝒋 𝜂 (𝑡) = 𝑗 𝑧 (𝑡) 𝑧ˆ + 𝑗 𝑃,𝐺 𝒆ˆ (𝑡) + [ 𝑗 𝑁,𝐺 + 𝑗 𝑁𝐻 ( 𝜒)] 𝑧ˆ × 𝒆ˆ (𝑡). (D.43) Photocurrent along the 𝑧-axis in the weak field limit 𝑒 𝐴/ℏ ≪ 𝑘 𝐹 When the Weyl node exhibits a tilt 𝜒 ≠ 0 along the 𝑧-axis, the topological photocurrent is insensitive to weak anisotropy 𝜒 ≪ 1, as demonstrated numerically in Figure 1(b) in the main text. However, the anisotropy allows for the group velocity term in the semiclassical equations of motion [see Eq. (4)] to contribute a nonzero, laser helicity-independent photocurrent [see Figure 3(d) in the main text], given by   ∫ 𝑇dr ∫ 3𝒌 1 𝑑 1 𝜕𝜀 (𝑡) 𝒌1 𝑧 𝑗¯𝐺 = 𝑑𝑡 𝑔1 ( 𝒌, 𝑡). (D.44) 𝑇dr 0 (2𝜋) 3 ℏ 𝜕𝑘 𝑧 In the slow relaxation limit 𝜏 𝑓 ≫ 1, we can rewrite ∫ 𝑇dr 1 eq 𝑑𝑡 ′ 𝑓1 ( 𝒌, 𝑡 ′) 𝑔1 ( 𝒌, 𝑡) ≈ 𝑇dr 0 (D.45) [see Eq. (D.6)] and express the instantaneous equilibrium occupation function in terms of the instantaneous eigenenergies eq 𝑓1 ( 𝒌, 𝑡) = 𝜃 (𝜀 𝒌1 (𝑡) − 𝜇). (D.46) To provide intuition for the emergence of the helicity-independent photocurrent, eq we note that the tilt produces an asymmetric 𝑓1 ( 𝒌, 𝑡) that preferentially populates electronic states with 𝑧-momentum 𝑘 𝑧 antiparallel to the direction of anisotropy, i.e., sign(𝑘 𝑧 ) = −sign( 𝜒). The steady state occupation 𝑔1 ( 𝒌, 𝑡), which time-averages the instantaneous electronic occupation, therefore acquires larger values for electronic states whose 𝑧-momentum has sign −sign( 𝜒), leading to a net group velocity in the same direction. 