Numerical Methods for Many-Body Quantum Dynamics Citation White, Christopher David (2019) Numerical Methods for Many-Body Quantum Dynamics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/VC0P-3K15. https://resolver.caltech.edu/CaltechTHESIS:05292019-112014488 Abstract This thesis describes two studies of the dynamics of many-body quantum systems with extensive numerical support. In Part I we first give a new algorithm for simulating the dynamics of one-dimensional systems that thermalize (that is, come to local thermal equilibrium). The core of this algorithm is a new truncation for matrix product operators, which reproduces local properties faithfully without reproducing non-local properties (e.g. the information required for OTOCs). To the extent that the dynamics depends only on local operators, timesteps interleaved with this truncation will reproduce that dynamics. We then apply this to algorithm to Floquet systems: first to clean, non-integrable systems with a high-frequency drive, where we find that the system is well-described by a natural diffusive phenomenology; and then to disordered systems with low-frequency drive, which display diffusion — not subdiffusion — at appreciable disorder strengths. In Part II, we study the utility of many-body localization as a medium for a thermodynamic engine. We first construct a small ("mesoscale") engine that gives work at high efficiency in the adiabatic limit, and show that thanks to the slow spread of information in many body localized systems, these mesoscale engines can be chained together without specially engineered insulation. Our construction takes advantage of precisely the fact that MBL systems do not thermalize. We then show that these engines still have high efficiency when run at finite speed, and we compare to competitor engines. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Numerical methods, condensed matter theory, thermalization, many body localization, quantum thermodynamics, matrix product states, dynamics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Refael, Gil Group: Institute for Quantum Information and Matter Thesis Committee: Motrunich, Olexei I. (chair) Mong, Roger S. Endres, Manuel A. Refael, Gil Defense Date: 16 May 2019 Funders: Funding Agency Grant Number NSF DGE1144469 Gordon and Betty Moore Foundation UNSPECIFIED NSF DMR-1410435 Packard Foundation UNSPECIFIED NSF PHY0803371 NSF DMR-1653271 Institute for Theoretical Atomic, Molecular, and Optical Physics at Harvard University UNSPECIFIED Smithsonian Astrophysical Observatory UNSPECIFIED Walter Burke Foundation UNSPECIFIED Defense Advanced Research Projects Agency (DARPA) UNSPECIFIED Alfred P. Sloan Foundation UNSPECIFIED W. M. Keck Foundation UNSPECIFIED NSF DGE1745301 Record Number: CaltechTHESIS:05292019-112014488 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05292019-112014488 DOI: 10.7907/VC0P-3K15 Related URLs: URL URL Type Description https://doi.org/10.1103/physrevb.97.035127 DOI Article adapted for Ch. II. http://arxiv.org/abs/1902.01859v1 arXiv Article adapted for Ch. III. https://doi.org/10.1103/physrevb.97.035127 DOI Article adapted for Part II. ORCID: Author ORCID White, Christopher David 0000-0002-8372-2492 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 11558 Collection: CaltechTHESIS Deposited By: Christopher White Deposited On: 10 Jun 2019 22:30 Last Modified: 28 Feb 2023 19:09 Thesis Files Preview PDF - Final Version See Usage Policy. 5MB Repository Staff Only: item control page

Numerical methods for many-body quantum dynamics Thesis by Christopher David White In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2019 Defended 16 May 2019 ii c 2019 Christopher David White ORCID: 0000-0002-8372-2492 Some rights reserved. This thesis is distributed under a Creative Commons Attribution NonCommercial-ShareAlike License. iii ACKNOWLEDGEMENTS cum caritate et gratia magna parochiae et populo Sancti Thomas Apostoli Holivudensis. “congregavit nos in unum Christi amor; exsultemus, et in ipso jucundemur.” iv ABSTRACT This thesis describes two studies of the dynamics of many-body quantum systems with extensive numerical support. In Part I we first give a new algorithm for simulating the dynamics of onedimensional systems that thermalize (that is, come to local thermal equilibrium). The core of this algorithm is a new truncation for matrix product operators, which reproduces local properties faithfully without reproducing non-local properties (e.g. the information required for OTOCs). To the extent that the dynamics depends only on local operators, timesteps interleaved with this truncation will reproduce that dynamics. We then apply this to algorithm to Floquet systems: first to clean, nonintegrable systems with a high-frequency drive, where we find that the system is well-described by a natural diffusive phenomenology; and then to disordered systems with low-frequency drive, which display diffusion—not subdiffusion— at appreciable disorder strengths. In Part II, we study the utility of many-body localization as a medium for a thermodynamic engine. We first construct a small (“mesoscale”) engine that gives work at high efficiency in the adiabatic limit, and show that thanks to the slow spread of information in many body localized systems, these mesoscale engines can be chained together without specially engineered insulation. Our construction takes advantage of precisely the fact that MBL systems do not thermalize. We then show that these engines still have high efficiency when run at finite speed, and we compare to competitor engines. v PUBLISHED CONTENT AND CONTRIBUTIONS This thesis contains material from: White, C.D., Zaletel, M.,, Mong, R.S.K., and Refael, G. “Quantum Dynamics of Thermalizing Systems.” Phys. Rev. B 97, 035127 (2018). DOI: 10.1103/physrevb.97.035127. CDW constructed the algorithm used, wrote the algorithm-specific code, constructed the benchmarking protocol, carried out the simulations, and performed the analysis. Ye, B., Machado, F., White, C.D., Mong, R.S.K., and Yao, N. “Emergent hydrodynamics in Floquet quantum systems”. arXiv:1902.01859. CDW participated in the conception of the project and the analysis of the results (in particular the framing in terms of hydrodynamics). In addition, the project used CDW’s algorithm-specific code. Yunger Halpern, N., White, C.D., Gopalakrishnan, S., and Refael, G. “Quantum engine based on many-body localization”. Phys. Rev. B. 99, 024203. DOI: 10.1103/physrevb.99.024203 CDW participated in the conception of the project, carried out all of the numerical calculations and many of the analytical calculations, and participated in constructing the physical arguments. vi TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Published Content and Contributions . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I: Background: MBL and MPS . . . . . . . . . . . . . . . . . 1.1 Many body localization . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Many-body localization in perturbation theory . . . . . 1.1.2 Long-time memory in MBL systems . . . . . . . . . . . 1.1.3 Level spacing . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Local dynamics at finite time . . . . . . . . . . . . . . 1.2 Matrix product states . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Graphical notation for tensors and tensor contractions 1.2.2 Expectation values and correlations in MPSs . . . . . . 1.2.3 Canonical forms and orthogonality centers . . . . . . . 1.3 Dynamics and matrix product states . . . . . . . . . . . . . . . 1.4 Matrix product operators . . . . . . . . . . . . . . . . . . . . . 1.4.1 MPO representation of an exponential interaction . . . 1.4.2 MPO representation of a power-law Hamiltonian . . . . 1.4.3 MPDOs . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Matrix product density operator representations of Gibbs states . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Some essentials of quantum thermodynamics . . . . . . . . . . iii iv v vi ix 1 1 1 5 7 10 10 14 15 16 20 21 22 23 24 26 26 I Thermalizing systems and matrix product density operators 28 Chapter II: Simulating quantum dynamics of thermalizing systems . . 2.1 Background and intuition . . . . . . . . . . . . . . . . . . . . . 2.1.1 Background: matrix product state methods . . . . . . 2.1.2 Intuition: thermalization and computation . . . . . . . 2.2 Method: truncation of MPDOs . . . . . . . . . . . . . . . . . . 2.2.1 Setting, notation, and tools . . . . . . . . . . . . . . . 2.2.2 Re-writing the MPDO to expose properties whose preservation we guarantee . . . . . . . . . . . . . . . . . . . . 2.2.3 Modifying the MPDO . . . . . . . . . . . . . . . . . . 2.2.4 Preservation of l-site operators . . . . . . . . . . . . . . 2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 29 30 30 33 36 37 39 41 44 46 46 vii 2.3.2 Application: pure-state evolution . . . . . . . . . . . . 47 2.3.3 Application: mixed-state evolution (near equilibrium) . 52 2.3.4 Application: mixed-state evolution (far from equilibrium) 52 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Chapter III: Heating, prethermalization, and hydrodynamics in a highfrequency Floquet system . . . . . . . . . . . . . . . . . . . . . . . 59 3.1 Model and Floquet phenomenology . . . . . . . . . . . . . . . 60 3.2 Benchmarking DMT . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2.1 Benchmarking Krylov subspace dynamics . . . . . . . . 65 3.2.2 DMT and Krylov . . . . . . . . . . . . . . . . . . . . . 66 3.2.3 Entropy in DMT . . . . . . . . . . . . . . . . . . . . . 69 3.2.4 Effect of Trotter step size on DMT numerics . . . . . . 69 3.3 Analysis and results . . . . . . . . . . . . . . . . . . . . . . . . 71 3.3.1 Approach to Gibbs Ensemble . . . . . . . . . . . . . . 73 3.3.2 Heating . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.3.3 Extracting diffusion coefficients of a spatially uniform, static Hamiltonian . . . . . . . . . . . . . . . . . . . . 78 3.3.4 Classical diffusion equation . . . . . . . . . . . . . . . . 82 3.3.5 Hydrodynamics in large spin chains . . . . . . . . . . . 86 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Chapter IV: Hydrodynamics in a disordered low-frequency Floquet system 88 4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.2.1 Convergence testing . . . . . . . . . . . . . . . . . . . . 91 4.2.2 Comparison with exact simulations . . . . . . . . . . . 93 4.2.3 Quantity of interest . . . . . . . . . . . . . . . . . . . . 95 4.2.4 Disorder averaging . . . . . . . . . . . . . . . . . . . . 96 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 II MBL-mobile: Many-body-localized engine 101 Chapter V: MBL and thermodynamic engines . . . . . . . . . . . . . . Chapter VI: The MBL-Mobile in its adiabatic limit . . . . . . . . . . . 6.1 Qubit toy model for the mesoscale engine . . . . . . . . . . . . 6.2 Set-up for the mesoscale MBL engine . . . . . . . . . . . . . . 6.3 Notation and definitions: . . . . . . . . . . . . . . . . . . . . . 6.4 Quantitative analysis of the adiabatic mesoscale engine: some easy limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Quantitative analysis of the adiabatic mesoscale engine: a more detailed calculation . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Partial-swap model of thermalization . . . . . . . . . . 6.5.2 Average heat hQ2 i absorbed during stroke 2 . . . . . . 6.5.3 Average heat hQ4 i absorbed during stroke 4 . . . . . . 6.5.4 Average per-cycle power hWtot i . . . . . . . . . . . . . 102 105 105 107 110 111 113 113 114 116 118 viii 6.5.5 Efficiency ηMBL in the adiabatic approximation . . . . . 120 6.6 MBL engine in the thermodynamic limit . . . . . . . . . . . . 121 6.7 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . 125 6.7.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 125 6.7.2 Scaling factor . . . . . . . . . . . . . . . . . . . . . . . 126 6.7.3 Representing states and Hamiltonians . . . . . . . . . . 129 6.7.4 Strokes 1 and 3: tuning . . . . . . . . . . . . . . . . . . 129 6.7.5 Stroke 2: Thermalization with the cold bath . . . . . . 130 6.7.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Chapter VII: The MBL-Mobile at finite speed . . . . . . . . . . . . . . 133 7.1 Diabatic corrections . . . . . . . . . . . . . . . . . . . . . . . . 133 7.1.1 Fractional-Landau-Zener transitions . . . . . . . . . . . 134 7.1.2 Landau-Zener transitions . . . . . . . . . . . . . . . . . 135 7.1.3 APT transitions . . . . . . . . . . . . . . . . . . . . . . 136 7.2 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . 137 7.2.1 Simulating finite-time tuning . . . . . . . . . . . . . . . 138 7.2.2 Enlightenment from numerical methods . . . . . . . . . 139 7.3 Precluding communication between subengines . . . . . . . . . 140 7.4 Lower bound on the cycle time from cold thermalization . . . . 141 Chapter VIII: Racing the MBL-mobile . . . . . . . . . . . . . . . . . . 144 8.1 Comparisons with competitor Otto engines . . . . . . . . . . . 144 8.1.1 Comparison with bandwidth engine . . . . . . . . . . . 144 8.1.2 Comparison with MBL engine tuned between same-strength disorder realizations . . . . . . . . . . . . . . . . . . . 145 8.1.3 Quantum-dot engine . . . . . . . . . . . . . . . . . . . 147 8.1.4 Anderson-localized engine . . . . . . . . . . . . . . . . 147 8.2 Order-of-magnitude estimates . . . . . . . . . . . . . . . . . . . 148 8.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 ix INTRODUCTION Numerical methods for many-body quantum systems go back to the earliest days of electronic computers. The physicists of Los Alamos during World War II relied heavily on numerical simulations; the first calculations were done by human “computers” or with mechanical calculators,1 but soon after the war they began using ENIAC, the first programmable electronic computer [84]. We need simulations because the phenomena we study are complex. This was certainly the case for the pioneers of computational physics, simulating weapons physics at Los Alamos, and it is the case for us now. We may have a microscopic Hamiltonian, but analytically solving the Schrödinger equation with that Hamiltonian is out of the question, and tightly-controlled approximations are frequently unavailable. So we are forced to rely on heuristic approximations justified by physical arguments. But these approximations need to be checked: do they describe the model’s behaviour well? At the very least, do they describe some essential phenomenon at work in the system? Numerical simulations of a model catch hidden assumptions and subtle mistakes in your physical arguments—and they suggest further approximations to make and phenomena to investigate. (The work of Part II of this thesis proceeded in exactly this fashion, as an ongoing three-part conversation between physical argument, careful analytical work, and numerical simulation.) But numerical methods are not simply devices for checking your intuitions and arguments. Developing a new numerical method and teasing out a characterization of where it succeeds, where it fails, and why it succeeds or fails provides important insight into the physics of the systems that method simulates Take the history of the density matrix renormalization group (DMRG) and the matrix product state (MPS).2 Steve White developed the method in 1993 [226, 227] as an extension of Wilson’s numerical renormalization group (NRG) [230], and it quickly saw wide application [82, 100, 127, 180], not only for computing ground states but also for simulating low-energy dynamics. Meanwhile, in 1995, Ostlund and Rommer [248] showed that “infinite-system” DMRG could be formulated in terms of matrix product states, which had existed for some time, e.g. as a mechanism for writing down the ground 1 The Los Alamos physicists required numerical stability from their algorithms because dust caused frequent errors. 2 This paragraph draws heavily on the review [187] of Schollwöck. x state of the AKLT Hamiltonian [9, 10, 59], and in 1999 Takasaki et al. [203] showed that “finite-system” DMRG is precisely a variational procedure, which iteratively minimizes the energy over the manifold of matrix product states. The notion of truncation was crucial to the practical developments. Matrix product states offer not only a compact representation of low-entanglement states, but also an obvious procedure for “lossy compression”—for optimally approximating one matrix product state by another that requires less memory and computation time to store and use. The growing understanding that ground states of gapped Hamiltonians have efficient matrix product state representations (in particular the work of Hastings, e.g. [86]) gave physicists the confidence to use DMRG as, essentially, a black-box method. Since most translation-invariant local Hamiltonians are gapped (see [131] for a recent proof) and ground states of gapped Hamiltonians have efficient matrix product representations, one expects DMRG to reliably find the ground state of an arbitrarily-chosen Hamiltonian. And DMRG is generally self-diagnosing: even if it does fail, you can tell by checking convergence in the simulation parameters, without having to compare to an exact result, or argue without empirical evidence that it should give the right answer.3 Even if you are not a specialist, you can choose from a number of carefully-optimized, well-engineered MPS implementations, e.g. ITensor [1] or TeNPy3 [87], and expect that either DMRG will work or you will see clear evidence that it fails. The long list of papers using ITensor (http://itensor.org/docs.CGI?page=papers) testifies to the impact of this development. But that understanding also led to a number of analytical results, notably Xie Chen and friends’ classification of bosonic symmetry protected topological phases (SPTs) [36]; cf Pollmann and friends’ [169, 170]. [170] is particularly close to the spirit of this thesis: in it, Pollmann and his collaborators ask how symmetry operations act on the vector space in which the entanglement spectrum of an MPS in mixed canonical form lives. They find that the decomposition of that vector space into irreducible projective representations of the symmetry group strongly constrains the entanglement spectrum. 3 A few of DMRG’s failure modes, notably difficulty capturing rare long-range correlations as in a random-singlet ground state, are not self-diagnosing. See [184] for some of this problem and the performance of an algorithm RRG that performs better for these states; for more examples of failures in disordered systems see todo schmitteckert Brenden. xi And DMRG also gave rise to a variety of numerical methods for time evolution, notably TEBD [213, 214, 228]. (I discuss TEBD in some detail in Sec. 1.3, and give an opinionated discussion of a variety of MPS-based time evolution methods in Sec.2.1.1.) These numerical methods in turn shed light on one of the fundamental questions of computational physics—what physical processes can one efficiently simulate on a classical computer?