167 To estimate the helicity-independent photocurrent, we rewrite Eq. (D.44) as ∫ ∫ 𝑑𝑘 𝑥 𝑑𝑘 𝑦 𝑇dr ′ 𝑧 𝑗¯𝐺 = 𝑒 𝑓 𝑑𝑡 J𝑧 (𝑘 𝑥 , 𝑘 𝑦 , 𝑡, 𝑡 ′), (D.47) 2 (2𝜋) 0 where ∫ 𝑑𝑘 𝑧 1 𝜕𝜀 𝒌1 (𝑡) ′ 𝜃 (𝜀 𝒌1 (𝑡 ′) − 𝜇). (D.48) J𝑧 (𝑘 𝑥 , 𝑘 𝑦 , 𝑡, 𝑡 ) ≡ 2𝜋 ℏ 𝜕𝑘 𝑧 As discussed previously, the slow relaxation time 𝜏 gives rise to a lagging of the Fermi volume along the trajectory of the Weyl node traced out by the vector potential. This is captured in Eq. (D.47) because the current at time 𝑡 is given by the average of the occupation-weighted group velocity J𝑧 (𝑘 𝑥 , 𝑘 𝑦 , 𝑡, 𝑡 ′) set by the instantaneous thermal equilibrium distribution at all other times 𝑡 ′. Crucially, J𝑧 (𝑘 𝑥 , 𝑘 𝑦 , 𝑡, 𝑡 ′) is nonzero with sign given by sign( 𝜒) only when 𝜒 ≠ 0 and 𝑡 ≠ 𝑡 ′, as illustrated in Figure D.4(a), which is the origin of the helicity-independent photocurrent. When 𝜒 < 0, the electronic momenta with nonzero instantaneous equilibrium occupation 𝜃 (𝜀 𝒌1 (𝑡 ′) − 𝜇) (thick orange line) at previous time 𝑡 ′ < 𝑡 preferentially populates electronic states with negative 𝑧-group velocity 𝜕𝜀 𝒌1 (𝑡)/𝜕𝑘 𝑧 < 0 at time 𝑡, allowing the integral in Eq. (D.48) to attain a nonzero negative value. Eq. (D.48) can be solved by integration by parts, yielding i 1 h (D.49) J𝑧 (𝑘 𝑥 , 𝑘 𝑦 , 𝑡, 𝑡 ′) = 𝜀 𝒌1 (𝑡)| 𝑘 𝑧 =𝑘 +𝑧 (𝑡 ′ ) − 𝜀 𝒌1 (𝑡)| 𝑘 𝑧 =𝑘 𝑧− (𝑡 ′ ) , 2𝜋ℏ where 𝜀 𝒌1 (𝑡 ′)| 𝑘 𝑧 =𝑘 ±𝑧 (𝑡 ′ ) = 𝜇. To compute the values of 𝑘 ±𝑧 (𝑡 ′), we consider the instantaneous eigenenergy at time 𝑡 ′, given by 𝜀 𝒌1 (𝑡 ′) = ℏ𝑣 𝐹 | 𝒌 + 𝑒 𝑨(𝑡 ′)/ℏ| + ℏ𝑣 𝐹 𝜒𝑘 𝑧 . (D.50) Defining 𝑘 𝐹 ≡ 𝜇/(ℏ𝑣 𝐹 ) and solving for the instantaneous Fermi surface 𝜀 𝒌1 (𝑡 ′) = ℏ𝑣 𝐹 𝑘 𝐹 gives rise to two roots √︃ 𝜒𝑘 𝐹 ∓ ( 𝜒2 − 1) [ 𝒌˜ ⊥ (𝑡 ′)] 2 + 𝑘 