—by carving out a set of quantum problems—namely those that generate little entanglement—that can be efficiently simulated. This question in turn is intimately related to the extended Church-Turing thesis. Other work along these lines includes the Gottesman-Knill theorem [73], which shows that computations involving only Clifford gates are efficiently simulable on classical computers. A feedback loop between development and characterization of numerical methods, on the one hand, and analytical work of varying degrees of rigor on the other, is clearly a fruitful way to do research: it leads to both practical numerical methods and to a deeper analytical understanding of the phenomena we started out trying to simulate. Until recently, this feedback loop had been applied in only the most limited of ways to questions of thermalization, disorder, and localization. Generic (viz., non-integrable) systems are expected to come to thermal equilibrium in finite time; we call these systems thermalizing. Thermalization is understood in terms of the eigenstate thermalization hypothesis (ETH): because the time to approach a state that is effectively random except to the extent that it has a certain energy expectation value—that is, a Gibbs ensemble—is given by the exponentially long dephasing time between near-in-energy eigenstates, each eigenstate must have the properties of a Gibbs state at that eigenstate’s eigenenergy [44, 54, 182, 183, 200]. Interestingly, the ETH is not enough. Some systems can satisfy the ETH but have very slow thermalization times, either because they have only weak integrability-breaking terms, or because they give rise to emergent local quasi-conserved quantities [20, 113, 136]; when I say “thermalizing” I implicitly mean “quickly thermalizing”, for some notion of “quickly” determined by the other timescales of the problem at hand and by the resources (memory and computation time) available for simulations. And not all “generic” systems thermalize at all: sufficiently strongly disordered systems undergo many-body localization (MBL), in the Hamiltonian commutes with local conserved quantites, essentially dressed onsite occupations. Near the transition between the ETH and MBL phases, a regime displaying rich Griffiths physics, e.g. subdiffusion, is expected to occur. xii Because thermalizing systems generate entanglement quickly, TEBD is of limited utility, and numerical studies were for some time restricted to exact diagonalization (e.g. [115]), with occasional forays into special-purpose MPS or matrix-product operator (MPO) calculations (e.g. [113]). But in the last few years, three methods—TDVP [133], CTWA [233, 234], and my own density matrix truncation, or DMT—have density matrix truncation come into use; they promise to shed light on the physics of thermalizing systems. This thesis sits in the tradition I have sketched of using both sides of this feedback loop to advance the discipline. I present necessary background in Chapter 1. The later chapters assume some basic knowledge of many-body localization, matrix product states, and algorithms for time evolution. Chapter 1 communicates that knowledge. I aim more for intuition than rigor. Nothing in 1 is original; most of the contents are ambient cultural knowledge, so I include few citations. Part I describes DMT and the physics we can use it to study. DMT is a new describe a new algorithm for approximating matrix product representations of density matrices that takes advantage of our understanding of thermalization. It succeeds at simulating previously unsimulable systems, and its convergence properties already prompt interesting insights into those systems. In Ch. 2, we argue that truncating MPDOs in such a way that the expectation values of the conserved quantities of some thermalizing Hamiltonian do not change should not change the long-time dynamics of that Hamiltonian. We construct a truncation procedure that does this, and find that it does indeed capture the long-time dynamics of a fruit-fly thermalizing Hamiltonian. In Ch. 3 we apply DMT to a high-frequency Floquet system. We find that it accurately captures not only the system’s hydrodynamics but also the Floquet heating process; the degree to which it does this illuminates the details of that process. We also find that the long-time prethermal dynamics of the system are well-described by a phenomenological diffusion equation In Ch. 4, we apply DMT to a low-frequency disordered Floquet system, and find that even for substantial disorder its hydrodynamics are diffusive. We also comment on some subtleties in evaluating DMT’s accuracy, and the degree to which it is self-diagnosing. In Part II I turn to a very different work: on turning MBL systems to thermodynamic advantage. After briefly contextualizing the problem in Ch. 5, we construct a thermodynamic engine from an MBL system and analyze it in a xiii relatively straightforward adiabatic limit (Ch. 6); we then analyze finite-speed effects (Ch. 7), and compare it to competitor engines (Ch. 8). At first sight the material of Part II does not fit so well into the tradition of using methods development to comment on physics, and vice versa. But this impression understates the degree to which the numerical simulations shaped our numerical and even conceptual work, and it also understates the degree to which the limits of the numerical methods we used give us insight into the physics we use and the robustness of our protocols. I briefly comment on this point in Sec. 7.2 1 Chapter 1 BACKGROUND: MBL AND MPS 1.1 1.1.1 Many body localization Many-body localization in perturbation theory Consider a random paramagnet—a system with Hamiltonian Hrpara = X hzj σjz , (1.1) where the hzj are drawn independently from the uniform distribution on [−h, h]. A system governed by this state has “perfect memory” in the sense that if it starts out in some product of σjz eigenstates, say |0xC0FFEEi = |↑↑↓↓↓↓↓↓↑↑↑↑↑↑↑↑↑↑↑↓↑↑↑↓i , (1.2) it will remain in that state forever: e−iHt |0xC0FFEEi = |0xC0FFEEi, up to a phase, for all t. In terms of operators d z hσ i = −ih[Hrpara , σjz ]i = 0 . dt j (1.3) Indeed [Hrpara , σjz ] = 0: the σjz are conserved quantities of the Hamiltonian Hrpara . Now suppose we change the Hamiltonian. Suppose in particular that we add a nearest neighbor interaction: HRFH = Hrpara + t =t X j X x x ]+∆ [σjx σj+1 + σjy σj+1 j x x [σjx σj+1 + σjy σj+1 ]+∆ X j X z σjz σj+1 ] j z σjz σj+1 + X hzj σjz , (1.4) j hzj still random in [−h, h]. (I choose this Hamiltonian with malice aforethought. Since the work of Pal and Huse [160] this Hamiltonian, called the random-field Heisenberg Hamiltonian, has served as the fruit-fly model for the study of many body localization: see inter alia multa [246].) Then [HRFH , σjz ] 6= 0, (1.5) 2 i.e. the σjz are no longer conserved quantities. Disaster! But if h is large, the σjz are almost conserved. First, take ∆ = 0 and add h 6= 0. Then the Hamiltonian is H& = t X X x x [σjx σj+1 + σjy σj+1 ]+ hzj σjz . j (1.6) j On Jordan-Wigner transformation this is the single-particle Hamiltonian H& = 2t X † X [cj cj+1 + h.c.] + hzj nj . j (1.7) j I’m going to take as read the fact that this Hamiltonian displays Anderson localization, so its single-particle eigenstates are localized with some localization length, call it ζ. To put that statement about localized single-particle eigenstates in a more useful form, write ñj for the occupation number operator of the eigenstate z centered at site j, and write τ&j = 2ñj . (Similarly Jordan-Wigner transform † + z c̃ ∼ τ .) τ&j is called an (Anderson) l-bit. Then H& = X z h0j τ&j , (1.8) j where the h0j are the eigenenergies of the single-particle eigenstates. There is some unitary U& (namely the unitary diagonalizing H& ) such that z U&† τ&j U& = σ z (1.9) + and U&† τ&j U& = σ + , etc. Since the single-particle eigenstates are localized with localization length ζ, z τ&j ∼ σjz + X j −(|k−j|+|j−l|)/ζ + − Rkl e σk σl (1.10) kl j for some appropriate random Rkl . How do interactions change this picture? P z z j σj σj+1 , for HRFH = H& + V = X Add an interaction term V = ~σj · ~σj+1 + X j hj σjz . (1.11) 3 We would hope to treat this as a perturbation. We can only do so if hα| V |αi  1. Dα − Dβ (1.12) To estimate this, suppose two states |αi , |βi differ on a subsystem of diameter l: that is, + + − |αihβ| = τ&a . . . τ&a τ − . . . τ&b , (1.13) m n &b1 1 where the sites a1 , . . . an , b1 , . . . , bm are all in a subsystem l. Then   z z |αi hβ| |αi = tr σjz σj+1 hβ| σjz σj+1   + + − − z . . . τ τ . . . τ = tr σjz σj+1 τ&a &a &b &b n m 1 1 h i † z z + + − = tr (U& σj σj+1 U& )σa1 . . . σan σb1 . . . σb−m . (1.14) z U ? We What is the operator content of this strange construction U † σjz σj+1 know that conjugation by U maps X j 1 U& σjz U&† ∼ σjz + Rkl e−(|k−j|+|j−l|)/ζ σk+ σl− ζ kl (1.15) j z with Rkl ∼ ζ −1 random, so it is reasonable to suppose that U † σjz σj+1 U is z U † σjz σj+1 U= X R~νj 2−2`[~ν ] e−`[~ν ]/ζ σ~ν , (1.16) ~ ν where R~νj ∼ ζ1 , ~ν = ν1 ν2 ν3 . . . νl , νj ∈ 0 . . . 3 label Pauli strings σ~ν , we take each ~ν to have at least one νj 6= 0, and `[~ν ] is the natural notion of the diameter of the Pauli string σ~ν `[~ν ] ≡ max{j : νj 6= 0} − min{j : νj 6= 0} . The factor of 2−2`[~ν ] ensures normalization. Then   z z hβ| σjz σj+1 |αi = tr (U † σjz σj+1 U )σa+1 . . . σa+n σb−1 . . . σb−m ∼ 2−l/2 e−l/ζ . (1.17) (1.18) The energy difference is typically not less than the spacing of levels on l sites: Dα − Dβ & hl2−l . (1.19) 4 So our condition for V to act perturbatively is that for most α, β 1 hα| V |βi −1 ∼ el(ln 2−ζ ) Dα − Dβ (1.20) or equivalently that the bare model H& have localization length ζ −1 > ln 2 . (1.21) (This is a heuristic sketch of certain elements of de Roeck et al.’s “avalanche picture”; see also [5, 49, 145, 171, 206, 207]. The chief element I have not treated is the possibility—near certainty, in fact—that some regions will be resonant in the sense in the sense that the condition (1.20) is not satisfied; de Roeck and friends give an RG-like scheme for determining whether these “resonant regions” destabilize the Anderson-localized system.) If you carry this diagonalization out in full you will find a unitary U such that z U † = U U& σjz U&† U&† is a conserved quantity of HRFH : τjz = U τ&j [HRFH , τjz ] = 0 . (1.22) Based on the condition (1.20) and the expectation that higher-order terms will scale as  n l(ln 2−ζ −1 ) [nth-order term] ∼ e , (1.23) one expects these l-bits to have the form localization length X j 1 Rkl e−(|k−j|+|j−l|)/ξ σk+ σl− τjz ∼ U † U&† σjz U& U ∼ σjz + ξ kl (1.24) where ξ = (ln 2 − ζ −1 )−1 . (1.25) (Again this is morally consistent with the results of the avalanche picture, but they run this quantity through an RG scheme.) We have therefore constructed extensively many local integrals of motion. These are l-bits in the usual sense [34, 92]. A couple of notes about the condition (1.21) on the Anderson localization length are in order. The right hand side of that condition, which ultimately comes from an estimate of the level spacing, is actually an entropy density. This has consequences: 5 Figure 1.1: Left: Notional density of states (hence entropy density) for a disordered many-body system. Right: Notional phase diagram, per (1.21). Compare the phase diagram found in [146] for the isotropic random-field Heisenberg model. 1. Systems with larger onsite Hilbert spaces, from the prosaic (spin-3/2 sites) to the exotic (SYK puddles), will require larger disorder to localize (if they localize at all: constructing this Anderson route to MBL will require creativity). 2. Since for real systems the entropy density is a function of energy density, the condition (1.21) (as well as the more sophisticated treatments of de Roeck et al.) naı̈vely predict a mobility edge. See Fig. 1.1. But de Roeck et al. argue in [48] that a mobility edge is impossible, in the sense that it will destabilize localized states at lower energies. Per this arguement, interacting systems must be fully MBL (no mobility edge) or not MBL at all. On the other hand, numerics (e.g. [146]) consistently show a mobility edge consistent with the naı̈ve prediction of (1.21)). My own belief is that that the mobility edge exists, but this will require a great deal of work to sort out. In addition, the full avalanche picture predicts that MBL cannot exist in d ≥ 2 dimensions. You can already glimpse how this comes about: if you construct a perturbative scaling in some length l based on (1.20), the level spacing shrinks exponentially in ld , while the matrix element only shrinks exponentially in l. So in d ≥ 2, couplings are always resonant at sufficiently long length scales. 1.1.2 Long-time memory in MBL systems We started with a simple question about a system’s memory: supposing I started out in state |0xCOFFEEi, does the system remain recognizably in that state? If the system is a random paramagnet, it obviously does: [H, σjz ] = 0. 6 If the system is goveerned by a random-field Heisenberg model, the situation is more complicated. Then [HRFH , σjz ] 6= 0, but as we have just seen there are operators τjz not so different from σjz that are conserved. Instead of worrying about the instantaneous decay dtd σjz = −i[HRFH , σjz ], attack the long-time limit directly. Work in the fermion picture, with nj = 21 (σjz + 1). If Eα are the eigenenergies of HRFH , then lim nj (t) = lim t7→∞ t→∞ X αβ e−i(Eα −Eβ )t |αi hα| nj |βi hβ| . (1.26) This is strongly oscillatory, so the outcome could depend strongly on when exactly we take the measurement. Clearer would be to start a measurement after some time t0 and let the measurement go on for some very long time T : 1 lim T t0 ,T →∞ Z t0 +T t0 X 1 |αihα| nj |βihβ| lim nj (t) = t0 ,T →∞ T αβ X tr [|αihα| nj ] . = Z t0 +T e−i(Eα −Eβ )t t0 (1.27) α The eigenstates |αi are specified by l-bit occupations, e.g. |αi = |10011101 . . .i, so eigenstate projections are |αihα| = Y ñj j∈α Y ñk , (1.28) k6∈α where j ∈ α indicates that l-bit j is occupied in state |αi, and I write X 1 1 ñj = (τjz + 1) ∼ nj + e−ξ|j−k| c†j ck + . . . 2 ξ jk (1.29) for the occupation operator of the l-bit centered at j. That occupation operator looks like X j 1 (1.30) ñj ∼ nzj + Rkl e−(|k−j|+|j−l|)/ξ c†k cl . ξ kl Then 1 lim T t0 ,T →∞ Z t0 +T t0 nj (t) = X α tr " Y k∈α ñk Y k0 6∈α # ñk0 nj |αihα| . (1.31) 7 Since 1 nj ∼ ñj + [multiple l-bit flips with small coeff’s] , ξ (1.32) this is roughly 1 lim T t0 ,T →∞ Z t0 +T t0 " # Y Y 1X nj (t) = tr ñk ñk0 nj |αihα| ξ α 0 k∈α k 6∈α (1.33) ≃ ñj : the long-time expectation value is given by the nearest l-bit. 1.1.3 Level spacing This long-time memory, while conceptually satisfying, is too blunt a knife either for diagnostics or for detailed considerations of dynamics. Level spacing statistics—in particular, the distribution of gaps between nearest-in-energy eigenstates, is a much finer tool: it is a standard diagnostic of thermalization and localization, and we will use it extensively in Part II. To understand the level-spacing statistics of a disordered Hamiltonian, think once again about a random-field Heisenberg model, this time on a system of size L with closed boundaries. To estimate the distribution of smallest gaps, start in an eigenbasis of the underlying Anderson-localized system. Once again, we’ll proceed by perturbation theory, but along a different route. Suppose for the moment that the system is many-body localized, and take two eigenstates |αi , |α + 1i of H& with energies E&α , E&α+1 . Nearest-in-energy eigenstates will differ on most of the system’s Anderson l-bits, so on the one hand δ& ≡ E&α+1 − E&α ∼ 2−L , (1.34) and on the other z λ ≡ hα| σjz σj+1 |α + 1i ∼ 2−L/2 e−L/ζ . (1.35) We can then write an effective two-level Hamiltonian on these two levels " # δ& λ H= . (1.36) λ −δ& 8 Figure 1.2: Heuristic derivation of level-spacing distribution The eigenvalues of (1.36) are so the gap is δ = p 2 δ& + λ2 . q 2 E = ±h δ& + λ2 , (1.37) What is probability distribution of gaps δ? Suppose now that the system is MBL, and indeed far from the transition. Both δ& and λ are uniformly distributed on their respective scales, but λ  δ& : this is the condition (1.20) for the system to be MBL. To see the distribution of δ, Imagine plotting δ& , λ in a plane (cf Fig. 1.2). For small δ (i.e. δ . λ) the ensemble-averaged levelspacing distribution is lim p(δ)dδ ∼ δ dδ, s→0 δ.λ; (1.38) λ.δ.h. (1.39) for δ & λ, p(δ)dδ ∼ const. × dδ, We are therefore unlikely to see gaps smaller than λ ∼ 2−L/2 e−L/ζ . (1.40) This logic was for gaps δ  2−L . If the gap is larger, δ ∼ 2−L , the reasoning above still holds—but δ ∼ δ& , and you must concern yourself with the fact 0 that δ& is not uniform. Take two Anderson levels separated by a gap δ& . 0 As δ& increases, the two levels with which you have concerned yourself are increasingly unlikely to be nearest-neighbors in specrum: there is likely to be a level in between. Taking this into account, one finds that p(δ)dδ ≃ p(δ& )dδ& = µe−µδ& dδ& , (1.41) 9 where µ = 2L is [average Anderson gap]−1 . (In Part II, we will frequently concern ourselves with the detail that the rate µ can vary with energy.) So p(δ) ∼  δ δ.λ µe−δ δ&λ . (1.42) It is convenient to have a single parameter to characterize this level repulsion. Your first thought is probably to compute moments of the level-spacing distribution (1.42), but none are suitable: hsi = 1 by construction, and higher moments capture the large-s behavior—not the small-s behavior we’re after. Moreover, estimating the level-spacing distribution requires that finicky normalization by the density of states. Instead, it is conventional to use the so-called gap ratio, frequently called the r-parameter. Given two adjacent level spacings δj = Ej − Ej−1 and Ej+1 − Ej , the gap ratio is min (δj , δj+1 ) . (1.43) rj = max (δj , δj+1 ) The gap ratio characterizes the distribution (1.42), so one should average it both over levels of a single Hamiltonian and over Hamiltonians drawn from whatever ensemble you’re studying. To get some intuition, consider two limiting cases: • If the system has an exact two-fold degeneracy (e.g. a global spin-flip) then small gaps are frequent, and min (δj , δj+1 ) = 0 max (δj , δj+1 ) 6= 0 (1.44) for r=0. (1.45) • If the eigenvalues are uniformly spaced then small gaps never show up, and δj = δj+1 (1.46) so min (δj , δj+1 ) = max (δj , δj+1 ) for r=1. (1.47) 10 1.1.4 Local dynamics at finite time We can use the level spacing to tease out details of the intermediate-time dynamics of MBL systems. Think about the evolution of the Heisenberg operator σ z (t) . (1.48) How many sites does the operator touch? That is, how many sites in the initial state does hσjz (t)i depend on? This is essentially a time-dependent correlation length: σjz (t) and σkz (t) will be correlated to the extent that they depend on the same sites in the initial state. To compute this correlation function, note that on a timescale t the dynamics is insensitive to gaps smaller than t−1 : if two levels are separated by a smaller gap they will not have had time to dephase. All levels differing on a subsystem of diameter ln t ∼ ξ ln t, (1.49) l(t) ∼ ln 2 − ζ −1 then, will have had time to dephase, but many levels separated on longer length scales will not—so σ z (t) will have support l(t) ∼ ξ ln t. (Cf [109, 158].) This in turn predicts that the entanglement entropy should be S ∼ ls ∝ ln t, (1.50) where s is an entropy density; this is exactly what MPS simulations see [16, 194]. This reasoning will make a reappearance in Chs. 6 and 7. 1.2 Matrix product states The long-time state of an MBL system is characterized by the expectation values of its conserved quantities, the l-bits. l-bits are local, so expectation values of physical operators here are only weakly dependent on values of l-bits there, and the long-time state should in some sense be “simple”. Moreover, as we just saw, an MBL system that starts out uncorrelated stays uncorrelated for a long time—its correlation length grows as ln t. There should be some nice numerical representations of the kinds of state that result from time evolution with MBL Hamiltonians. 11 So take an arbitrary state |ψi = X s1 ...sN ψs1 ...sN |s1 . . . sN i (1.51) with short-range correlations on an N -site system with closed boundary conditions. We want a “computationally efficient” representation for the state—that is, an approximation to |ψi such that that 1. the representation requires little memory, as long as |ψi has a short correlation length; 2. typical operations require little time (polynomial in system size), again as long as |ψi has a short correlation length; 3. the representation can “expand” to represent even states that are not short-range entangled. Such a representation exists: the matrix product state or MPS. The standard reference on matrix product states is the review of Schollwöck [187] . The field has seen many new developments since that review (some described in this thesis), but none have achieved the widespread acceptance of DMRG or TEBD have. For a very different perspective, see the book of Xie Chen et al. [242] . To get some intuition for matrix product states work, start with a Schmidt decomposition. Imagine cutting the state in two at some bond l (that is, the bond to the right side of site l). One slightly roundabout way to find the reduced density matrix of the left part of the system is to take the tensor ψs1 ...sN of Eq. (1.51), group indices to turn it into a matrix,1 and do a singular value decomposition (SVD) ψ(s1 ...sl )(sl+1 ...sN ) = X U(s1 ...sl )α sα V(sl+1 ...sN ) . (1.52) α This gives |ψi = 1 X α |xLα i sα |xRα i (1.53) In this case, “grouping indices” is exactly the reshape operation in NumPy or Julia. In other cases, “grouping indices” will correspond to reshape preceded by a reshuffling of indices: np.transpose or Julia’s permutedims. 12 with |xLα i = |xRα i = X s1 ...sl U(s1 ...sl )α |s1 . . . sl i X sl+1 ...sN V(sl+1 ...sN )α |sl+1 . . . sN i (1.54) Since U, V are isometries hxLα |xRβ i = hxLα |xRβ i = δαβ . (1.55) (Eq. (1.53) is therefore a Schmidt decomposition.) The reduced density matrices left and right of bond l are then immediately ρL = ρR = X α s2α |xLα ihxLα | α s2α |xRα ihxRα | . X (1.56) So the singular values sα are (the square roots of the) eigenvalues of the density matrices ρ{L,R} , and entirely specify the state’s entanglement properties: the 2nd Rényi and von Neumann entanglement entropies, for example, are S2 = X s4α α SN = X 2s2α lg sα . (1.57) α If the state has low entanglement entropy at bond l, it should require only a few Schmidt vectors to capture—that is, we should be able to write a pretty good approximation by truncating the sum in Eq. (1.53) If we follow that line of attack, we’re still faced with the problem of representing the states |x{L,R}α i. What we require is a generalization of Eq. (1.53) that in some sense offers Schmidt decompositions at every bond simultaneously. A matrix product state offers exactly this property. It consists of a decomposition of the tensor ψs1 ...sN into a product of matrices (hence the name) ψs1 ...sN = As11 As22 . . . AsNN s (1.58) where each of the Aj j is a matrix of dimension χ × χ for some χ not too large (except for As11 and AsNN , which are 1 × χ and χ × 1 respectively). χ is called 13 the bond dimension of the MPS; it controls the computational properties and the entanglement of the MPS. The state is |ψi = X s1 ...sN As11 As22 . . . AsNN |s1 . . . sN i (1.59) (relabeling α 7→ α1 —in the future such relabelings will be silent) for ψs1 (s2 ...sN ) = Asα1 rα Vα(s2 ...sN ) . (1.60) The index α1 has dimension χ1 given by the rank of the d × dN −1 matrix ψs1 (s2 ...sN ) . Now apply roughly the same process to rV : regroup indices so that it is a (χ1 d) × dN −2 and perform an SVD rα V(α1 s2 )(s3 ...sN ) = U(α1 s2 )α2 rα2 Vα2 (s3 ...sN ) . (1.61) Asα21 α2 = U(α1 s2 )α2 (1.62) X (1.63) Identify for ψs1 s2 ...sN = Asα11 Asα21 α2 rα2 Vα2 (s3 ...sN ) . α1 α2 The index α2 has dimension χ2 given by the rank of the χ1 d × dN −1 matrix rα V(α1 s2 )(s3 ...sN ) Proceed in this way until you find (after doing N − 1 SVDs) X ψs1 s2 ...sN = Asα11 Asα21 α2 . . . AsαNN−1 V ; −2 αN −1 αN −1 sN ) (1.64) Asα11 Asα21 α2 . . . AsαNN−1 AsαNN −1 −2 αN −1 (1.65) α1 α2 ...αN −1 call VαN −1 sN ) ≡ AsαNN −1 , so ψs1 s2 ...sN = X α1 α2 ...αN −1 —this is exactly the form required for Eq. (1.59). We can bound the bond dimension inductively. χ1 is the rank of a d × dN −1 matrix, so χ1 ≤ d, χ1 ≤ dN −1 . (1.66) χ2 is the rank of a (χ1 d) × dN −2 matrix, so χ2 ≤ (χ1 d) ≤ d2 , χ1 ≤ dN −2 . (1.67) 14 Proceeding in this way χ n ≤ dn , χn ≤ dN −n , (1.68) and in general χn ≤ dbn/2c . (1.69) In fact one rarely constructs an MPS in this way. The utility of MPSs is in the fact that they are efficient representations of states too large to represent otherwise. If the system size is small enough that one can store the whole coefficient tensor ψs1 ...sN you are almost certainly better off working with the exact state in an ordinary representation. But the fact that all states can be represented exactly is useful. By taking large bond dimensions, one can perform exact simulations and use them to check your low-bond dimension approximate simulations (we will do this over and in Part I). More generally, this motivates convergence testing. Suppose you are using an approximation method that cuts MPSs down to some low bond dimension cap. If doubling (say) that bond dimension cap doesn’t change the results, you can argue that— since the method will give the exact result for sufficiently large bond dimension cap—this insensitivity means that you have the right answer. 1.2.1 Graphical notation for tensors and tensor contractions Writing out indices in sums like (1.65) is flexible but cumbersome, while the matrix-multiplication form of Eq. (1.59) is compact but inflexible. It is customary to use a compact graphical notation for tensors and tensor contractions. Tensors are represented by boxes with legs, each leg representing a tensor index, so for example Asα21 α2 = . (1.70) The mapping between leg and index is generally to be understood from context and implicit convention; in this case, the left and right legs represent α1 , α2 respectively, and the bottom leg s2 . Frequently the box is omitted, and a tensor is represented by a vertex: Asα21 α2 = = . (1.71) 15 Tensor contraction—in which one sets indices equal and sums—is indicated by joining the lines corresponding to the indices, so X Asα11 Asα21 α2 = . (1.72) α1 In this notation the coefficient tensor ψ is ψs1 s2 ...sN = X AsαNN −1 = Asα11 Asα21 α2 . . . AαsNN−1 −2 αN −1 . (1.73) α1 α2 ...αN −1 Complex conjugation is represented by flipping the diagram top-to-bottom ψs∗1 s2 ...sN , (1.74) so a state’s normalization is hψ|ψi = 1.2.2 X ψs1 s2 ...sN ψs∗1 s2 ...sN = . (1.75) s1 ...sN Expectation values and correlations in MPSs One is not typically satisfied with computing the normalization of a state. Suppose one has an onsite operator B with matrix elements Bss0 . Then Bss0 = (1.76) and hψ|B|ψi = . (1.77) (Ordinarily the operators are Hermitian; if they are not, some care must be taken in interpreting diagrams like this one.) This is less difficult to evaluate than it may appear. One can perform the contractions in a diagram like Eq. (1.77) by “sweeping in” from left and right— this is much more efficient than forming ψs1 ...sN explicitly. Reordering the contractions corresponds to reshuffling sums and multiplications in an explicit representation analogous to Eq. (1.65), so one can iteratively construct left 16 and right environment tensors ELj ≡ = ELj = ERj ≡ (1.78) = ELj = Then the expectation value is hψ| B |ψi = . (1.79) Multi-site operators are an easy generalization: if O is a two-site operator, it has matrix elements Oss0 = (1.80) and expectation value hψ| B |ψi = (1.81) None of these computations require constructing the coefficient tensor ψ, which would require memory (and time) exponential in system size. Rather, they require time and memory linear in system size and polynomial in the maximum bond dimension χ, and if you want many expectation values you can store the environment tensors and amortize the cost of computing them. 1.2.3 Canonical forms and orthogonality centers The time-consuming part of the expectation value constructions of Sec. 1.2.2 is the computation of the environment tensors of Eq. (1.78). But if we choose the A matrices to have appropriate properties, then computing the environment tensors (1.78) becomes immediate. Suppose that the MPS is constructed as in Sec. ??. Then, because the A matrices are (for j < N ) the left unitary matrices of SVDs, we have X ∗ sj ∗ Aαsjj−1 αj Aαj−1 αj 0 = δαj α0j , j<N (1.82) sj αj−1 or in graphical notation = where the curved line is = δαα0 . , (1.83) 17 So EL1 = EL2 = = (1.84) = ELn = = . To find the analogous result for the right side, return to decompositions of the form Eq. (1.63). After n − 1 steps we have ψs1 s2 ...sN = X r V Asα11 Asα21 α2 . . . Asαn−1 n−1 αn αn αn (sn ...sN ) (1.85) α1 α2 so, comparing with Eq. (1.65), X Asαnn αn+1 . . . AsαNN −1 = X rαn Vαn (sn ...sN (1.86) αn αn+1 ...αN −1 and E= = X rαn rα0n Vα∗n (sn ...sN ) Vα0n (sn ...sN ) = rα2 δαα0 . (1.87) sn ...sN Working from right to left one can deduce the right-side analogue of the local condition Eq. (1.83): X s j ∗ Asαjj−1 αj Aα0 j−1 αj = rα2 j−1 δαj−1 α0j−1 , j<N (1.88) s j αj or in graphical notation = . (1.89) The diamond conventionally indicates a diagonal matrix, here with diagonal entries rα2 j . Now a single-site expectation value is .= . (1.90) MPSs with the property Eq. (1.83) are called left canonical. If a state is called “left canonical” one generally expects it also to have the property (1.90), but arranging this can require some effort. One can define right canonical 18 MPSs analogously: these are MPSs X |ψi = such that s1 ...sN X sN |s1 . . . sN i . B1s1 B2s2 . . . BN s j ∗ 2 Bαsjj−1 αj Bαj−1 α0 = rαj α0j , j (1.91) (1.92) sj αj−1 and typically X s j ∗ Bαsjj−1 αj Bαj−1 α0 = δαj−1 α0j−1 , j (1.93) s j αj j < N , or = (1.94) Expectation values are then hψ| |B |ψi = . (1.95) How does one switch between left and right canonical forms? One’s first thought is to write (j) Bαsjj−1 αj = (rα(j−1) )−1 Asαjj αj−1 rj , j−1 (1.96) (j−1),(j) (where I write rα to emphasize that these are two different sets of singular values r). This gives the same state, because the r(j−1) on the right side of B (j−1) will multiply the [r(j−1) ]−1 on the left side of B (j) . In principle this gives the desired result, but it is numerically unstable because it involves dividing (j−1) by the singular values rαj−1 , which generically range in magnitude from ∼ 1 to numerical precision. Better is to sweep from right to left, performing SVDs at each site. Start with a left-canonical MPS |ψi = X s1 ...sN As11 As22 . . . AsNN |s1 . . . sN i . (1.97) SVD AαN sN = UαN α0N rα0N Vα0N sN . (1.98) 19 Identify BαsN0 ≡ Vα0N sN ; N (1.99) now |ψi = X s s1 ...sN s sN −2 −1 ANN−1 U rBN |s1 . . . sN i , As11 As22 . . . ANN−2 (1.100) where BN satisfies the right canonical relation (1.92) Now multiply that U r into the A to its left, defining XαsNN−1 = AαsNN−1 r . −2 αN −1 −2 αN −1 αN −1 (1.101) SVD XαN −2 (αN −1 sN −1 ) = UαN −2 γN −2 rγN −2 VγN −2 (αN −1 sN −1 ) . (1.102) Identify BαsNN−1 = VαN −2 (αN −1 sN −1 ) ; −2 αN −1 (1.103) this B again satisfies the right canonical relation (1.92). |ψi = X s1 ...sN s s sN −2 −1 As11 As22 . . . ANN−2 U r BNN−1 BN |s1 . . . sN i . (1.104) Proceeding in this way, after N − n SVDs |ψi = X s1 ...sN and after N − 1 s sN n+1 As11 As22 . . . Asnn U r Bn+1 . . . BN |s1 . . . sN i |ψi = X s1 ...sN sN B1s1 . . . BN |s1 . . . sN i (1.105) (1.106) with all the B satisfying (1.92): we have put the MPS into right canonical form, as desired, without ever dividing by singular values. A state in the intermediate form (1.105) is said to have a (zero-site) orthogonality center at bond n2 , or to be in mixed canonical form. This is equivalent to the Schmidt decomposition. 2 One sometimes uses “zero-site” to distinguish MPSs like (1.105) from those in which the U r has been multiplied into the A to its left; this is called a “one-site” orthogonality center. I do not use one-site orthogonality centers in this thesis, so I do not say “zero-site”. 20 1.3 Dynamics and matrix product states In Sec. 1.1 we asked what happened to a state |0xC0FFEEi = |↑↑↓↓↓↓↓↓↑↑↑↑↑↑↑↑↑↑↑↓↑↑↑↓i , (1.107) under evolution by a Hamiltonian HRFH = X ~σj · ~σj+1 + X hj σjz . (1.108) j We found that at long times the system retained some memory of that initial state |0xC0FFEEi, and that it develops correlations slowly—with a correlation length l ∼ ξ log t, ξ the localization length of HRFH . Then I argued (briefly) that matrix product states are natural tools for numerically simulating MBL, and described (at length) natural operations on those matrix product states. Now I need to describe one more “natural operation”—time evolution. A Hamiltonian like HRFH acts locally. If I time evolve |0xCOFFEEi by HRFH for some time δt  1, then it is unreasonable to expect that the state of site 1 will come to depend on the state of site 20 in any substantial way—except to the extent that they depended on each other to start out with. Hamiltonian does not quickly introduce new correlations. This is doubly true of an MBL system. We can attempt to quantify the notion that the Hamiltonian does not quickly introduce long-distance correlations by noticing that e−iHδt = 1 − iHδt + O(Hδt2 ) (1.109) is local. Eq. 1.109 is of dubious suitability for describing dynamics, though, because not unitary. We can remedy this defect, while preserving insight into locality, by splitting the Hamiltonian up into even and odd terms and time evolving first by the odd terms, and then by the even terms. Write two-site gates    1 z z (h + hj+1 ) + ~σj · ~σj+1 . (1.110) Fj (δt) = exp −iδt 2 j Then—as one can see by comparing Taylor series— −iHRFH δt e L/2 L/2 Y Y = Uδt + O(δt ) Uδt ≡ [F2j (δt)] [F2j+1 (δt)] 2 j=1 j=1 (1.111) 21 and Uδt (t) ≡ [Uδt ]t/δt + O(δt) . e−iHRFHt = Uδt (t) + O(δt), (1.112) There exist Trotter decomposition’s with faster convergence, e.g. the secondorder decomposition (2) e−iHRFH δt = Uδt + O(δt3 ) , L/2 L/2 L/2 Y Y Y [F2j (δt/2)] [F2j+1 (δt)] [F2j (δt/2)] (2) Uδt = j=1 j=1 (1.113) j=1 which takes little more time than (1.112) to compute, thanks to a convenient telescoping property. This higher-order convergence is of secondary (practical) importance for times t & 1 in “natural units” (some coupling). In fact the error estimate O(δt) in Eq. (1.112) is extraordinarily pessimistic: one can show that Uδt (t) approximates e−iHRFH t well to times exponentially large in 1/δt [4, 6, 7, 57, 126, 155]. Alternatively, we can decompose −iHδt e ≃ L−1 Y [Fj (δt/2] j=1 | {z 1 Y [Fj (δt/2]; j=L−1 } | {z leftward sweep after rightward sweep (1.114) } the error is again of order Lkhj k3 δt3 , where khj k is an estimate of the typical magnitude of the terms hj . We call this a “boustrophedon” Trotter decomposition. If you wish to time evolve an MPS in mixed-canonical form and truncate at the orthogonality center, as we will in 2, this makes it easy to move the orthogonality center. 1.4 Matrix product operators In Sec. 1.3, I represented time evolution in terms of Trotter gates—that is, of local unitaries. This is a restrictive representation: if you wish to work with more general operators than local Hamiltonians, like long-range Hamiltonians or density operators, you need a matrix product representation of an operator. To write a matrix product operator (MPO) representation of some generic 22 operator O, decompose it in a basis of local operators O= X Oµ1 ...µN σ µ1 . . . σ µN , (1.115) µ1 ...µN and decompose the tensor Oµ1 ...µN as a product of matrices Oµ1 ...µN = Aµ1 . . . AµN . (1.116) All of the tricks for manipulating MPSs (gauge transformations, canonical forms, truncations, etc.) apply to MPOs, but frequently with altered interpretation. The operators are vectors in a larger Hilbert space than the system’s physical Hilbert space—but their inner product is hA|Bi = tr A† B . 1.4.1 (1.117) MPO representation of an exponential interaction Take a (portion of a) Hamiltonian Hexp = X σiz σjz e−s|j−i| (1.118) i<j for some constant s. We can build this Hamiltonian by walking in from the right end of the chain with the recursion relation       HRi+1 0 Ii σix HRi        hRi  =  0 e−s Ii σix  ⊗  hRi+1  IRi+1 0 0 Ii IRi (1.119) starting with a right-end base case     HRL+1 0      hRL+1  = 0 . IRL+1 1 (1.120) The Hamiltonian is then HR0 —that is, the thing in the first slot when you’ve walked all the way to the left end of the chain. 23 The matrices constituting the MPO are then   I1  z W1 = σ1  0   Ii σiz 0   W1<i<L =  0 s−1 Ii σiz  0 0 Ii   0  z WL = σi  Ii 1.4.2 (1.121) MPO representation of a power-law Hamiltonian This is from [241]; cf [240]. Take for concreteness H = Hlocal + Hpower , X σiz σjz , Hpower = |j − i|q i<j (1.122) where Hlocal is 2-local. 