2𝐹 𝑘 ±𝑧 (𝑡 ′) ≡ , (D.51) 𝜒2 − 1 where 𝒌˜ ⊥ (𝑡 ′) ≡ (𝑘 𝑥 + 𝑒 𝐴𝑥 (𝑡 ′)/ℏ, 𝑘 𝑦 + 𝑒 𝐴 𝑦 (𝑡 ′)/ℏ) (D.52) denotes the in-plane momentum shifted by the drive-induced vector potential via minimal coupling. Therefore, using Eq. (D.49), we find that √︃ 𝑣𝐹 ′ J𝑧 (𝑘 𝑥 , 𝑘 𝑦 , 𝑡, 𝑡 ) = −[ 𝒌˜ ⊥ (𝑡)] 2 + 𝑘 2𝐹 𝜋 ª © 𝑘𝐹 ® ­ × ­1 − √︃ ® 𝜃 (𝑘 2𝐹 − [ 𝒌˜ ⊥ (𝑡)] 2 ) 𝜒 + 𝑂 ( 𝜒3 ). ® ­ 2 −[ 𝒌˜ ⊥ (𝑡)] 2 + [ 𝒌˜ ⊥ (𝑡 ′)] 2 + 𝑘 𝐹 ¬ « (D.53) 168 To make further progress, we take the limit of weak fields, given by 𝑒 𝐴/ℏ ≪ 𝑘 𝐹 , in which case we can take the limit |[𝒌 ⊥ (𝑡 ′)] 2 − [𝒌 ⊥ (𝑡)] 2 | ≪ 𝑘 𝐹 and obtain √︃ 𝑣𝐹 [ 𝒌˜ ⊥ (𝑡 ′)] 2 − [ 𝒌˜ ⊥ (𝑡)] 2 J𝑧 (𝑘 𝑥 , 𝑘 𝑦 , 𝑡, 𝑡 ′) ≈ −[ 𝒌˜ ⊥ (𝑡)] 2 + 𝑘 2𝐹 𝜃 (𝑘 2𝐹 − | 𝒌˜ ⊥ (𝑡)| 2 ) 𝜒. 2 2𝜋 𝑘𝐹 (D.54) 𝑧 ′ Note that after integration over 𝑡 , the photocurrent 𝑗¯𝐺 is time-independent. Therefore, without loss of generality, we set 𝑡 = 0. To estimate | 𝒌 ⊥ (𝑡)| and | 𝒌 ⊥ (𝑡 ′)|, we transform to the polar coordinates 𝑘 𝑥 = 𝑘 ⊥ cos 𝜑 and 𝑘 𝑦 = 𝑘 ⊥ sin 𝜑, where   𝑒𝐴 𝑒𝐴 𝑒𝐴 ′
′
′ 2 2 −2 cos 2𝜋 𝑓 𝑡 + 2𝑘 ⊥ sin 𝜑 + 2𝜋 𝑓 𝑡 . [𝒌 ⊥ (𝑡 )] − [𝒌 ⊥ (𝑡)] = ℏ ℏ ℏ (D.55) 2 2 2 2 Furthermore, in the limit 𝑒 𝐴/ℏ ≪ 𝑘 𝐹 , 𝑘 𝐹 − [𝒌 ⊥ (𝑡)] ≈ 𝑘 𝐹 − 𝑘 ⊥ . Using these approximations,  2 ∫ 𝑘𝐹 ∫ 𝑇dr ∫ 2𝜋 √︃ 𝑒𝐴 𝑣𝐹 ′ 2 2 ¯𝑗 𝑧 = 𝑒 𝑓 𝑘 𝐹 − 𝑘⊥ 𝑑𝑘 ⊥ 𝑘 ⊥ 𝑑𝑡 𝑑𝜑 𝐺 2 2 ℏ (2𝜋) 0 2𝜋𝑘 𝐹 0 0 (D.56)   𝑘⊥ ′
′
sin 𝜑 + 2𝜋 𝑓 𝑡 . × 1 − 2 cos 2𝜋 𝑓 𝑡 + 2 𝑒 𝐴/ℏ Upon integration over 𝜑, 𝑘, and 𝑡 ′, we find that  2 𝑒𝜇 𝑒𝐴 𝑧 𝑗¯𝐺 = 𝜒 f𝑜𝑟 12𝜋 2 ℏ ℏ 𝑘 𝐹 ≫ 𝑒 𝐴/ℏ. (D.57) Photocurrent in the 𝑥-𝑦 plane We finally discuss how band anisotropy modifies the photocurrent induced by the circularly polarized