1.4.2.1 Exact representation To understand how to represent this as an MPO, first note that there exist unique Dk , k = 1 . . . L, such that L X 1 = Cs e−jsk /L , jq k=1 j ∈ 1 . . . L, sk = k/L (1.123) —this simply by linear independence of the vectors e−jsk /L . Then Hpower = X k Ck X i<j e−|j−i|sk /L σiz σjz , (1.124) 24 which has a recursion relation representation   ... 0 Ii C1 σiz C2 σiz       0 ... σiz   0 e−s1 Ii HRi HRi+1     ..  ..  .. .  hRi  =  . .   ⊗  hRi+1    IRi IRi+1  e−sL Ii σiz  0 ... 0 Ii (1.125) and resulting MPO representation. The Hamiltonian has an MPO representation given by adding the MPO representations for Hpower and Hlocal . 1.4.2.2 Approximate representation The key insight in [241] is that you can get away with many fewer than L exponentials if you’re willing to accept small errors in your interaction. This is apparently well-studied in the control system literature under the name model reduction. The approximation is given by the recursion relation with     HRi HRi+1      hRi  = Wi ⊗  hRi+1  IRi+1 IRi (1.126)   ... 0 Ii C1 σiz C2 σiz   0 ... σiz   0 e−s1 Ii . ..  ... .. Wi ≡  .      e−sn Ii σiz  0 ... 0 Ii (1.127) (with resulting MPO representation) where the n < L decay wavevectors s1 . . . sn are chosen by the algorithm. Note that the sk can be complex. 1.4.3 1.4.3.1 MPDOs Closed-system dynamics of an MPDO Consider a Hamiltonian H= X Hj (1.128) (for instance (2.40)) where Hj is supported on sites j, j + 1. Call the onsite Hilbert space Hj and its dual Hj∗ . Pure states then live in a Hilbert space decay rate sk = ln|d| 25 3 2 1 0 0 10 20 index k 30 Decay length (sk) 1 Figure 1.3: Decay rates sk for a power law 1/j 1.5 100 50 0 0 10 20 index k 30 1.5 Figure 1.4: Decay lengths s−1 k for a power law 1/j H = Hj⊗L , and density operators ρ in a Hilbert space H ⊗ H∗ = [Hj ⊗ Hj∗ ]⊗L . Closed-system Hamiltonian evolution, then, is d ρ = −i[H, ρ] ≡ −iH ] ρ dt (1.129) with a linear superoperator Hamiltonian defined H ] : H ⊗ H∗ → H ⊗ H∗ , H ] = H ⊗ 1 − 1 ⊗ H. (1.130) For a spin-1/2 chain, we can write Sjx,y,z for operators on the ordinary (“ket”) space Hj and Tjx,y,z for operators on the dual (“bra”) space Hj∗ ; in this notation, the superoperator corresonding to our fruit-fly Ising Hamiltonian (2.40) is X Hj] Xh z z ≡ (Sjz Sj+1 − Tjz Tj+1 ) H] = i 1 1 + hx (Sjx − Tjx ) + hz (Sjz − Tjz ) . 2 2 (1.131) 26 With H ] in hand, we can proceed by Trotter decomposition, as with an MPS. 1.4.4 Matrix product density operator representations of Gibbs states 1.4.4.1 Construction In sections 2.3.3 and 2.3.4 we will require a Gibbs MPDO as our initial state. The Gibbs state is ρ ∝ e−βH = e−βH/2 Ie−βH/2 , (1.132) which is precisely the imaginary-time evolution of the product MPDO I by the Hamiltonian superoperator ] Htherm =H ⊗1+1⊗H (1.133) out to time β/2. We approximate this imaginary-time evolution not by tDMRG with the boustrophedon Trotter decomposition (1.114), but by ordinary TEBD using the trick of Hastings for numerical stability (as described in 7.3.2 of Schollwöck’s magisterial review [187], q.v.). 1.4.4.2 Estimating thermal expectation values In analyzing the time evolution of ETH states, one naturally requires Gibbs state expectation values as a function of total energy (or, equivalently, energy density): the long time limit of an expectation value is given by its expectation value in a Gibbs state whose energy density matches that of the initial state. We tabulate energy densities and observables of interests for Gibbs states at a variety of temperatures using MPDOs as described above; to find the longtime limit of an expectation value, we measure the energy density of the initial state and linearly interpolate between the two nearest Gibbs energy densities. Note that this does not account for Trotter heating (that is, the fact that— because the system actually simulated is a Floquet system with period given by the Trotter step δt, its energy as measured by the Hamiltonian simpliciter gradually increases). 1.5 Some essentials of quantum thermodynamics The classical Otto engine consists of a gas that expands, cools, contracts, and heats [179]. During the two isentropic (constant-entropy) strokes, the gas’s volume is tuned between values V1 and V2 < V1 . The compression ratio is defined as r := VV12 . The heating and cooling are isochoric (constant-volume). 27 The engine outputs a net amount Wtot of work per cycle, absorbing heat Qin > 0 during the heating isochore. A general engine’s thermodynamic efficiency is η := Wtot . Qin (1.134) The Otto engine operates at the efficiency ηOtto = 1 − 1 rγ−1 < ηCarnot . (1.135) A ratio of the gas’s constant-pressure and constant-volume specific heats is denoted by γ := CCPv . The Carnot efficiency ηCarnot upper-bounds the efficiency of every thermodynamic engine that involves just two heat baths. A quantum Otto cycle for harmonic oscillators was discussed in Refs. [3, 51, 53, 102, 123, 190, 215, 244]. The quantum harmonic oscillator’s (QHO’s) gap plays the role of the classical Otto engine’s volume. Let ω and Ω > ω denote the values between which the angular frequency is tuned. The ideal QHO Otto cycle operates at the efficiency ηQHO = 1 − ω . Ω (1.136) This oscillator model resembles the qubit toy model that informs our MBL Otto cycle (Sec. 6.1). The heat and work exchanged by slowly tuned systems are defined as   dH W := dt Tr ρ dt 0   Z τ dρ Q := dt Tr H dt 0 Z τ and (1.137) (1.138) in quantum thermodynamics [215]. This Q definition is narrower than the definition prevalent in the MBL literature [39, 44, 70, 137]: here, all energy exchanged during unitary evolution counts as work. Part I Thermalizing systems and matrix product density operators 28 29 Chapter 2 SIMULATING QUANTUM DYNAMICS OF THERMALIZING SYSTEMS Questions about how (and whether) hydrodynamic behavior emerges from microscopic quantum physics arise frequently in condensed matter physics. The exploration of this physics is hampered by the limitations of existing numerical methods (cf Fig. 2.1). Numerically exact methods (like exact diagonalization and Krylov subspace methods) can treat the dynamical properties of small systems at arbitrary times, but require memory and computation time exponential in system size. Matrix product state methods, on the other hand, can treat large systems—but only when the systems have little entanglement entropy. This means that for thermalizing systems, whose entanglement entropy grows linearly with time, MPS methods can only treat short-time behavior (see Figure 2.1). This chapter describes a numerical method (“density matrix truncation” or “DMT”) based on matrix product representations of density operators. The al- time Hydrodynamics ED DMT MPS system size Figure 2.1: For small systems, exact diagonalization and related methods can treat time evolution of small thermalizing systems to long times. Matrix product state methods, on the other hand, can treat time evolution of large thermalizing systems, but only to short times, and hydrodynamic effective theories can phenomenologically describe the long-time limit, but not intermediate times. Our method, DMT, can treat large ETH systems at all times. 30 gorithm can accurately simulate not only short-time, low-entanglement behavior and long-time, hydrodynamic behavior, but also the complex intermediatetime behavior from which the hydrodynamics emerges. While using matrixproduct representations of mixed states is not new, the core of our algorithm is a new method for truncating matrix product density operators (MPDOs). This truncation exactly preserves the energy density of the system and other local conserved quantities, with the aim of leaving the hydrodynamics unaffected. It also avoids generating unphysical density matrices with negative entropy. At the same time, it is efficient enough that by taking large (but constant in system size) bond dimension one can capture the thermalization process. We first (in Sec. 2.1) offer some background on matrix product state methods and intuition for why a method using MPDOs should be able to efficiently simulate time evolution governed by Hamiltonians satisfying the eigenstate thermalization hypothesis (ETH) to arbitrary times. We also motivate certain properties of our method. We then describe (in Sec. 2.2) our algorithm for time evolution. This algorithm consists of the time-evolution framework described in Sec. 1.3 (cf [214]), paired with a novel scheme for truncating MPDOs. We then apply our algorithm to time evolution starting from a pure state (Sec. 2.3.2) and find that it qualitatively improves upon existing methods. Applied to a mixed state (Sec. 2.3.3 and Sec. 2.3.4), we find that our algorithm DMT matches or exceeds the state of the art. We conclude with directions we hope will improve on the method. 2.1 2.1.1 Background and intuition Background: matrix product state methods Simulating real-time evolution of many-body quantum-mechanical systems is hard: a system of L sites generically requires storage and computation time exponential in L. One line of attack, e.g. time-evolving block decimation (TEBD) [213, 214, 228], proceeds by representing unitary time evolution as a series of small time-steps applied to a matrix-product state (MPS) representation of a pure state. These matrix-product structures offer efficient representations of certain states (broadly speaking, “low-entanglement states”) in the sense that typical operations require polynomial time and memory. Matrix product state simulations of time-evolving pure states are stymied by the fast increase of entanglement entropy with time, which grows linearly in time for a typical global quench. When one compresses a pure state as a matrix 31 product state, memory and computation time requirements grow exponentially in the entanglement entropy among the subsystems, so this linear growth in entanglement entropy sets a hard upper limit on the timescales on which matrix product states are useful, though a variety of methods have been used in attempts to circumvent this limit. [14, 58, 64, 78–81, 90, 157, 204, 223, 231]. One case in which entanglement growth does not limit the useful timescale of matrix product state simulations is the dynamics of many-body localized systems, which exhibit a modest logarithmic entanglement growth [16, 192]. On the thermal side of the localization transition, however, entanglement grows quickly, and even the transition itself is expected to show extensive (volumelaw) entanglement [76]—consequently, pure-state time evolution under Hamiltonians remains restricted to short times. Research into mixed-state time evolution and Lindblad dynamics has also progressed. It has been proven that density matrices (and purifications) of Gibbs states of local Hamiltonians have efficient matrix product representations [85, 154, 163, 247]. Two schools of thought have used this insight to develop a series of methods for simulating time evolution. One school employs density matrices [21, 41, 95, 95, 101, 151, 152, 167, 168, 175–178, 212, 221, 222, 247]. They note that the space of operators on a spin chain is the tensor product of onsite operator spaces, just as the space of many-body pure states being a tensor product on onsite Hilbert spaces; the chief difference (in this view) is merely the dimensionality of the onsite space. For example, on a spin-half chain, the space of onsite operators is four-dimensional, while the space of pure states is two dimensional. This school then applies familiar pure state methods, including the creation and truncation of matrix product states and time evolution by TEBD, to density matrices—which are, after all, vectors in a larger space. The resulting truncation algorithms minimize the error according to the Hilbert-Schmidt (Frobenius) norm. In certain situations—in particular, dynamics near thermal equilibrium or a non-equilibrium steady state—this approach works well. In other situations, however—in particular, time evolution starting from a pure state—the density matrices suffer from a catastrophic loss of positivity. (Even checking positivity is NP-hard in the system size [117].) The second school [17, 18, 23, 58, 60, 103–106, 208] uses purifications instead of density matrices to represent mixed states. They pair each site in the system with an ancilla, a notional site representing the bath. The mixed nature of the system is represented by entanglement between sites and their ancillae, 32 so the system and ancillae together are in a pure state. Grouping each site and its ancilla into a larger onsite Hilbert space, one can write a matrix product representation for this pure state and apply the usual methods (truncation, TEBD, etc.) This solves the positivity problem: unlike density matrices, where many operators with reasonable matrix product representations are not positive and hence are invalid as density matrices, every representable vector is a valid state. Moreover, since one can act with a unitary on the space of ancillae without changing the physical state, one can try to exploit this freedom to reduce the matrix product state bond dimension of the purification [17, 104]. There is also a hybrid approach which locally unzips a density matrix into a purification, which preserves positivity by construction [225]. These purification methods employ truncations which minimize error according to the inner product h·|·i defined on the whole (system with ancillae) state. Neither the Frobenius norm on density matrices nor the quantum-mechanical norm on purifications is the correct notion of error. In the case of density matrices, the Frobenius norm fails to account for the fact that truncations that change the component of the density-matrix vector along the identity (i.e. which are not trace-preserving) are disastrous, because they can lead to loss of positivity. Moreover, neither notion of error captures spatial locality: a good notion of error should prioritize short-range properties of the model and guarantee that certain quantities (the local conserved quantities of the model under consideration, like energy density or spin) are unchanged. Since the methods of both the density-matrix and purification schools generically change the model’s conserved quantities at every gate application, they are unable in principle to approach the known “hydrodynamic” long-time behavior of systems which thermalize. This may be the reason that existing density-matrix methods lose accuracy over time, even though one would expect the accuracy of the matrix-product representation to improve as the state approaches equilibrium. In this work we propose a truncation of density matrices that ameliorates the positivity problem of Frobenius truncation and exactly preserves the expectation values of all operators on all regions of up to three sites in diameter. (We choose three sites for notational and practical convenience. In Sec. 2.2.4, we discuss how to generalize to regions of arbitrary diameter.) 33 2.1.2 Intuition: thermalization and computation Why should one be able to efficiently simulate the dynamics of a local Hamiltonian satisfying the eigenstate thermalization hypothesis? In the long-time limit, the system is well described (as far as local observables are concerned) by a Gibbs state, which has an efficient matrix product density operator (MPDO) representation [85, 154]. Moreover, the system will (one expects) locally thermalize before it reaches global equilibrium, and indeed after some short local thermalization time ttherm expectation values of local operators will be well approximated by the expectation values of those operators in a Gibbs state with spatially varying temperature, chemical potential, etc. Heuristically, one can imagine keeping the state exactly out to the local thermalization time and then truncating to an efficient representation. This would require a maximum bond dimension ca. (d2 )vttherm , where v is some entanglement speed and d is the dimension of the onsite Hilbert space. If vttherm is not too large, this approach itself may be workable. In practice, however, one will wish to efficiently represent the state even at early and intermediate times t < ttherm —and also to avoid dependence on the hard-to-define and likely-unknown constant ttherm . Having decided upon an MPDO representation, then, one is faced with the problem of writing a truncation algorithm: an algorithm that will approximate a given MPDO by another, more compact MPDO. The natural approach, by analogy with matrix product states, is to discard low-weight Schmidt vectors. (This approach turns out to be an imperfect solution, but it is a useful first step.) We call this truncation “Frobenius truncation.” A density operator is a vector in a space with the same tensorproduct structure as a state, but a larger onsite dimension. We can therefore cut the chain at bond j into two sections L and R, Schmidt decompose, and truncate it: ρ= χ−1 X α=0 χ0 −1 x̂Lα sα x̂Rα 7→ X x̂Lα sα x̂Rα , χ0 < χ, (2.1) α=0 where x̂Lα , x̂Rα are operators supported on L and R, respectively, and tr x†Lα xLβ = tr x†Rα xRβ = δαβ . Explicitly, one starts with a matrix-product representation 34 of the density operator χ−1 X  µj+1  µ1  µ  A1 · · · Aj j α sα Bj+1 · · · BLµL α σ̂1µ1 · · · σ̂LµL ρ= (2.2) α=0 where Aµl l , Blµl are χ × χ matrices—with the exception of Aµ1 1 and BLµL , which are 1 × χ and χ × 1 respectively. We suppress for compactness a sum on µ (that is, on a basis for the space on onsite operators, here the Pauli matrices σ µ with σ 0 = I). The truncation (2.1) is then ρ= χ−1 X  α=0 χ0 −1 7→  µj+1  µ  · · · Aj j α sα Bj+1 · · · α σ̂1µ1 · · · σ̂LµL X α=0   µj+1 µ  · · · Aj j α sα Bj+1 · · · α σ̂1µ1 · · · σ̂LµL , (2.3) with χ0 < χ. This approximation minimizes the Frobenius (Hilbert-Schmidt) norm distance—but a priori that is not the only norm one could use. The Frobenius approximation scheme gives poor results for initial states far from equilibrium. One can see why by considering the expectation values of the operator Olty = U (t)σly U (t)† for a system that starts at t = 0 in a y product of σ y eigenstates. (Note that Ol,t is a Schrödinger-picture operator parametrized by time t. We work in the Schrödinger picture throughout, y y except where noted.) At time t, hψ(t)|Ol,t |ψ(t)i = ±1—but generically Ol,t will be a large, complicated operator (if we choose t larger than the whole-system entanglement time, as we are free to do, it will have support throughout the system) and essentially unrelated to the local operators we wish to measure. There are 2L such operators Oly1 l2 ...,t = U (t)σly1 σly2 · · · U (−t), all corresponding to long-range operators with expectation value ±1. These operators Oly1 l2 ...,t form part of an orthonormal basis for