laser in the 𝑥-𝑦 plane. In Section D.3, we discuss the effect of weak band anisotropy (𝜒 ≪ 1) on the occupation function. In Section D.3, we use the occupation function to understand the emergence of an additional topological current 𝑗 𝑁𝐻 ( 𝜒) in the direction perpendicular to the laser electric field, produced by the anomalous velocity of electrons. This mechanism gives rise to the chiral plasmons discussed in the main text [see Figure 8.1(e)]. In Section D.3, we demonstrate that the nonzero Weyl node tilt does not modify the group velocity contribution to the in-plane current to first order in 𝜒. As in Section D.2, we analyze the photocurrent in the small Fermi volume limit 𝑘 𝐹 ≪ 𝑒 𝐴/ℏ. Occupation function for 𝜒 ≪ 1 and 𝑘 𝐹 ≪ 𝑒 𝐴/ℏ The tilt along the 𝑧-axis reshapes the instantaneous equilibrium occupation function eq 𝑓1 ( 𝒌, 𝑡) in the conduction band of the Weyl node, which in turn modifies the steady 169 Figure D.4: Origin of the laser helicity-independent photocurrent in a tilted WN. In the limit 𝜏 𝑓 ≫ 1, the steady state occupation of bands at any given time 𝑡 is the time average of the instantaneous equilibrium occupation function 𝜃 (𝜀 𝒌1 (𝑡 ′) − 𝜇) at all other times 𝑡 ′. The black solid and dashed gray curves represent the electronic bands at representative times 𝑡 and 𝑡 ′, respectively. The instantaneous equilibrium distribution at time 𝑡 ′ (orange) preferentially fills states with 𝜕𝜀 𝒌1 (𝑡)/𝜕𝑘 𝑧 < 0, producing a net helicity-independent current. state ocupation function 𝑔1 ( 𝒌, 𝑡). To recalculate 𝑔1 ( 𝒌, 𝑡), we revisit the derivation in Section D.2, using the approximate expression for the occupation function in the slow relaxation limit 𝜏 𝑓 ≫ 1 given by ∫ 𝑇dr 1 eq 𝑑𝑢 𝑓1 ( 𝒌, 𝑢). (D.58) 𝑔1 ( 𝒌, 𝑡) ≈ 𝑇dr 0 In the presence of a tilt, we again have eq 𝑓1 ( 𝒌, 𝑡) ≈ 𝜃 (min |𝜑−2𝜋 𝑓 𝑡+2𝜋𝑛|−𝛿𝜑 𝜒 (𝑘 ⊥ , 𝑘 𝑧 ))𝜃 (𝑘 2𝐹 −𝑘 2𝑧 −(𝑘 ⊥ −𝑒 𝐴/ℏ) 2 ) (D.59) 𝑛∈Z [see Eq. (D.30)], but with a modified 𝛿𝜑 𝜒 (𝑘 ⊥ , 𝑘 𝑧 ) defined by 𝛿𝜑 𝜒 (𝑘 ⊥ , 𝑘 𝑧 ) = 𝜑 𝜒,+ (𝑘 ⊥ , 𝑘 𝑧 ) − 𝜑 𝜒,− (𝑘 ⊥ , 𝑘 𝑧 ), where ℏ𝑣 𝐹 |(𝑘 ⊥ cos[𝜑 𝜒,± (𝑘 ⊥ , 𝑘 𝑧 )] − 𝑒 𝐴/ℏ, 𝑘 ⊥ sin[𝜑 𝜒,± (𝑘 ⊥ , 𝑘 𝑧 )]| + ℏ𝜒𝑘 𝑧 = ℏ𝑣 𝐹 𝑘 𝐹 (D.60) parameterizes the edge of the Fermi volume [see Eq. (D.31)] which may be rewritten in the form 𝑒 2 𝐴2 𝑒𝐴 2 (1 + 𝜒2 )𝑘 2𝐹 = 𝑘 ⊥ − 2 𝑘 ⊥ cos[𝜑 𝜒,± (𝑘 ⊥ , 𝑘 𝑧 )] + ℏ ℏ √︁ √︁ 2 2 + ( 1 − 𝜒 𝑘 𝑧 + 𝜒𝑘 𝐹 ) + 2(1 − 1 − 𝜒2 ) 𝜒𝑘 𝐹 𝑘 𝑧 . (D.61) Expanding cos[𝜑 𝜒,± (𝑘 ⊥ , 𝑘 𝑧 )] ≈ 1 − [𝜑 𝜒,± (𝑘 ⊥ , 𝑘 𝑧 )] 2 /2 in the limit 𝑘 𝐹 ≪ 𝑒 𝐴/ℏ and evaluating 𝛿𝜑 𝜒 (𝑘 ⊥ , 𝑘 𝑧 ) = 𝜑 𝜒,+ (𝑘 ⊥ , 𝑘 𝑧 ) − 𝜑 𝜒,− (𝑘 ⊥ , 𝑘 𝑧 ), we find that [𝛿𝜑 𝜒 (𝑘 ⊥ , 𝑘 𝑧 )] 2 ≈ (ℏ/𝑒 𝐴) 2 [𝑘 2𝐹 − (𝑘 𝑧 + 𝜒𝑘 𝐹 ) 2 − (𝑘 ⊥ − 𝑒 𝐴/ℏ) 2 ], (D.62) 170 where we have discarded terms of 𝑂 ( 𝜒2 ) or higher. Using Eq. (D.58), we time-average the equilibrium occupation function, Eq. (D.59), to find   √︃ 1 ℏ 𝑔1 ( 𝒌, 𝑡) ≈ 𝑘 2𝐹 − (𝑘 𝑧 + 𝜒𝑘 𝐹 ) 2 − (𝑘 ⊥ − 𝑒 𝐴/ℏ) 2 𝜃 (𝑘 2𝐹 −(𝑘 𝑧 +𝜒𝑘 𝐹 ) 2 −(𝑘 ⊥ −𝑒 𝐴/ℏ) 2 ). 𝜋 𝑒𝐴 (D.63) Note that 𝑔1 ( 𝒌, 𝑡) is time-independent. We see that the primary effect of the tilt in the limit of weak anisotropy 𝜒 ≪ 1 is the shift of the Fermi volume along the 𝑘 𝑧 axis by 𝜒𝑘 𝐹 . Topological current 𝑗 𝑁𝐻 ( 𝜒) for 𝜒 ≪ 1 and 𝑘 𝐹 ≪ 𝑒 𝐴/ℏ Due