the space of operator space. Errors along the dimensions Oly1 l2 ... will be penalized by the Frobenius-norm notion of distance with precisely the same severity as errors along more physically relevant dimensions, like σly . A more reasonable metric for truncation error should be willing to “forget” this information, in favor of more accurately keeping local operators, once they are no longer expected to feedback into the hydrodynamics. (There are more worrying problems still with the naı̈ve Frobenius truncation, which rapidly leads to a dramatic loss of positivity in the supposed density matrix for many far-from-equilibrium initial-conditions, 35 but these problems can be remedied by considering purifications.) A BBGKY-like hierarchy for the dynamics of reduced density matrix of a spin chain offers a clue as to how to proceed. In a system governed by a P Hamiltonian that is the sum of two-site terms H = j hj,j+1 , the dynamics of one-site reduced density matrices (say ρj , the reduced density matrix on site j) depends on the two site density matrices: d ρj = −i tr{j 0 6=j} [H, ρ] dt = −i[hj,j+1 ρj,j+1 ] − i[hj−1,j ρj−1,j ] , (2.4) where we write ρj,j+1 for the two-site reduced density matrix on sites j, j + 1. Meanwhile the two-site reduced density matrices depend on three-site density matrices, the three-site on four-site, and so on up the sequence. One can imagine truncating this hierarchy at some length l—that is, tracking l-site reduced density matrices for some l, say 2 or 3 or 6, and writing the dynamics for the l-site density matrices in terms of some approximation for the l + 1-site density matrices. A natural choice for such an approximation is to replace the l + 1-site matrices by their disconnected component, e.g., for l = 1, take y ρj,j+1 ∼ ρj ρj+1 . The problem of the operators Ol,t then never arises. The BBGKY-like approach fixes certain problems with Frobenius truncation, but comes with its own set of problems. It is not obvious that the l-site density matrices will even be consistent: it may not be possible to write them as reduced density matrices for a density matrix on the whole system. (Checking this, for a given set of reduced density matrices, is QMA-hard [139].) Moreover, the hard truncation at l sites may not be appropriate to capture the dynamics of the system. Longer-range operators may feed back into the dynamics of few-site density matrices via the hierarchy starting with (2.4)—and conversely, some short-range operators may have a negligible effect on the dynamics of operators of physical interest. The Frobenius norm attempts to keeps all operators with equal weight, while BBGKY keeps connected components only up to a hard cutoff. A natural compromise is to interpolate between the two by weighting the connected components of an operator according to some measure of locality. In the current work, we take the first step in this direction: we approximate the whole-system density matrices in such a way that the dynamics of reduced density matrices on up to three sites matches the BBGKY hierarchy, but 36 instead of straightforwardly closing the BBGKY hierarchy at that level, we approximate larger connected components using a method similar in spirit to the Frobenius truncation (2.1). Our method zeros out long-range correlations, replacing entanglement between different parts of the system by entanglement with a notional bath. Although we have used thermalizing Hamiltonians to motivate our method, our method does not in fact assume that the Hamiltonian governing the dynamics is thermalizing. We expect to be able to use it to treat MBL Hamiltonians, with a more stringent accuracy vs. bond dimension tradeoff. 2.2 Method: truncation of MPDOs Given an MPDO and a particular bond j, we wish to truncate the rank of the MPDO. How can we modify the Frobenius truncation (2.1) in such a way that it does not change the trace of the density matrix, nor the expectation values of local operators? The trick is to start by Schmidt decomposing the whole density matrix, and then cleverly choose basis changes on the spaces of left and right Schmidt vectors that put the data we want to avoid changing in specific, easily-understood locations (see Fig. 2.2). We can then slot our new truncation into a (slight modification of a) standard time-evolution framework, TEBD (see Sec.1.3). Our truncation algorithm guarantees that the following will not change, up to the precision of the numerical linear algebra involved, in a truncation on bond j (cf Fig. 2.3): 1. the trace of the density matrix, tr ρ; 2. the reduced density matrix ρ1···j+1 on sites 1, . . . , (j + 1); and 3. the reduced density matrix ρj···L on sites j, . . . , L. Consequently, no truncation will change the expectation of any operator on three contiguous sites, because any such operator is always contained within one of the guaranteed-preserved density operators. These guarantees do not fully specify our truncation method. To do so, define a matrix of connected correlators across the cut j: M̃αβ = hŷLα ŷRβ i − hŷLα ihŷRβ i (2.5) 37 spanf^ σjµg truncate spanf^ xLα g Figure 2.2: DMT, viewed in the space of operators on the left half of the chain. We truncate perpendicular to certain physically-relevant operators (the σjµ ). with ŷLα and ŷRβ operators supported on sites 1, . . . , j and j + 1, . . . , L respectively. (The ŷ’s spans the set of observables and are defined below.) We wish to replace this matrix M̃ by another M̃ 0 with lower rank such that tr(M̃ − M̃ 0 )† (M̃ − M̃ 0 ) is minimized subject to the constraints above. The focus on connected components is itself an important improvement. 2.2.1 Setting, notation, and tools The concept of the method may be straightforward, but it is obscured by a flurry of notation. We start truncation on bond j with an MPDO of the form χ−1 X X µ   µj+1 µ  ρ= A1 1 · · · Aj j α sα Bj+1 · · · BLµL α σ̂1µ1 · · · σ̂LµL (2.6) α=0 {µ} on an L-site chain. The Aµl l , Blµl are χ×χ matrices—with the exception of Aµ1 1 and BLµL , which are 1×χ and χ×1 respectively. (χ, called the bond dimension, will, in fact, vary between bonds and between steps of time-evolution and truncation, but for the moment we suppress this variation.) In writing our truncation algorithm, we hat our operators 1 . We use Roman 1 We hat all operators except for the density matrix ρ, and identity I. 38 Truncation at one bond preserves this density matrix this density matrix Truncation at each bond in succession preserves this density matrix this density matrix etc. Figure 2.3: The reduced density matrices that are guaranteed to be preserved under truncation. letters (frequently j, l) to index sites and bonds; a bond inherits the index of the site to its left. We use Greek letters (frequently α, β, γ—but excepting µ and χ) for the virtual index labeling the Schmidt vectors. The Greek letter µ = 0, 1, 2, 3 is used to label Pauli matrices, i.e., σ̂jµ is an operator at site j (with σ̂ 0 = I, σ̂ 1 = σ̂ x , etc.). Following the standard notation [187], the MPDO is in mixed-canonical form with an orthogonality center at site j—that is, for any j1 ≤ j and j2 ≥ j, the operators x̂Lα [j1 ] = X µ µ  µ A1 1 · · · Aj1j1 α σ̂1µ1 · · · σ̂j1j1 , {µ} X  µj +1  µ 2 +1 2 Bj2 +1 · · · BLµL α σ̂j2j+1 · · · σ̂LµL x̂Rα [j2 ] = (2.7) {µ} are orthogonal with respect to the Frobenius inner product tr[x†Lα xLβ ] = tr[x†Rα xRβ ] = δαβ . This mixed-canonical form gives the Schmidt decomposition of the density matrix ρ at bond j: ρ= χ−1 X α=0 x̂Lα [j] sα x̂Rα [j]. (2.8) 39 Henceforth, we will implicitly always be working with Schmidt vectors at bond j, and drop the bond label as follows: x̂Lα = x̂Lα [j], x̂Rα = x̂Rα [j]. (2.9) The two vector spaces span{x̂Lα } and span{x̂Rα } are the setting in which we work. We frequently abuse notation by replacing s for a diagonal matrix whose entries are sα . This allows us to shorten Eq. (2.8) into ρ = x̂L sx̂R , (2.10) where x̂L,R are row and column vectors of operators, respectively. With stage set and notation defined, we can walk through the algorithm. 2.2.2 Re-writing the MPDO to expose properties whose preservation we guarantee We wish to take the MPDO ρ= χ−1 X x̂Lα sα x̂Rα (2.11) ŷLα Mαβ ŷRβ , (2.12) α=0 (cut along bond j) and re-write it as ρ= χ−1 X α,β=0 with the new bases {ŷLα }, {ŷRα }. The bases {ŷLα }, {ŷRα } and M are chosen such that the properties we wish to avoid changing are characterized by certain easily-identifiable blocks of M : 1. tr ρ is independent of Mαβ for α, β 6= 0. 2. The reduced density matrix on sites 1, . . . , (j + 1), ρ1···j+1 = tr{(j+2)···L} ρ, is independent of Mαβ for β ≥ 4. 3. The reduced density matrix on sites j, . . . , L, ρj···L = tr{1···(j−1)} ρ, is independent of Mαβ for α ≥ 4. 40 Once we have made this basis change we will be able to modify Mαβ , α, β ≥ 4 with impunity: no such modification will violate our guarantees. Consider a change of basis ŷLβ ≡ (x̂L Q∗L )β ≡ = χ−1 X x̂Lα Q∗Lαβ α=0 X µ µ µ  A1 1 . . . Aj j α Q∗Lαβ σ̂1µ1 · · · σ̂j j , α,{µ} ŷRβ ≡ (Q†R x̂R )β ≡ = χ−1 X Q̂∗Rαβ xRα α=0 X (2.13) α,{µ}  µj+1  µj+1 · · · σ̂LµL Q∗Rαβ Bj+1 . . . BLµL α σ̂j+1 with QL,R unitary χ × χ matrices. Now write ρ = x̂L sx̂R = [x̂L Q∗L ][QTL sQR ][Q†R x̂R ] = ŷL M ŷR ; (2.14) we can see that M is related to s via M = QTL sQR . (2.15) The requisite basis transformations QL,R are given by QR decompositions QLαβ RLβ µ = tr[x̂Lα σ̂jµ ]   ∝ A01 · · · A0j−1 Aµj α ∈ Cχ×4 , µ QRαβ RRβ µ = tr[x̂Rα σ̂j+1 ]   µ 0 ∝ Bj+1 · · · BL−1 BL0 α ∈ Cχ×4 (2.16) (here we use the Einstein summation convention). In this context, the fact that the fact that the RLβ µ is upper triangular is exactly the statement that RLβ µ = 0 for β > µ. (Similarly for RRβ µ .) 41 To see that this is in fact the basis change we seek, first note the trace relations tr[σ̂jµ ŷLβ ] = X tr[σ̂jµ x̂Lα ]Q∗Lαβ = RLβ µ , α µ tr[σ̂j+1 ŷRβ ] = X µ x̂Lα ] = RRβ µ . Q†Rβα tr[σ̂j+1 (2.17) α The trace of the density matrix is tr ρ = χ−1 X (tr ŷLα )Mαβ (tr ŷRβ ) = RL0 0 M00 RR0 0 . (2.18) α,β=0 0 (Recall that σ̂j0 = σ̂j+1 = I and that the R’s are upper triangular.) This shows that tr ρ is independent most of the elements of M , as desired. Similarly, the density matrices on sites 1, . . . , j + 1 and j, . . . , L are ρ1···j+1 = χ−1 X ŷLα Mαβ α,β=0 µ 3 µ X ]σ̂j+1 tr[ŷRβ σ̂j+1 µ=0 2 χ−1 3 1X µ X = ŷLα Mαβ RRβ µ , σ̂ 2 µ=0 j+1 α,β=0 (2.19a) and ρj···L = χ−1 3 X X σ̂jµ tr[σ̂jµ ŷLα ] α,β=0 µ=0 2 Mαβ ŷRβ χ−1 3 1X µ X RLα µ Mαβ ŷRβ . = σ̂ 2 µ=0 j α,β=0 (2.19b) That is, they depend only on the left four columns and top four rows of M , again as desired. 2.2.3 Modifying the MPDO Working in the {y} bases, we can modify Mαβ for α, β ≥ 4 at will without violating our guarantees. We wish to do so in a way that reduces the rank of M while doing the least violence, in some sense, to the connected components of correlations across the cut. Explicitly, we wish to change the quantities C(AL , BR ) = hÂL B̂R i − hÂL i hB̂R i, where ÂL , B̂R have support on the left, right portions of the chain respectively, as little as possible. First, let us see what the correlator involves. Using the definition hÂi = 42 1 tr Âρ, with Z ≡ tr ρ = RL0 0 RR0 0 M00 , it is easy to see that Z hÂL i = 1 1 tr[ŷLα ÂL ]Mαβ RRβ 0 = aα Mα0 RR0 0 Z Z (2.20a) hB̂R i = 1 1 RRα 0 Mαβ tr[ŷRβ B̂R ] = RL0 0 M0β bβ . Z Z (2.20b) and We define aα = tr[ŷLα ÂL ], bα = tr[ŷRα B̂R ] for convenience. (Throughout this subsection, we employ Einstein summation notation over the Greek indices.) The expectation value of the product is then 1 tr[ŷLα ÂL ]Mαβ tr[ŷRβ B̂R ] Z 1 = aα Mαβ bβ . Z hÂL B̂R i = (2.21) Putting all these together we find   1 Mα0 M0β C(ÂL , B̂R ) = aα Mαβ − bβ . Z M00 (2.22) Since we wish to only alter pieces of the density matrix which affect correlations accross the cut at bond j, we will modify the matrix in the parenthesis, and denote it by M̃αβ = Mαβ − Mα0 M0β . M00 (2.23) At this point we have pushed one step further the process of re-writing the density matrix so that its structure explicitly reflects the distinction between information we are willing to change and information we are not willing to change, but we still have not truncated it. To carry out the truncation, perform P an SVD on the lower right block of M̃ , writing M̃αβ = γ Xαγ rγ Yγβ for 0 α, β ≥ 4. Choose an integer χ (we will see shortly how it relates to the bond 0 dimension of the final truncated MPDO) and insert a projection P (χ ) onto 43 the largest χ0 elements of r to form a new matrix     M̃ =    0 0 XrP (χ ) Y   .   (2.24) 0 M̃ 0 differs from M̃ only in those elements M̃αβ with α, β ≥ 4—that is, in those elements that encapsulate correlations with range ≥ 2. Moreover, only small elements of M are changed, since we take small (connected) correlations and set them identically to zero. 0 to replace Mαβ : This truncation results in a new matrix Mαβ 0 0 Mαβ = M̃αβ + Mα0 M0β M00 (2.25) and then 0 ρ0 = ŷLα Mαβ ŷRβ , (2.26) where the matrix M 0 has rank at most 8 + χ0 (vide infra). Since we know M 0 and (matrix-product representations of) the xLα and xRβ , putting this into MPDO form like (2.6) with bond dimension 8 + χ0 is a matter of rearrangement. To rearrange into MPDO form perform a second singular value decomposition, this time on M 0 , for M 0 = U s0 V. (2.27) Since M 0 has rank at most χ0 + 8, there will be at most χ0 + 8 nonzero singular values s. The density matrix after truncation is then ρ 7→ X µ  0µ   0µj+1 A1 1 · · · Aj j s0 Bj+1 · · · BLµL σ̂1µ1 · · · σ̂LµL , (2.28) {µ} with 0µ Aj j = Aj Q∗L U, 0µ j Bj+1 = V Q∗L Bj+1 ; (2.29) and the rest of matrices A1 , . . . , Aj−1 , Bj+2 , . . . , BL are untouched. Note that, regardless of our choice of χ0 , the reduced density matrices on sites 1, . . . , j +1, 44 j, . . . , L are exactly as they were before the truncation. One loose end needs to be tied up. We claimed that the rank of the matrix M 0 (hence the bond dimension of the MPDO after truncation) is χ0 + 8. To see this, first decompose M̃ 0 as M̃ 0 = M̃ 0A + M̃ 0B + M̃ 0C (2.30) with M̃ 0A,B,C the left, upper, and lower right blocks of M 0 respectively: 0A 0 M̃αβ = M̃αβ ,0 ≤ α ≤ 3 0B 0 M̃αβ = M̃αβ , 3 < α, 0 ≤ β ≤ 3 (2.31) 0 0C = M̃αβ , 3 < α, β M̃αβ (other elements zero). These have ranks rank M̃ 0A ≤ 4, rank M̃ 0B ≤ 4, (2.32) rank M̃ 0C ≤ χ0 , so rank(M̃ 0 ) ≤ rank M̃ 0A + rank M̃ 0B + rank M̃ 0C ≤ 8 + χ0 . Since range Mα0 M0β ⊆ range M̃ 0A M00 (2.33) (2.34) (the range of an operator is also known as its column space) we have rank M 0 = rank M̃ 0 ≤ χ0 + 8, (2.35) as desired. 2.2.4 Preservation of l-site operators 2 The preservation of `-site operators for any ` is a generalization of this preservation of 3-site operators. Suppose we truncate at the bond between site i and i + 1. Then the reduced density matrix of the whole system can once again be 2 This section was work with Bingtian Ye. 45 written ρ= χ−1 X x̂Lα sα x̂Rα (2.36) α=0 with x̂Lα = X µ [A1 1 . . . Aµi i ]α σ̂1µ1 . . . σ̂iµi {µ} X µ µi+1 i+1 x̂Rα = [Bi+1 . . . BLµL ]α σ̂i+1 . . . σ̂LµL . (2.37) {µ} As before, we perform a basis transformation before SVD decomposition and truncation: ŷLβ = χ1 X x̂Lα Q∗Lαβ α=0 ŷRβ = χ1 X (2.38) Q∗Rαβ x̂Rα . α=0 But now the tranformations QL,R are n λ λ QLαβ RLβ = tr[x̂Lα Ôi+1−n,i ] ∈ Cχ×4 n λ λ QRαβ RRβ = tr[x̂Rα Ôi+1,i+n ] ∈ Cχ×4 , (2.39) where n is an integer to control the size of the preserved operators, and the λ Ôj,k form a basis for operators on the subsystem [i + 1 − n, i] indexed by λ. After the transformation, we proceed by analogy with the previous section. This procedure will preserve the reduced density matrices of the subsystem [1, i + n] and the subsystem [i + 1 − n, L]. To guarantee this requires the bond dimension χ ≥ χpreserve = 2 × 4n , where 4n is the number of all possible λ λ Ôi+1−n,i (or Ôi+1,i+n ), i.e. the number of operators living in the subsystem [1, i + n] (or the subsystem [i + 1 − n, L]). Suppose we keep in this way n sites on either side of the cut. What is the largest-diameter operator that is preserved? For an operator on ` consecutive sites, we only need to consider the case when the truncation seperates it into two parts. Let `lef t and `right denote the size of the left and the right parts (relative to the truncation) of this operator respectively. Since `lef t + `right = `, we have min{`lef t , `right } ≤ dl/2e. If `lef t ≤ n, the `-site operator will live on the subsystem [1, i + n], and is thus preserved (the marginal case is the blue frame in Fig. 2.4a). Similarly, if `lef t ≤ n, the `-site operator will 46 (a) … i-n i-n+1 i-n+2 … subsystem [i+1-n,L] (preserved) i-1 i i+1 i+2 … … i+n-1 i+n i+n+1 subsystem [1,i+n] (preserved) (b) … i-n i-n+1 i-n+2 … subsystem [i+1-n,L] (preserved) i-1 i i+1 i+2 … … i+n-1 i+n i+n+1 subsystem [1,i+n] (preserved) Figure 2.4: The operators preserved during the truncation. (a) 2n + 1 is the maximum size of the operators that can be preserved. (b) A truncation can change the expectation of a (2n + 2)-site operator. also be preserved (the marginal case is shown by the red frame in Fig. 2.4a). Therefore, any l-site operator with d`/2e ≤ n is preserved during a truncation on any bond i, which means for a given n, we can preserve all (2n + 1)-site operators. However, a (2n + 2)-site operator can be changed by the truncation at the middle of it (Fig. 2.4b). Combining the previous expression for χpreserve and ` = 2n + 1, we prove that to preserve all `-site operators requires bond dimension χpreserve = 2` . 2.3 Results We apply the method to a variety of initial states, pure and mixed. We time evolve by the boustrophedon Trotter decomposition (1.114) of a Hamiltonian known to satisfy the eigenstate thermalization hypothesis with Trotter step δt = 1.0 (except where specified). We work at a maximum bond dimension cutoff (i.e., at each gate application we truncate to this cutoff using the algorithm described in section 2.2) and measure performance by varying this cutoff. 2.3.1 Hamiltonian We take as our Hamiltonian the spin-1/2 transverse-field Ising model on an L-site chain with open boundary conditions: H= L−1 X j=1 1 z Sjz Sj+1 + hx 2 L X L 1 zX z S + h S . 