to the asymmetry of the electronic occupation function along the 𝑘 𝑧 -axis, and the combination of rotation and time-translation symmetry of the problem, the anomalous velocity in Eq. (D.25) contributes a photocurrent given by "   #  ∫ 𝑒 𝑬 (0) 𝑑3 𝒌 𝑎 𝑒𝛀1 𝒌 + 𝑨(0) · 𝑧ˆ 𝑧ˆ × 𝑔1 ( 𝒌, 0). (D.64) 𝒋⊥ = 𝑒 ℏ ℏ (2𝜋) 3 𝑎 The current flows perpendicular to 𝑬 (0) and is given by 𝒋 ⊥ = 𝑗 𝑁𝐻 ( 𝜒) 𝑧ˆ × 𝑬 (0), where ∫ 𝑑3 𝒌 𝑘𝑧 𝑒2 𝐻 𝑔1 ( 𝒌, 0). (D.65) 𝑗 𝑁 ( 𝜒) = 𝜉 𝐸 3 2ℏ (2𝜋) {[𝑘 𝑥 + 𝑒 𝐴/ℏ] 2 + 𝑘 2𝑦 + 𝑘 2𝑧 }3/2 To evaluate the integral, we again transform to the dimensionless coordinates given by 𝑥 𝐹 = 𝑘 𝐹 /(𝑒 𝐴/ℏ), 𝑥 ⊥ = 𝑘 ⊥ /(𝑒 𝐴/ℏ), and 𝑥 𝑧 = 𝑘 𝑧 /(𝑒 𝐴/ℏ), such that G(𝑥 ⊥ , 𝑥 𝑧 , 𝜙) ≡ 𝑔1 [𝑒 𝐴/ℏ(𝑥 ⊥ cos 𝜙, 𝑥⊥ sin 𝜙, 𝑥 𝑧 ), 0]. This leads to  2 ∫ ∞ ∫ 2𝜋 ∫ ∞ 𝑒 𝑓 𝑒𝐴 𝑥𝑧 𝐻 𝑗 𝑁 ( 𝜒) = 𝜉 2 𝑑𝑥⊥ 𝑥⊥ 𝑑𝜙 𝑑𝑥 𝑧 2 G(𝑥 ⊥ , 𝑥 𝑧 , 𝜙). ℏ 8𝜋 (𝑥 ⊥ + 1 − 2𝑥 ⊥ cos 𝜙 + 𝑥 𝑧2 ) 3/2 0 0 −∞ (D.66) Next we caluclate G(𝑥⊥ , 𝑥 𝑧 , 𝜙) using Eq. (D.63): √︃ 1 2 G(𝑥 ⊥ , 𝑥 𝑧 , 𝜙) ≈ 𝑥 − (𝑥 𝑧 + 𝛿) 2 − (𝑥 ⊥ − 1) 2 𝜃 (𝑥 2𝐹 − (𝑥 𝑧 + 𝛿) 2 − (𝑥 ⊥ −1) 2 ), (D.67) 𝜋 𝐹 where 𝛿 ≡ 𝜒𝑥 𝐹 . This leads to   2 ∫ 𝑥 𝐹 −𝛿 ∫ 1+√𝑥 2 −(𝑥 𝑧 +𝛿) 2 𝐹 𝑒 𝑓 𝑒 𝐴 𝑗 𝑁𝐻 ( 𝜒) ≈ 𝜉 3 𝑑𝑥 𝑧 𝑑𝑥 ⊥ 𝑥⊥ × √ ℏ 8𝜋 −𝑥 𝐹 −𝛿 1− 𝑥 2𝐹 −(𝑥 𝑧 +𝛿) 2 √︃ ∫ 2𝜋 𝑥 2𝐹 − (𝑥 𝑧 + 𝛿) 2 − (𝑥⊥ − 1) 2 𝑑𝜙 2 𝑥𝑧 . [𝑥 ⊥ + 1 − 2𝑥 ⊥ cos 𝜙 + 𝑥 𝑧2 ] 3/2 0 (D.68) 171 We first perform integration over 𝜙 by expanding the integrand around 𝜙 ∼ 𝜋, where the integrand is largest ∫ 2𝜋 1 1 2 𝑑𝜙 2 ≈ 3/2 2 , (D.69) 2 2 3/2 [𝑥 ⊥ + 1 − 2𝑥 ⊥ cos 𝜙 + 𝑥 𝑧 ] 0 𝑥 ⊥ 𝑥 𝑧 + 𝛿𝑥 ⊥ where 𝛿𝑥⊥ ≡ 𝑥⊥ − 1. Therefore, √︃   2 ∫ 𝑥 𝐹 −𝛿 2 ∫ √𝑥 2 −(𝑥 𝑧 +𝛿) 2 𝑥 2𝐹 − (𝑥 𝑧 + 𝛿) 2 − 𝛿𝑥 ⊥ 𝐹 𝑒 𝑓 𝑒𝐴 𝐻 𝑑𝛿𝑥 𝑑𝑥 𝑧 √ 𝑥𝑧 . 𝑗 𝑁 ( 𝜒) ≈ 𝜉 3 2 ℏ 4𝜋 𝑥 𝑧2 + 𝛿𝑥 ⊥ − 𝑥 2𝐹 −(𝑥 𝑧 +𝛿) 2 −𝑥 𝐹 −𝛿 (D.70) ′ Transforming to the shifted coordinates 𝑥 𝑧 = 𝑥 𝑧 + 𝛿, we find √︃   2 ∫ 𝑥𝐹 2 ∫ √𝑥 2 −(𝑥 ′𝑧 ) 2 𝑥 2𝐹 − (𝑥 ′𝑧 ) 2 − 𝛿𝑥 ⊥ 𝐹 𝑒 𝑓 𝑒𝐴 ′ 𝐻 𝑑𝛿𝑥 (𝑥 ′𝑧 − 𝛿). 𝑗 𝑁 ( 𝜒) ≈ 𝜉 3 𝑑𝑥 𝑧 √ 2 ℏ 4𝜋 (𝑥 ′𝑧 − 𝛿) 2 + 𝛿𝑥 ⊥ −𝑥 𝐹 − 𝑥 2𝐹 −(𝑥 ′𝑧 ) 2 (D.71) We now use the integral identity "√︂ # ∫ 𝑎 √ 2 2 2 𝑎 −𝑥 𝑎 𝑑𝑥 2 =𝜋 (D.72) 1+ 2 −1 2 𝑏 +𝑥 𝑏 −𝑎 to obtain ∫ √𝑥 2 −(𝑥 ′𝑧 ) 2 √︃ 2 𝑥 2𝐹 − (𝑥 ′𝑧 ) 2 − 𝛿𝑥 ⊥ 𝐹 √ 𝑑𝛿𝑥 − 𝑥 2𝐹 −(𝑥 ′𝑧 ) 2 2 (𝑥 ′𝑧 − 𝛿) 2 + 𝛿𝑥⊥ √︄ = 𝜋 1+ 𝑥 2𝐹 − (𝑥 ′𝑧 ) 2 (𝑥 ′𝑧 − 𝛿) 2 − 𝜋. (D.73) Therefore, 𝑒𝑓 𝑗 𝑁𝐻 ( 𝜒) = 𝜉 2 4𝜋  𝑒𝐴 ℏ  2 ∫ 𝑥𝐹 𝑑𝑥 ′𝑧  sign(𝑥 ′𝑧 − 𝛿)  √︃ ′ 𝑥 2𝐹 − (𝑥 ′𝑧 ) 2 + (𝑥 ′𝑧 − 𝛿) 2 − (𝑥 𝑧 − 𝛿) . −𝑥 𝐹 (D.74) Finally, upon integrating over 𝑥 ′𝑧 and expanding in 𝜒, we find  2 𝑒 𝑓 𝑒𝐴 𝐻 𝑥 2𝐹 𝜒 f𝑜𝑟 𝑘 𝐹 ≪ 𝑒 𝐴/ℏ. 𝑗 𝑁 ( 𝜒) ≈ −𝜉 2 4𝜋 𝜂 ℏ (D.75) 𝜂 Current 𝑗 𝑁,𝐺 and 𝑗 𝑃,𝐺 for 𝜒 ≪ 1 and 𝑘 𝐹 ≪ 𝑒 𝐴/ℏ Lastly, we consider the effect of the tilt on the in-plane current components produced by the group velocity of electrons. The group velocity contribution to the currents perpendicular and parallel to the electric field 𝑬 (𝑡) are respectively given by ∫ 𝑑3 𝒌 𝑘 𝑥 − 𝑒 𝐴/ℏ 𝜂 𝑗 𝑁,𝐺 = 𝑐 𝜂 𝑒𝑣 𝐹 𝑔1 ( 𝒌, 0), (D.76) √︃ 3 (2𝜋) [𝑘 𝑥 + 𝑒 𝐴/ℏ] 2 + 𝑘 2𝑦 + 𝑘 2𝑧 ∫ 𝑘𝑦 𝑑3 𝒌 𝜂 𝑗 𝑃,𝐺 = 𝑐 𝜂 𝑒𝑣 𝐹 𝑔1 ( 𝒌, 0), (D.77) √︃ (2𝜋) 3 [𝑘 + 𝑒 𝐴/ℏ] 2 + 𝑘 2 + 𝑘 2 𝑥 𝑦 𝑧 172 as predicted by the semiclassical equations of motion [see Section D.2 for details]. Transforming Eq. (D.76) into the dimensionless cylindrical coordinates parameterized by 𝑥 ⊥ , 𝑥 𝑧 , and 𝜙, we obtain, for 𝛼 = 𝑁, 𝑃, 𝑒𝑣 𝐹 𝜂 𝑗 𝛼,𝐺 ≈ 𝑐 𝜂 (2𝜋) 3  