2 j=1 j=1 x (2.40) 47 hSjz i 0.1 0.0 −0.1 0 5 10 site j 15 20 25 Figure 2.5: hS z i for the initial state described in (2.42) on a 24-site chain. At hz = 0 the model is integrable (by Jordan-Wigner and Bogoliubov transP formations); the longitudinal field hz S z breaks integrability, and at hz = 0.8090, hx = 0.9045 the model is known to satisfy the eigenstate thermalization hypothesis in a strong sense [115]. We work at onsite fields hz = 0.8090, hx = 0.9045 (2.41) (except where otherwise specified). Despite their ETH nature, TFIM Hamiltonians like this can display ill-understood pre-thermalization behavior, thought to be related to long-range emergent conserved or nearly-conserved quantities [13, 113, 135]. We do not expect DMT to be able to capture this emergentintegrable behavior (see Section 2.1), so we choose our initial conditions to avoid it. 2.3.2 Application: pure-state evolution We engineer an initial state with a long-wavelength energy variation by taking a product of σ y eigenstates and rotating blocks of four spins alternately towards +z and −z (cf Fig. 2.5). The initial state is L Y   |nearY i = 1 + i(1 + gj )σj+ |↓↓ . . . ↓i (2.42) j=1 (suitably normalized), where  −1 j mod 8 = 1, 2, 7, or 0, gj = 0.1 × +1 j mod 8 = 3, 4, 5, or 6. (2.43) 48 DMT χmax = 64 DMT χmax = 128 DMT χmax = 256 DMT χmax = 512 Frobenius χmax = 64 Frobenius χmax = 128 Frobenius χmax = 256 Frobenius χmax = 512 106 104 [trρ(t)]/ p tr[ρ(t)2] 108 102 100 10−2 10−4 0 20 40 60 80 100 time t (units of S z S z coefficient) p Figure 2.6: Normalization Z = [tr ρ]/ tr[ρ2 ] as a function of time, comparing DMT (solid) and Frobenius (dashed), for the initial pure state (2.42) evolving under the Hamiltonian (2.40). Note that the second Rényi entropy of the whole chain (64-sites long) is S2 = 2 ln Z. As Z = 1 for a pure state, any deviations from Z = 1 result from the truncation. (We choose the state to be near the σ y product state in order that we may avoid the pre-thermalization behavior found in [13, 113, 135].) Since the initial state is a product state, it may be represented exactly as an MPO with bond dimension χ = 1. Trotter time evolution increases the bond dimension with each time step δt, but truncation (whatever the algorithm) kicks in only at a time ttrunc (χmax ) when χ(t) reaches χmax . Thus for each χmax the time evolution is semiexact (that is, exact up to error resulting from the Trotter decomposition of the Hamiltonian) for t < ttrunc (χmax ), at which time it begins to deviate from the semiexact value. This effect appears in all of our results; we also use it to benchmark our method (vide infra). Figure 2.6 shows the normalization Z = √tr ρ 2 as a function of time. The tr ρ normalization is related to the second Rényi entropy of the entire chain S2 ≡ tr[ρ2 ] via S2 = 2 ln Z. Time evolution by DMT produces bath entropy for − ln [tr ρ]2 the system, and this is reflected in the increase of Z as a function of time. In contrast, we find that the Frobenius method produces non-physical states with Z < 1 over the course of time-evolution, which results from negative eigenvalues of the density matrix generated in the truncation. The observation that Z ≥ 1 for DMT does not imply positive semi-definitity, but suggests that any error arising from the negative eigenvalues is small and well-controlled. Figure 2.7 shows the second Rényi entropy of the left half of the chain—that is, the subsystem consisting of sites 1 to L/2 = 32. (The von Neumann entropy is difficult to calculate for MPDOs, while second Rényi entropies are nearly 49 DMT χmax = 64 DMT χmax = 128 DMT χmax = 256 DMT χmax = 512 MPS χmax = 64 MPS χmax = 128 MPS χmax = 256 MPS χmax = 512 S2 of left half (bits) 35 30 25 20 15 10 5 0 0 20 40 60 80 100 time t (units of S z S z coefficient) Figure 2.7: Second Rényi entropy (in units of bits) of left half of a 64-site chain for an initial pure state (2.42) evolving under the Hamiltonian (2.40), in matrix product state (MPS) and DMT simulations. The largest entropy we see (χ = 64 at t = 100) is S2 ≈ 31.4 bits, very close to the theoretical maximum of 32 bits. trivial.) In contrast to matrix product states, MPDOs can represent states with arbitrarily large entropy by replacing system entanglement entropy with bath entropy. Note that once truncation starts, the entropy in the DMT simulation increases above that in MPS. This is not unexpected: in ordinary matrix product state TEBD the entanglement entropy of the left half is exactly its entropy with the right half and is a property of the matrix product state at bond bL/2c, so it can only increase when we apply a gate at bond bL/2c. In the DMT algorithm, on the other hand, the entanglement entropy of the left half of the chain is entanglement entropy not only with the right half but also with a notional bath, and it increases with every truncation on bonds within the left half. Figure 2.8 shows the system’s total energy over time as simulated by ordinary matrix product state TEBD and our density-matrix TEBD. In the DMT simulation, the energy is constant. Matrix produt state time evolution, however, imposes an additional ‘heating’ whereas DMT is designed to conserve total energy. Because the matrix product state representation is biased towards low-entanglement states, the system drifts towards the extrema of the energy spectrum over time. (The simulation begins with positive energy, and hence drifts towards the negative temperature state T → 0− .) Onsite spins are easy to measure but hard to analyze: their expection values 50 1.4 DMT δt = 0.0625 MPS δt = 0.0625 1.2 E(t) = hH(t)i 1.0 0.8 0.6 0.4 0.2 0 5 10 15 20 25 30 35 time t (units of S z S z coefficient) 40 Figure 2.8: Energy over time at fixed χ = 16 for the initial pure state (2.42) evolving under the Hamiltonian (2.40) at system size L = 64. By design, the total energy remain constant under DMT. are noisy and they decay quickly with time. Instead, we measure a Fourier component of the energy density. The energy density j , defined over a pair of sites, is hz z 1 z
hx x 1 x
S + S + S + S + S1z S2z , 2 1 2 2 2 1 2 2 hz 1 z 1 z
z S + 2 Sj+1 + (x ↔ z) + Sjz Sj+1 , 1<j<L−1 = 2 j 2
hz 1 z z z L−1 = S + S + (x ↔ z) + SL−1 SLz . L−1 L 2 2 1 = (2.44) We measure a Fourier component of the energy density 1 X ikj e j , L j=1 L−1 k=π/4 = − k = π/4 (2.45) with a wavelength of 8 sites. Fourier components are eigenmodes of the diffusion equation which should govern the system’s long-time non-equilibrium behavior. We choose this particular component (and choose the initial state accordingly) because its wavelength is long enough that it should not immediately decay, but not so long as to be longer than accessible system sizes. Figure 2.9 shows the the Fourier component of energy density (2.45) as simulated by matrix product states (dashed) and DMT (solid)for L = 16, 20, 24. 51 Energy density fourier component hk=π/4 (t)i 0.03 0.02 L = 16 DMT χmax = 16 MPS χmax = 16 DMT χmax = 32 MPS χmax = 32 DMT χmax = 64 MPS χmax = 64 DMT χmax = 128 MPS χmax = 128 DMT χmax = 256 MPS χmax = 256 DMT χmax = 512 MPS χmax = 512 DMT χmax = 1024 MPS χmax = 1024 0.01 0.00 0.03 0.02 L = 20 0.01 0.00 0.03 0.02 L = 24 0.01 0.00 0 20 40 60 80 100 L = 16 Error in energy density fourier component hk=π/4 (t)i time t (units of S z S z coefficient) 0.001 0.000 DMT χmax = 16 MPS χmax = 16 DMT χmax = 32 MPS χmax = 32 DMT χmax = 64 MPS χmax = 64 DMT χmax = 128 MPS χmax = 128 DMT χmax = 256 MPS χmax = 256 DMT χmax = 512 MPS χmax = 512 DMT χmax = 1024 MPS χmax = 1024 −0.001 L = 20 0.001 0.000 −0.001 L = 24 0.001 0.000 −0.001 0 20 40 60 80 100 time t (units of S z S z coefficient) Figure 2.9: (a) Fourier component of energy density for the initial pure state (2.42) evolving under the Hamiltonian (2.40) on chains of length L = 16, 20, 24. (b) The ‘error’ of the energy density, measured by comparing each data set with the semiexact result as simulated by matrix product states at χ = 2L/2 . Note we do not show the MPS simulations for χmax = 2048, 4096. At fixed maximum bond dimension χ  2bL/2b , DMT is more accurate than matrix product state TEBD, which illustrates the power of DMT in both shortand long-time dynamics. Moreover, where matrix product state TEBD error increases with system size, DMT error decreases. This is due to finite-size deviations from thermalizing behavior: oscillations about the local equilibrium values for local operators result from long-range coherences that we do not expect to be able to capture. Any pure state on a system of length L can be represented exactly by matrix product states with bond dimension χmax = 2bL/2c . At this bond dimension the evolution by matrix product state TEBD becomes semiexact for all times; no truncation occurs for the simulation. We can therefore simulate pure-state evolution of a matrix product state using exactly the same Hamiltonian (2.40) and boustrophedon Trotter decomposition (1.114), and compare the results to those of DMT, shown in Fig. 2.9. The data is a measure of the error introduced 52 by the truncation; these are small ≈ 10−3 for a wide range of bond dimensions in the DMT simulations. 2.3.3 Application: mixed-state evolution (near equilibrium) To probe the behavior of our algorithm near equilibrium, we take as our initial state a Gibbs state with a spatially-varying temperature ρ0 ∝ exp X βj j j  (2.46) with j the energy density of Eq. (2.44), and βj = β0 1 + gj0 where
 0 j mod 8 = 1, 2, 7, or 0, 0 gj = 0.1 × 1 j mod 8 = 3, 4, 5, or 6. (2.47) (2.48) This temperature profile is broadly similar to the S z profile we impose on the pure initial state (see Eq. (2.42) and Fig. 2.5). See Sec. 1.4.4.1 for details of the construction of the Gibbs state. In Figs. 2.10 and 2.11 we compare DMT to the purification method of Karrasch, Bardarson and Moore [104], which we label “purification”. This method takes advantage of the freedom to apply unitaries to the ancillae by timeevolving the ancillae backwards even as it evolves the physical system forwards. The time-evolution framework is therefore very similar to ours; the 2 chief differences are in the interpretation of the vector space (Cd )L in which one works and in the truncation algorithm. The similarity is magnified by our choice to use the boustrophedon Trotter decomposition (1.114) not only for DMT but for purification. Both DMT and this purification time evolution converge very quickly, as one might expect: the results are essentially identical between the methods and between bond-dimension cutoffs, even to quite small bond dimensions. (Note that we subtract the thermal value in each case.) 2.3.4 Application: mixed-state evolution (far from equilibrium) One might worry that the two initial states (2.42) from Sec. 2.3.2 and (2.46) from Sec. 2.3.3 are each special cases in their own ways: the first is a pure hk=π/4(t)i − hk=π/4ithermal 53 hk=π/4(t)i − hk=π/4ithermal 0.0030 0.0025 0.0020 0.0015 0.0010 0.0005 0.0000 0.0030 0.0025 0.0020 0.0015 0.0010 0.0005 0.0000 purification χ = 16 purification χ = 128 purification χ = 512 0 20 40 60 80 t (units of S z S z coefficient) 100 DMT χ = 16 DMT χ = 128 DMT χ = 512 0 20 40 60 80 t (units of S S coefficient) z z 100 Figure 2.10: Fourier component of energy density for (a) purification time evolution and (b) DMT for the near-equilibrium mixed state (2.46) evolving under the Hamiltonian (2.40) on a 128-site chain. The thermal value is hk=π/4 ithermal = −0.00038. state, and the second is very near equilibrium. In order to probe the performance of DMT for in more generic situations, we quench from a Gibbs state of the TFIM (2.40) with hx0 = 0.5 hz0 = 0.5 and where
βj = β0 1 + gj0 ,  0 j mod 8 = 1, 2, 7, or 0, gj0 = 0.1 × 1 j mod 8 = 3, 4, 5, or 6 (2.49) (2.50) (2.51) (as in section 2.3.3), to a TFIM (2.40) with hx1 = 2.0 hz1 = 0.5. (2.52) We again compare to the purification method [104], and find that our method z hSbL/2c (t)i z − hSbL/2cithermal 54 0.0000 −0.0005 purification χ = 16 purification χ = 128 purification χ = 512 −0.0010 −0.0015 −0.0020 z hSbL/2c (t)i z − hSbL/2cithermal −0.0025 0 20 40 60 80 t (units of S z S z coefficient) 100 0.0000 −0.0005 DMT χ = 16 DMT χ = 128 DMT χ = 512 −0.0010 −0.0015 −0.0020 −0.0025 0 20 40 60 80 t (units of S S coefficient) z z 100 Figure 2.11: Expectation value of S z at the midpoint of the chain for (a) purification time evolution and (b) DMT for the near-equilibrium initial state (2.46) evolving under the Hamiltonian (2.40) on a 128-site chain. Both methods converge very quickly, so they give nearly identical results (cf. Fig. 2.15). This expectation value fails to approach the thermal value due to the large z ithermal = −0.0622. Trotter step we use (dt = 1.0). The thermal value is hSbL/2c and that purification time evolution both converge quickly (see Figs. 2.12a and 2.13). Even very small bond dimensions (e.g. χ = 16) can accurately treat long-time, hydrodynamic behavior; accurately treating short-time behavior requires somewhat higher bond dimension. hk=π/4(t)i − hk=π/4ithermal 0.0030 0.0025 0.0020 0.0015 0.0010 0.0005 0.0000 −0.0005 DMT χ = 16 DMT χ = 128 DMT χ = 512 0 20 40 60 t (units of S z S z coefficient) 80 100 (a) Fourier component of energy density for purification time evolution and DMT starting from a far-from-equilibrium initial state on a 128-site chain. The thermal value is hk=π/4 ithermal = −0.00031. 55 z hSbL/2c (t)i z − hSbL/2cithermal 0.020 0.015 0.010 0.005 0.000 −0.005 −0.010 −0.015 −0.020 0 20 40 60 t (units of S z S z coefficient) z hSbL/2c (t)i z − hSbL/2cithermal 0.020 0.015 0.010 0.005 0.000 −0.005 −0.010 −0.015 −0.020 purification χ = 16 purification χ = 128 purification χ = 512 80 100 DMT χ = 16 DMT χ = 128 DMT χ = 512 0 20 40 60 t (units of S z S z coefficient) 80 100 Figure 2.13: Expectation value of S z at the midpoint of the chain for purification time evolution and DMT starting from a far-from-equilibrium initial z ithermal = −0.021. state on a 128-site chain. The thermal value is hSbL/2c 2.3.4.1 Convergence of mixed-state evolution It is difficult to judge convergence of any of the three algorithms from plots like Figure 2.10 or 2.12a. In Figure 2.14 we take a near-equilibrium initial state and plot the deviation in k=π/4 , as measured for a series of bond dimensions χ, from the last (largest) χ in the series. In Figure 2.16 we do the same z for a far-from-equilibrium mixed state, and in Figures 2.15, 2.17 for SL/2 for near-equlibrium and far-from-equilibrium mixed states. Our method converges with approximately the same bond dimension vs. accuracy tradeoff as purification time evolution for both the near-equilibrium initial state (Fig. 2.14) and the far-from-equilibrium initial state (Fig. 2.16). In both cases, Frobenius time evolution converges more slowly than either method. 2.4 Conclusion We have presented an algorithm for approximating density operators by lowrank matrix product operators suitable for simulating long-time quantum dynamics. The method exactly preserves expectation values of operators on up to three contiguous sites, and it slots neatly into a standard Trotterdecomposition framework for time evolution of matrix product structures (TEBD), allowing time evolution by an ETH Hamiltonian of a variety of initial states. k=π/4 [χ] − k=π/4 [χ = χmax ] 56 0.00004 0.00003 0.00002 χ = 64 χ = 128 χ = 256 χ = 512 0.00001 0.00000 −0.00001 −0.00002 0 20 40 60 80 100 0 20 t DMT 40 60 80 100 0 20 t purification 40 60 80 100 t Frobenius z z SbL/2c [χ] − SbL/2c [χ = χmax ] Figure 2.14: Convergence of k=π/4 for three algorithms. Initial state is a near-equilibrium mixed state (cf Section 2.3.3 and Figure 2.10) on a 128-site chain. For each algorithm, we plot k=π/4 [χ] − k=π/4 [χ = χmax ]—that is, how far the measurement during a run with a certain bond dimension χ deviates from measurement during a run with some high bond dimension. 0.00010 0.00005 χ = 64 χ = 128 χ = 256 χ = 512 0.00000 −0.00005 −0.00010 −0.00015 0 20 40 60 t DMT 80 100 0 20 40 60 80 t purification 100 0 20 40 60 80 100 t Frobenius Figure 2.15: Convergence of S z at site L/2 for three algorithms. Initial state is a near-equilibrium mixed state (cf Section 2.3.3 and Figure 2.11) on a 128-site z z [χ = χmax ]—that is, how far [χ] − SL/2 chain. For each algorithm, we plot SL/2 the measurement during a run with a certain bond dimension χ deviates from measurement during a run with some high bond dimension. Our algorithm, DMT, qualitatively outperforms its nearest competitor (ordinary matrix product state TEBD) for pure initial states. We use the fact that matrix product density operators with small bond dimension can represent states with high entropy to circumvent the area-law entanglement bound on matrix product states. Thus far the work is unoriginal: Zwolak and Vidal realized this was possible more than a decade ago. Our key insight is that we can preserve the trace of the density matrix and the expectation values of conserved quantities by appropriately rotating the Schmidt spaces at the bond at which we truncate. Consequently, DMT can simulate time evolution by ETH Hamiltonians to arbitrary times using memory and computation time polynomial in system size. k=π/4 [χ] − k=π/4 [χ = χmax ] 57 0.00010 0.00005 χ = 64 χ = 128 χ = 256 χ = 512 0.00000 −0.00005 −0.00010 −0.00015 0 20 40 60 80 100 0 20 t 40 60 80 100 0 20 t DMT 40 60 80 100 t purification Frobenius z z SbL/2c [χ] − SbL/2c [χ = χmax ] Figure 2.16: Convergence of k=π/4 for three algorithms. Initial state is a farfrom-equilibrium mixed state (cf Section 2.3.4 and Figure 2.12a) on a 128-site chain. For each algorithm, we plot k=π/4 [χ] − k=π/4 [χ = χmax ]—that is, how far the measurement during a run with a certain bond dimension χ deviates from measurement during a run with some high bond dimension. 0.0008 0.0006 0.0004 χ = 64 χ = 128 χ = 256 χ = 512 0.0002 0.0000 −0.0002 −0.0004 −0.0006 −0.0008 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 t t t DMT purification Frobenius 80 100 Figure 2.17: Convergence of S z at site L/2 for three algorithms. Initial state is a far-from-equilibrium mixed state (cf Section 2.3.4 and Figure 2.13) on a z z [χ = χmax ]—that [χ] − SL/2 128-site chain. For each algorithm, we plot SL/2 is, how far the measurement during a run with a certain bond dimension χ deviates from measurement during a run with some high bond dimension. In addition, DMT matches the current state of the art (purification time evolution) for near-equilibrium mixed initial states and outperforms it for farfrom-equilibrium initial states. The reader would be right to worry that our method does not converge: as we increase the bond dimension above a certain value (perhaps 25 –26 ), the accuracy of our method does not improve. We suspect that—once again—this y is a result of the operators Ol,t , whose large expectation values result from y the fact that we start near an Sl eigenstate. When we reduce the rank of the matrix M̃ in (2.24), we still do so in a way that minimizes error with respect to the Frobenius norm (even though we have arranged to exactly preserve veryy short-range operators). This means that the operators Ol,t again dominate 58 the error, and the matrix resulting from the truncation is pulled toward those operators. The obvious next step is to reduce rank in such a way that we minimize error with respect to a different norm, one that takes into account the spatial structure of the operator space: if we truncate at bond j, we should z z z z weight errors along σj−1 σj+2 more heavily than errors along σj−7 σj+6 . Such controlled-metric truncation is a natural extension of this work. One natural question to attack using our algorithm is the characterization of the ergodic side of the MBL transition. The random field Heisenberg model with small disorder appears to satisfy the ETH [160], but the nature of its dynamics is unclear (see the review of Luitz and Bar Lev [143]). Quantities z (t)Siz (t)i, from which one can compute a like the spin-spin correlation hSi+r number of diagnostics for subdiffusion, should be straightforward to calculate using our method. More interesting still are questions about interfaces between ETH and MBL systems. Besides being of inherent interest (how large must a bath be to thermalize an MBL system of a given size? how quickly does it thermalize?), answers to these questions will shed light on the phenomenological RG schemes of Potter, Vasseur, and Parameswaran [172] and Vosk, Huse, and Altman [220] for which ETH-MBL interfaces are fundamental building blocks. Because MBL systems display low entanglement in a wide variety of situations, we expect our algorithm to be able to simulate both bath and system out to large system sizes. 59 Chapter 3 HEATING, PRETHERMALIZATION, AND HYDRODYNAMICS IN A HIGH-FREQUENCY FLOQUET SYSTEM In the previous chapter we considered out-of equilibrium dynamics resulting from a quench. But, a many-body system can also be taken out of equilibrium via periodic (Floquet) driving—a strategy which has received recent attention in the context of novel Floquet phases of matter [38, 56, 110, 116, 138, 216– 218, 235, 243]. In this case, the non-equilibrium system is generically expected to absorb energy from the driving field (so-called Floquet heating) until it approaches a featureless infinite temperature state [29, 45, 128, 150, 173, 174]. Floquet heating, emergent hydrodynamics and microscopic thermalization are all fall naturally under the umbrella of non-equilibrium dynamics, but understanding the interplay between them represents an important step toward the characterization and control of non-equilibrium many-body systems [4, 6, 7, 30, 57, 126, 155, 224]. One can already see such connections; for example, in the limit of a high-frequency Floquet drive, energy absorption is set by an extremely slow heating rate. So one anticipates a relatively long timescale where the system’s stroboscopic dynamics can be captured by an effective static prethermal Hamiltonian. These expectations prompt one to ask: How do the late-time dynamics of driven systems account for both the prethermal Hamiltonian’s hydrodynamics and the energy absorption associated with Floquet heating? In this chapter, we investigate the dynamics of a non-integrable Floquet spin chain using DMT. Our main results are three fold. First, by benchmarking DMT against conventional Krylov subspace methods (for small systems, L = 20), we find that DMT efficiently simulates the system’s late-time physics, including both prethermalization and late-time Floquet heating to infinite temperature (Fig. 3.2). Second, by extending to larger systems (up to L = 100), we essentially eliminate the finite-size effect and demonstrate the exponentially slow Floquet heating (Fig. 3.1); such a calculation is fundamentally impossible for either exact diagonalization based methods (owing to the size of the Hilbert space) or conventional density matrix renormalization group (DMRG) 60 ⟨Hstatic⟩ /L 0.0 (a) hx -0.2 T -0.4 S2 (bits) 2T hy , hz 3 (b) Infinite temperature 2 10 0 -0.20 (c) -0.25 -0.30 Prethermal plateau ω= 6 ω= 7 ω= 8 ω= 9 1 0 Local Energy Density Infinite temperature hy , hz … … hx 10 3 10 1Time (1/J)10 2 ω = 5.5 ω = 6.0 ω = 6.5 ω = 7.0 ω = 7.5 hy , hz … … 0 25 50 Site Number L/2 75 hx 100 Figure 3.1: Floquet thermalization of an L = 100 spin chain (χ = 128). (a) (0) Average energy density measured with respect to Deff = Hstatic under a global drive. (b) The second Rényi entropy of the leftmost three sites. (c) Spatial profiles of energy density under a half-system drive with hHstatic i = −0.25. Insets: the drive’s time dependence (a) and schematics of the global drive (b) and the half-system drive (c). methods (owing to the large amount of entanglement at late times). Finally, we illustrate the emergence of a classical hydrodynamical description from the underlying quantum model. Having studied the late-time heating, we begin by considering transport in the associated static model and use DMT to directly measure the diffusion coefficient of the energy density. We then consider an inhomogeneously driven spin chain, where both heating and transport are crucial for understanding the late-time equilibration of the system (Fig. 3.8). Intriguingly, the dynamics of the local energy density is well described by a simple hydrodynamical equation. 3.1 Model and Floquet phenomenology We study the dynamics of a one-dimensional spin-1/2 chain whose evolution is governed by a time periodic Hamiltonian H(t) = Hstatic + Hdrive (t), where L−1 L X X z z x x Hstatic = [Jσi σi+1 + Jx σi σi+1 ] + hx σix , i=1 i=1 (3.1) 61 with σiα being the α ∈ {x, y, z} Pauli operator acting on site i.1 . We note that, while the bond terms can be mapped to a free-fermion integrable model, the additional field term breaks this integrability. The driven part, Hdrive (t) = Hdrive (t + T ), has a period T = 2π/ω and corresponds to an oscillating field in the ŷ and ẑ directions (Fig. 3.1a inset): Hdrive (t) = L X vi (t) (hy σiy + hz σiz ) . (3.2) i=1 In this work we consider two driving protocols: a global drive, where all the spins are driven [vi (t) = sgn cos(ωt) for all spins], and a half-system drive, where only the right half of the system is driven [vi≤L/2 (t) = 0 and vi>L/2 (t) = sgn cos(ωt)]. In our calculations we take the initial state to be the Néel state with a domain wall every four spins while using parameters {J, Jx , hx , hy , hz } = {1, 0.75, 0.21, 0.17, 0.13} within the high-frequency regime ω ≥ 5J. Based on previous studies we believe this choice of parameters and initial state to be generic and to capture the main features of Floquet heating [150]. The quenched dynamics of a high-frequency driven system is characterized by two timescales. The heating timescale, τ ∗ , determines the rate of energy absorption from the drive (thus the approach to infinite temperature) and has been proven to be at least exponential in the frequency of the drive, τ ∗ ≥ O(eω/Jlocal ), where Jlocal is a local energy scale [4, 6, 7, 57, 126, 155]. Until τ ∗ the dynamics of the system is, stroboscopically, well described by the static prethermal Hamiltonian Deff = Hstatic + O(ω −1 ). The prethermalization timescale, τDeff , determines the time at which the system approaches the equilibrium state with respect to Deff . When τDeff  τ ∗ , the system exhibits a well defined, long-lived prethermal regime with respect to the prethermal Hamiltonian. There is some ambiguity in the definition of the heating timescale τ ∗ . It is the characteristic timescale on which the system’s state decays towards the infinite-temperature state. But how does one measure that? Two options present themselves: the energy and the entropy. The energy is perhaps the most physically natural—in an ETH system, one expects that any state with energy E = 0 will be locally like the infinite temperature state, or at least 1 A Jordan-Wigner transform maps Hstatic to a p + ip superconductor with hopping t = J + Jx , pairing ∆ = J − Jx , and an integrability breaking term of order hx . 62 0.0 (a) ⟨Hstatic⟩ /L ω=6 ω=7 ω=8 ω=9 ω = 10 -0.2 -0.4 -0.6 0.0 (b) DMT Krylov 𝜒 9 10 z ⟩ ⟨𝜎9z 𝜎10 -0.2 |𝛿⟨𝜎 z𝜎 z ⟩| .05 512 0 -0.4 1024 2048 5 0 Time (1/J) 10 2 S 2 (bits) S2 (bits) -0.6 3 (c) 1 0 0 𝜒 = 512 𝜒 = 1024 𝜒 = 2048 1 10 0 10 1 Krylov 0 𝜏Deff 10 5 10 Time (1/J) 2 10 3 Time (1/J) Figure 3.2: Comparison of the evolution of a L = 20 spin chain under global drive between DMT and Krylov subspace methods. (a) Average energy density z . (c) The second Rényi entropy hHstatic i/L. (b) A typical local observable σ9z σ10 S2 of the leftmost 3 sites. Insets (early-time behaviors): (b) errors in DMT z z z z z i = hσ σ δhσ9z σ10 9 10 iDMT − hσ9 σ10 iKry , (c) early-time evolution of S2 . The blue lines are χ = 64; the red lines in the insets are ω = 10. evolve into a state locally like the infinite temperature state on a timescale like τDeff  τ ∗ , so one can measure the τE∗ such that ∗ hE(t)i ∼ −e−t/τE . (3.3) But the entropy, in particular the second Rényi entropy, is more natural from a formal point of view. Take a subsystem of length l and consider the Frobenius distance between the reduced density matrix ρl on the subsystem and the infinite-temperature density matrix 2−l I. This characterizes how far all operators are from their infinite temperature values. If we decompose the reduced density matrix into Pauli-string components " ρl = 2−l I + X ~ ν # R~ν σ~ν , (3.4) 63 where ~ν = ν1 ν2 ν3 . . . νl , νj ∈ 0 . . . 3] label Pauli strings σ~ν and we take each ~ν to have at least one νj 6= 0, then the R~ν are the expectation values of those Pauli strings tr[σ~ν ρl ] = R~ν (3.5) and kρl − Ik2 = 2−2l tr(ρl − I)2 = 2−l X R~ν2 = 2−l X tr[σ~ν ρl ] ~ ν ~ ν
2 . (3.6) But this is intimately related to the second Rényi entropy: kρl − 2−l Ik2 = tr ρ2l + 2−2l tr I − 2−l tr ρl = 2−S2 − 2−l
= 2−l eln 2(l−S2 ) − 1 (3.7) ≃ 2−l ln 2(l − S2 ) when S2 is close to maximum the entropy l. (We measure the entropy S2 in bits, not nits). Then the time τS0∗ such that for the subsystem 0∗ l ln 2 − S2 (t) ∼ e−t/τS (3.8) characterizes the decay of all observables But there turns out to be a factor of two between τE∗ and τS0∗ . This is already apparent from Eq. (3.6): τS0∗ actually characterizes the decay time of the square of the local expectation values, including energy. But we can see the factor more explicitly when ρl is a high-temperature Gibbs state. Define τS∗ = 21 τS0∗ , and consider a Gibbs ensemble at temperature T . The probability pi assigned to the i’th eigenstate (with i being its eigenenergy) of Hstatic can be approximated to the first order as e−βi 1 − βi pi = ≈ = 2−L (1 − βi ) , −βH tr[e ] tr[1 − βH] (3.9) where β = 1/T , 2−L is the dimension of the Hilbert space, and we use the fact that tr H = 0. The energy is then E= X i pi i ≈ 2−L X X (1 − βi )i = −2−L β2i ∝ β . i i (3.10) 64 The second Rényi entropy of the entire system is similar: Sentire = − log2 X " p2i ≈ − log2 2−2L = L − β 2 2−L X i X (1 − βi )2 # (3.11) 2i i for ∆Sentire ≡ L − Sentire ≃ β 2 2−L X i 2i ∝ β 2 . (3.12) Since the entropy is an extensive quantity for a Gibbs state, one expects this behavior to hold for any subsystem; thus ∆S ∝ β 2 . Therefore    1 dE 1 d∆S / E dt ∆S 2dt     d log ∆S d log |E| / = dt 2dt     d log |E| 1 d log ∆S = / d log β 2 d log β 1/τE∗ = 1/τS∗  (3.13) = 1. In Figs. 3.1a,b, we illustrate these two timescales τ ∗ and τDeff for an L = 100 initial state evolved with DMT (we defer the analysis of the validity of DMT until next section). In Fig. 3.1a, we highlight τ ∗ using the average energy density hHstatic (t)i/L. Until τ ∗ , the evolution is well described by Deff , and thus hHstatic i remains constant (up to ω −1 corrections). After τ ∗ , hHstatic i starts the approach to its infinite temperature value hHstatic iT =∞ = 0. To highlight the prethermalization timescale τDeff , a different diagnostic is needed. In Fig. 3.1b, we use the second Rényi entropy of the leftmost three spins of the system S2 = − log2 tr[ρ2sub ], where tr describes the trace operation, ρsub = trsite i>3 [ρ], and ρ is the density matrix describing the system. Our choice of a small subsystem ensures that S2 is not biased by DMT (see discussion below and [? ]). While the system begins in a product state with S2 = 0, its entropy quickly approaches a plateau, consistent with the Gibbs state of Deff at a temperature which matches the energy density [? ]; the system has approached the prethermal state. The timescale at which this occurs corresponds to τDeff and, indeed, we observe τDeff ∼ 1/Jlocal independent of 65 the frequency ω of the drive. At later times t > τ ∗ , parallel to hHstatic i/L, S2 begins the approach to the infinite temperature value of S2T =∞ = 3 bits. 3.2 Benchmarking DMT To ensure the reliability of DMT in the simulation of Floquet dynamics, we compare it Krylov subspace evolution [15, 89, 185] at small system sizes (L = 20). But before we compare DMT to Krylov, we verify that the Krylov evolution gives accurate results. This analysis not only gauges the applicability of DMT, but also leads to insights into the nature of the Floquet heating. Time evolution with DMT proceeds via two repeating steps, each acting on a matrix product representation of the system’s density operator. The two steps are a TEBD-like approximation of the time evolution unitary and a truncation via DMT. In the TEBD-like step, we Trotter decompose the time evolution operator into a series of gates which we then apply to the MPDO. Because each gate application increases the bond dimension of the MPDO, we must truncate it back to a fixed maximum bond dimension, which we call χ. During the truncation step, DMT separates χ into two contributions: χpreserve and χextra . χpreserve = 2` is used to guarantee the preservation of all observables on ` contiguous sites during the truncation [? ], and we call ` the preservation diameter. χextra is then used to preserve the remaining correlations with largest magnitude. We emphasize that, although truncation does not directly affect `-sized operators, their dynamics is affected by the truncation of larger sized operators via the evolution of the system. 3.2.1 Benchmarking Krylov subspace dynamics In this section we study the error of the Krylov subspace method by comparing to exact diagonalization (ED) at small system size. For each system size L ∈ {4, 6, 8, 10}, we consider a random initial product state of spins aligned in the ẑ direction. We then compute the evolution of the system under driving frequencies ω ∈ {5, 6, 7, 8, 9, 10}. In Fig. 3.3, we show the difference between the two numerical methods for the quantities of interest in our study. In particular we consider energy density {x,y,z} Hstatic /L, second Rényi Entropy S2 , onsite operators (σi ) and two-site x x z z operators (σi σi+1 and σi σi+1 ). In the top row of plots of Fig. 3.3, we consider the maximum error observed in the first 600/J time units of the evolution. Errorbars correspond to the standard deviation of the maximum error observed over 6 different initial states. We note that the maximum error in this regime 66 is not substantially affected by the system size. It is perhaps more enlightening to estimate the rate of error growth: for a given simulation parameters, Krylov subspace dynamics should induce a small constant error per timestep. This observation is borne out by the data for t < 600/J. In the bottom row of Fig. 3.3 we show the maximal rate of error growth, for some quantity O: R = max t≤600 |OED (t) − OKrylov (t)| . t (3.14) Rate of Error Growth Maximum Error This provides an estimate of the error growth as a function of the simulation time. Since the rate of growth is ≤ 10−7 , we believe Krylov subspace methods are suitable for benchmarking DMT, even at long times. 10 4 Hstatic /L Second Rényi Entropy Single Site Operators Two Site Operators 10 5 10 6 10 7 10 8 10 9 4 6 8 10 4 6 8 10 4 6 8 10 4 6 8 10 System Size System Size System Size System Size Figure 3.3: Difference of measured quantities between exact diagonalization and Krylov subspace methods over t ≤ 600/J physical time. In particular we consider energy density Hstatic /L, second Réniy Entropy S2 , single body oper{x,y,z} x z and σiz σi+1 ), in their respective ators (σi ) and two body operators (σix σi+1 columns. In the top row we consider the largest difference observed within the elapsed time. In the lower row, we consider the largest rate of error growth, defined as Eq. (3.14). 