𝑒𝑣 𝐹 𝜂 𝑗 𝛼,𝐺 ≈ 𝑐 𝜂 (2𝜋) 3  3 ∫ ∞ ∫ ∞ ∫ 2𝜋 ℎ𝛼 (𝜙) G(𝑥 ⊥ , 𝑥 𝑧 , 𝜙), 𝑑𝑥 𝑧 √︃ −∞ 0 0 2 2 𝑥 ⊥ + 1 − 2𝑥 ⊥ cos 𝜙 + 𝑥 𝑧 (D.78) where ℎ 𝑁 (𝜙) = 1 − 𝑥 ⊥ cos 𝜙 and ℎ 𝑃 (𝜙) = −𝑥 ⊥ sin 𝜙. Because G(𝑥 ⊥ , 𝑥 𝑧 , 𝜙) is only nonzero when |𝑥 𝑧 | ≲ 𝑘 𝐹 and we assume 𝑘 𝐹 ≪ 𝑒 𝐴/ℏ, we have 𝑥 𝑧 ≪ 1. As a result, we can approximate 𝑒𝐴 ℏ 𝑒𝐴 ℏ 𝑑𝑥⊥ 𝑥 ⊥ 3 ∫ ∞ 𝑑𝜙 ∫ 2𝜋 𝑑𝑥⊥ 𝑥 ⊥ 0 0 ℎ𝛼 (𝜙) 𝑑𝜙 √︃ 2 + 1 − 2𝑥 cos 𝜙 𝑥⊥ ⊥ ∫ ∞ 𝑑𝑥 𝑧 G(𝑥⊥ , 𝑥 𝑧 , 𝜙). −∞ (D.79) We now use the occupation function G(𝑥 ⊥ , 𝑥 𝑧 , 𝜙) in the presence of a tilt in Eq. (D.67) to evaluate the integral above. The integral over 𝑥 𝑧 reads ∫ ∞ ∫ √𝑥 2 −(𝑥⊥ −1) 2 −𝛿 √︃ 𝐹 1 2 𝑑𝑥 𝑧 G(𝑥 ⊥ , 𝑥 𝑧 , 𝜙) = √ 𝑑𝑥 𝑧 𝑥 𝐹 − (𝑥 𝑧 + 𝛿) 2 − (𝑥⊥ − 1) 2 , 2 2 𝜋 −∞ − 𝑥 𝐹 −(𝑥 ⊥ −1) −𝛿 (D.80) which evaluates to ∫ √𝑥 2 −(𝑥⊥ −1) 2 −𝛿 √︃ 𝐹 1 1 2 𝑑𝑥 𝑥 𝐹 − (𝑥 𝑧 + 𝛿) 2 − (𝑥 ⊥ − 1) 2 = [𝑥 2𝐹 − (𝑥 ⊥ − 1) 2 ], (D.81) 𝑧 √2 𝜋 2 − 𝑥 𝐹 −(𝑥⊥ −1) 2 −𝛿 which is identical to the result without a tilt, see Eq. (D.39). Thus, to first order 𝜂 𝜂 in 𝜒, the group velocity contribution to the in-plane currents 𝑗 𝑁,𝐺 and 𝑗 𝑃,𝐺 are unmodified relative to the isotropic case, and given by 𝜂 𝑗 𝑃,𝐺 = 0 a𝑛𝑑 𝑒𝑣 𝐹 𝜂 𝑗 𝑁,𝐺 ≈ 𝑐 𝜂 3 𝑥 3𝐹 3𝜋  𝑒𝐴 ℏ 3 f𝑜𝑟 𝑘 𝐹 ≪ 𝑒 𝐴/ℏ, (D.82) for the components parallel and perpendicular to the electric field 𝑬 (𝑡), respectively. 173 Bibliography [1] Dmitry Abanin, Wojciech De Roeck, Wen Wei Ho, and François Huveneers. A rigorous theory of many-body prethermalization for periodically driven and closed quantum systems. Communications in Mathematical Physics, 354(3): 809–827, June 2017. 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