3.2.2 DMT and Krylov In Figs. 3.2 and 3.4, we compare the time evolution under DMT and Krylov and observe good quantitative agreement for t > τDeff , even at small bond dimension χ = 64. In this regime, the system is close to a local thermal state and therefore has a natural low-bond-dimension MPDO representation. This behavior is clearest in the evolution of the energy density hHstatic i/L, as it is not sensitive to the prethermalization process (Fig. 3.2a). Moreover, the 67 (a) ⟨Hstatic⟩ /L 𝜂Γ ω=9 ω = 10 -0.4 -0.5 10-1 32 𝜒 64 128 Krylov 0 24 25 26 27 𝜒 (b) 0.2 ω=9 ω = 10 0.1 𝜂Γ -0.3 100 𝜒 0.02 3 ℓ5 7 𝜒 Krylov ℓ =3 250 500 750 0 Time (1/J) 3 5 ℓ 7 𝜒 = 256 250 500 750 Time (1/J) Figure 3.4: (a) Energy density at bond dimension χ = 32, 64, 128 and (b) the preservation diameter ` = 3, 5, 7. Insets: relative error in the heating rate ηΓ = |ΓDMT −ΓKrylov |/ΓKrylov , in which Γ is defined by dtd hHstatic i = −ΓhHstatic i. same late-time behavior emerges when studying local observables, as well as the second Rényi entropy S2 (Fig. 3.2b,c). Closer inspection, however, shows that the small long-time disagreement increases with the frequency of the drive. This trend reflects the nature of Floquet heating. As the frequency increases, absorbing an energy quantum from the drive requires the rearrangement of a larger number of sites [6, 7, 155]. However, the truncation in DMT, limited by χ, often destroys the necessary longer ranged correlations (habi − hai hbi 7→ 0 for operators a and b supported a distance r > `) suppressing the heating more severely at larger driving frequencies. In Fig. 3.4, we leverage this very high-frequency regime to highlight the convergence properties of DMT with the bond dimension χ and the preservation diameter `. As expected, increasing χ at fixed ` improves the accuracy of DMT since the amount of information preserved during each truncation is greater (Fig. 3.4a). Curiously, tuning ` at fixed χ can also affect the accuracy, despite not increasing the amount of preserved information (Fig. 3.4b). This suggests that, by carefully choosing which operators to preserve during truncation, one can achieve a high accuracy in DMT while keeping a low total bond dimension χ. Having studied the late-time heating via the energy density, we now focus on the early-time dynamics via local observables and the second Rényi entropy. Unlike energy density, which is insensitive to the prethermalization dynamics, both local observables and S2 can accrue significant errors for t < τDeff at small bond dimension. This reflects the importance of long-range coherences in the early-time thermalization dynamics. A clear example is the dips in S2 that 68 DMT fails to capture (arrows in Fig. 3.2c). We interpret these dips as the growth of operators bouncing back at the other edge of the system — DMT is unable to correctly capture the dynamics of the intermediate long-range operators and thus misses their return, resulting in a larger entropy value. We note that, by increasing χ, DMT better captures longer ranged operators and better captures the resonant dip, Fig. 3.2c inset. Curiously, although both S2 and local observables are affected by long-range coherences, DMT appears to better capture the entropy dynamics. We believe this arises from the higher sensitivity of local observables to the details of the underlying interactions. One might expect the large early-time deviations to lead to equally large latetime deviations. This is not what we observe. In particular, despite a considerable error in the early-time evolution under DMT, the system approaches the correct prethermal state at τDeff , and exhibits the correct energy density, local observables and entropy. To understand this behavior, we note that the prethermal Gibbs state can be fully determined by the system’s energy density, which is conserved until τ ∗  τDeff . Because both exact evolution and DMT conserve Deff , the dynamics are constrained to the same subspace of the Hilbert space. As a result, although the precise agreement between the dynamics is limited by the finite bond dimension χ, as the evolution explores this subspace, both methods reach the same equilibrium state. For t > τ ∗ , the system absorbs energy from the drive, and evolves between different thermal states of Deff at the heating rate (for further evidence see [? ]). In this regime, DMT accurately simulates observables to the extent it accurately simulates the heating process. Finally, we explain how our choice to study S2 on only three sites is motivated by the choice of preservation diameter ` ≥ 3. During the truncation procedure, by ensuring that any operator acting on ` contiguous sites is preserved, DMT also preserves the corresponding density matrix. As a result, truncation via DMT does not affect the entropy if the subsystem considered is smaller or equal to `. As such, errors in the entropy arise from errors in longer ranged operators propagated via the system’s dynamics. This ensures that by measuring the accuracy of S2 , we are measuring the accuracy of the dynamics rather than the impact of the truncation on the density matrix. In fact, when considering a subsystem larger than `, the truncation biases the measured entropy. Each truncation step reduces the purity of the system, generating thermal-like entropy which leads to an extensive rate of entropy growth (for a detailed discussion see [? ]). 69 3.2.3 Entropy in DMT In the main text, we show the evolution of the entropy of the leftmost three sites. Here we provide more details on the entropy of subsystems with different sizes that motivates that choice for the main text. 3.2.4 Effect of Trotter step size on DMT numerics In the main text, we considered the convergence of DMT with respect to the bond dimension χ and the size of preserved operators `. Here we complement that analysis by considering the convergence in the size of the Trotter step. We quantify the error by measuring the average error s X
2 δhÔi i ≡ L−1 hÔi iDMT − hÔi iKrylov , (3.15) i i.e., the error of a local observable averaged over all sites. z x In Fig. 3.5, we take Ôi ∈ {σiz σi+1 , σix σi+1 , σix }, the three local observables that contribute to energy Hstatic . By decreasing the Trotter step size from 4/J to 10/J, we observe an improvement of results, especially during the late-time heating. However, the simulation does not benefit from further decreasing the step size from 10/J to 20/J at late time, although the error at very early-time decreases. This is because to apply truncations too frequently will destroy the many-body correlators more severely. We therefore use Trotter step 10/J throughout this work. 3.2.4.1 Page-like correction at late time For a subsystem with size Lsub ∼ L/2, the bipartite entanglement entropy approahes the maximal value of Lsub bits in DMT simulations, but a smaller value in Krylov subspace simulations (see Fig. 3.6). This discrepancy arises because the Krylov method guarantees that the system remains in a pure state, while in DMT the MPDO representation is entangled with a notional bath arising from the truncation procedure. The difference between the two corresponds to a Page-like correction. At Lsub = L/2 and infinite temperature, this correction is exactly log 2 (1 bit), in agreement with the theoretical prediction [142]. As one decreases Lsub , the correction decreases exponentially, and the two methods agree better and better. 70 10-1 z δ⟨𝜎iz𝜎i+1 ⟩ (a) 10-2 10-3 10-1 x δ⟨𝜎ix𝜎i+1 ⟩ (b) 10-2 10-3 (c) δ⟨𝜎ix ⟩ 10-2 10-3 Number of Trotter steps per 1/J 5 10 0 10 10 20 10 2 Time (1/J) 1 10 3 Figure 3.5: Convergence of local observables with respect to Trotter steps z x (L = 20, ω = 8, χ = 128, ` = 3). (a) hσiz σi+1 i, (b) hσix σi+1 i, and (c) hσix i are all the three terms that have non-zero contribution to energy density. 10 Lsub = L/2 DMT Krylov S2 (bits) 8 1bit 6 4 Lsub 2 0 10 0 101 10 2 Time (1/J) 10 3 Figure 3.6: Late-time second Rényi entropy of subsystems with different sizes (L = 20, ω = 6). The subsystem sizes Lsub for the curves from the bottom to top are {1,2,3,4,5,6,7,8,9,10}. 3.2.4.2 Early-time behaviors For t < τDeff , the system is well described by the time-independent interacting Hamiltonian Deff . For an initial product state, the entropy of a subsystem is expected to increase linearly with time. Using Krylov subspace methods, we indeed observe the linear increase of entropy at early time. Curiously, the system often exhibits some non-monotonicity at intermediate times (from long-range coherences), but it eventually approaches its prethermal value at 71 S 2 (bits) (a) 8 L = 20 6 4 2 Lsub 0 (b) 40 S 2 (bits) 30 L = 100 DMT Krylov 20 10 0 Lsub 5 10 15 Time (1/J ) 20 25 Figure 3.7: Early-time second Rényi entropy of subsystems with different sizes (ω = 10). (a) Small system with L = 20, and the subsystem sizes Lsub for the curves from the bottom to top are {1,2,3,4,5,6,7,8,9,10}. (b) Large system with L = 100, and the subsystem sizes Lsub for the curves from the bottom to top are {1,9,17,25,33,41,49}. The black straight line sets the upper bond of the entropy, of which the rate is extracted from Krylov data with L = 20. τDeff (Fig. 3.7a). We can extract the early-time entropy production rate ΓSearly . Importantly, t · ΓSearly sets an upper bound on the of entropy production (black line in Fig. 3.7). However, DMT fails to capture both the effect of the many-body coherences, as well as the previously mentioned bound in entropy growth. On the one hand, the truncation destroys the long-range coherences necessary to capture the many-body revivals observed when using Krylov, resulting in a smoothed entropy curve. On the other hand, at subsystem sizes greater than the preservation diameter `, the entropy in DMT can exceed the upper bound t · ΓSearly (Fig. 3.7b). This is because the truncation in DMT can convert some entanglement entropy to thermal entropy, which does not care about the subsystem boundary. The entropy of a subsystem can then increase with the number of bonds truncated, so in larger subsystems we observe a higher rate of entropy growth (Fig. 3.7b). 3.3 Analysis and results Having established that DMT accurately captures the late-time thermalization of quantum systems, we now apply it in the study of the late-time emergent 72 hydrodynamics in large spin chains (L = 100), beyond what can be achieved with previous methods. Our ultimate goal is to write down a phenomenological description of the state and its dynamics, and to verify numerically that it describes our model. The phenomenology is easily written down. In general, the dynamics of the local energy density follows the continuity equation: ∂t (x, t) = ∂x j(x, t) + Q̇(x, t), where j is the energy current and Q̇ describes heating. Near equilibrium, the energy current is j ∝ ∂x T , where T is the local temperature. In contrast to homogeneous systems where ∂x T ∝ ∂x , the spatial inhomogeneity induced by the half-system drive forces the current to depend not only on ∂x , but also the position in the chain; thus a small correction in j is needed. At the same time, motivated by the exponential heating of the global drive, we ∗ ∗ , where τlocal expect the heating rate at the driven sites to be Q̇ = −(t)/τlocal is set by the global heating timescale τE∗ . We combine both these effects — heating and diffusion across the chain — in a simple hydrodynamical equation for the local energy density: ∂t (x, t) = D()∂x2   (x, t) [1 + ηg1 (x)](x, t) − g2 (x) ∗ , τlocal (3.16) where D() is the energy density dependent diffusion coefficient, while g1 (x) and g2 (x) characterize the regions where the drive affects diffusion and heating, respectively (the magnitude of the latter is controlled by a small parameter η). We set g1 (x) = g2 (x) = 21 + 12 tanh[(x − L/2)/ξ], a smoothed step function, and ξ = 5. (Our results are not sensitive to the particular choice of ξ, as long as ξ  L.) Verifying this phenomenology is less straightforward. To do so, we consider two protocols: Fig. 3.8 offers a précis of the process. After verifying that the system is in fact well-described by a Gibbs state with time-varying temperature (Sec. 3.3.1, not shown in Fig. 3.8), we verify that the system displays an appropriate Floquet heating rateh under both a half-system drive and a global drive (Fig. 3.8a). We then turn to the details of the dynamics of the local energy density. To provide a baseline for comparison, we measure the diffusion coefficients of the static Hamiltonian Hstatic (Sec. 3.3.3, Fig. 3.8b). We then apply the halfdriving protocol and fit the results to the modified diffusion equation (3.16) 73 (a) L = 20: 𝜏E∗ (global) 𝜏S∗ (global) 𝜏∗ Mode Amplitude 10 2 6 1 (b) L ∗= 100: 𝜏E (half) 𝜏∗E (global) 𝜏S∗ (half) 𝜏S∗ (global) 7 8 ω (1/J) 0.1 0 -0.20 -0.25 0.5 0.2 -0.15 Local Energy Density 10 3 -0.10 ε = -0.10 ε = -0.15 ε = -0.20 ε = -0.25 ε = -0.30 25 50 Time (1/J) (c) t = 500 t = 450 t = 400 t = 350 t = 300 t = 250 Initial state -0.30 t = 200 75 0 25 ω= 6.0 50 75 100 Site Number Figure 3.8: (a) Heating rate τ ∗ extracted from energy density and subsystem entropy for L = 20, 100, as well as global and half-system drive. (b) Evolving with the static Hamiltonian Hstatic , an initial spatial variance in energy density will decay. The diffusion coefficient (at different energy densities) can be extracted from the exponential decay of the amplitude of Fourier modes. The amplitudes are normalized. (c) Dynamics of the energy density. By starting with the data at some initial state (e.g. t = 200/J) the hydrodynamical description Eq. (3.16) (dashed black curve) is able to correctly capture the energy density at late times. (Fig. 3.8c, Sec. 3.3.4). 3.3.1 Approach to Gibbs Ensemble To show that the system approaches a Gibbs ensemble at late time, we compare the expectation value of local observables of the late-time Floquet state and the Gibbs state of the static Hamiltonian Hstatic , which approximates the prethermal Hamiltonian Deff to order ω −1 . By doing imaginary-time evolution with DMT, we obtain the Gibbs states of the Hstatic at different temperatures. The inset of Fig. 3.9 shows the averaged energy density hHstatic i at different temperatures; similarly we can obtain other physical quantities as a function of temperature. This provides the map between energy density and the expectation value of other observables in the Gibbs state, allowing us to directly compare the late-time Floquet state to the Gibbs state at the same averaged energy density. As shown in Fig. 3.9, the two states have the same entropy and local observables, given the same energy density, indicating that the Floquet system can be described by a Gibbs ensemble with respect to the prethermal 74 Hamitonian (for local observables). 0.0 -0.4 -0.8 -1.2 0.0 ⟨Hstatic⟩/L S2 (bits) 10 (a) 9 1.0 2.0 Temperature-1(1/J) z z ⟩ 𝜎11 ⟨𝜎10 0.0 (b) -0.1 -0.2 x ⟩ ⟨𝜎10 -0.3 0.00 (c) -0.02 -0.04 Gibbs state late-time Floquet state -0.4 -0.3 -0.2 ⟨Hstatic⟩ /L -0.1 0.0 Figure 3.9: Physical quantities as a function averaged energy density for Gibbs states and late-time Floquet states (L = 20, ω = 6). (a) second Rényi entropy of the half chain, (b) a two-site local observable, and (c) an one-site local observable. Inset: The averaged energy density as a function of inverse temperature. How can these results be consistent with our claim in the main text that the relationship between effective temperature and (e.g.) the energy density of Hstatic is itself frequency dependent? Essentially, consistency requires that the particular local observables of Fig. 3.9 depend weakly on frequency. We find that they depend on frequency only at second order or higher in ω −1 , because those observables and Hstatic are even under global spin flip, while our drive is odd. To see these symmetry considerations play out, suppose the system is in a Gibbs state 1 −1 (3.17) ρ(t; ω) = e−β(t)Deff (ω ) , Z where the effective temperature T (t) = 1/β(t) is given by hDeff (t)i ≡ Z −1 tr[Deff (ω −1 ) e−β(t)Deff (ω with Z = tr[e−β(t)Deff (ω −1 ) −1 ) ] (3.18) ]. For compactness we drop the time dependence of 75 the temperature. Expand Deff in powers of ω −1 : Deff = Hstatic + ω −1 D0 + O(h2 ω −2 ). (3.19) Per Ref. [150]: Z t Z 1 T dt0 [Hdrive (t0 ), Hstatic ] ω D = dt i T 0 0 Z t Z T X 1 z x = dt0 v(t) dt i [hz σjz + hy σjy , Jσkz σk+1 + Jx σkx σk+1 + hx σkx ] T 0 0 jk −1 0 πω −1 X z x + Jx σkx σk+1 + hx σkx ] [hz σjz + hy σjy , Jσkz σk+1 2 jk X y x z x = πω −1 −hz Jx (σky σk+1 + σkx σk+1 ) − hz hx σky − hy J(σkx σk+1 + σkz σk+1 ) =i k  x z +hy Jx (σkz σk+1 + σkx σk+1 ) + hy hx σkz . (3.20) It is immediately apparent that D0 is odd under a π rotation about the x axis, while Hstatic is even—more specifically, if X= Y σjx (3.21) j then XHstatic X = Hstatic XD0 X = −D0 , (3.22) so n ]. 0 = tr[D0 Hstatic (3.23) 76 Then the partition function is i h −1 0 Z = tr e−β(Hstatic +ω D ) + O(ω −2 )   −1 −1 − βω2 D0 −βHstatic − βω2 D0 ≈ tr e e e + O(ω −2 )     βω −1 0 βω −1 0 −βHstatic 1− + O(ω −2 ) ≈ tr 1 − D e D 2 2   ≈ tr e−βHstatic (1 − βω −1 D0 ) + O(ω −2 )   = tr e−βHstatic + O(ω −2 ) (3.24) = Z0 + O(h2 ω −2 )   with Z0 = tr e−βHstatic the partition function of the static Hamiltonian. Consider now some (local) operator Oj . With this fact about the partition function in mind its expectation value in the Gibbs state ρ is tr Oj ρ = Z0−1 tr  −1 − βω2 −1 D0 −βHstatic − βω2 D0  e e e × Oj + O(ω −2 )      βω −1 0 −βHstatic βω −1 0 −1 D e D × Oj + O(ω −2 ) = Z0 tr 1 − 1− 2 2    βω −1 −1  −βH static = Z0−1 tr e−βHstatic Oj − {D0 , Oj } + O(ω −2 ) , Z0 tr e 2 (3.25) where {·, ·} corresponds to the anti-commutator.   If Oj is even under X, as the operators of Fig. 3.9 are, tr e−βHstatic D0 Oj = 0 and, to first order in ω −1 , Oj takes the same expectation value it has in the Hstatic Gibbs state:     tr ρ Ojeven = Z0−1 tr e−βHstatic Ojeven + O(ω −2 ) . (3.26) If Oj is odd under X, then   βω −1  −βHstatic 0 odd  tr ρ Ojodd = −Z0−1 tr e {D , Oj } + O(ω −2 ) . (3.27) 2   Here we note that Z0−1 tr e−βHstatic {D0 , Ojodd } corresponds to the sum of expectation values of one and two-body operators. Taking Ojodd = σky , the sigx x nificant terms in D0 are hz Jx (σk+1 σky + σky σk+1 ) + hz hx σky . Using, β ∼ 0.2, hz = 0.13, hx = 0.21, Jx = 0.75, ω = 6 and the data from Fig. 3.9, we estimate 77 0.004 Site 20 (undriven) Site 80 (driven) 0.003 y ⟨𝜎i ⟩ 0.002 0.001 0.000 -0.4 -0.3 -0.2 ⟨Hstatic⟩ /L -0.1 0.0 Figure 3.10: Comparison of the expectation value of σ y for different sites on the half-driven chain, at frequency ω = 6 and using the parameters of the main text. In red, the site is located within the driven side of the chain, where the effective Hamiltonian is modified by the drive. In blue, the site is located within the undriven side of the chain, where the effective Hamiltonian is only modified by the drive at very large order. tr ρσky ∼ βπω −1 (2hz Jx hσkx i + hz hx ) ∼ 2.2 × 10−3 . In fact when we consider our half-driven chain and compare σ y for a site in the the undriven part (Deff = Hstatic ) to a site in the driven part (Deff = Hstatic + ω −1 D0 + O(ω −2 )), we find that in the driven part hσ y i ∼ 3 × 10−3 whereas the in the undriven region hσ y i ∼ 0 (Fig. 3.10), in agreement with our estimate. 3.3.2 Heating To ensure our results are not affected by finite-size effects, we revisit the global drive protocol in Figs. 3.1a,b. As in the L = 20 case shown in Figs. 3.2a,b, we observe a prethermal plateau in S2 followed by the late-time heating of both hHstatic i/L and S2 toward the infinite temperature state. We characterize the late-time dynamics by extracting the heating timescale associated with their evolution. More specifically, given the observed exponential decay of both hHstatic i/L and S2 to their infinite-temperature values, i.e. |hHstatic i/L| ∝ ∗ ∗ e−t/τE and (S2T =∞ − S2 ) ∝ e−2t/τS , we extract τE∗ and τS∗ as a measure of the heating timescale τ ∗ [? ]. Crucially, τE∗ and τS∗ agree with one another for both system sizes, Fig. 3.8a, demonstrating they capture the same heating process and are not affected by finite size effects. By repeating the same procedure for different frequencies of the drive, we observe the expected exponential E,S ∗ dependence in frequency, τE,S ∝ eω/Jlocal , suggesting previous bounds are tight E [4, 6, 7, 57, 126, 155]. The extracted effective local energy scales, Jlocal = 1.21± S 0.04 and Jlocal = 1.16±0.04, are in good agreement with the microscopic onsite (σ x +σ x ) z x energy scale of Hstatic (||ĥi || ≃ 1.26, where ĥi = Jσiz σi+1 +Jx σix σi+1 +hx i 2 i+1 78 is the local Hamiltonian on each bond). Having investigated Floquet heating with DMT, we now move to study the emergent diffusive behavior of local energy density. By performing imaginarytime evolution, we prepare the initial state to be a thermal state (Gibbs ensemble) with a small spatial variation in energy (taken to be different Fourier modes) [? ]. As the system evolves and approaches the global equilibrium state, the amplitude of this spatial variation decays. Indeed, we observe the exponential decay of the amplitude of these modes. Crucially, this decay timescale grows as q 2 , where 2π/q is the wavelength of the Fourier mode, corroborating the diffusive nature of the dynamics [? ]. By also changing the temperature of the initial Gibbs ensemble, the diffusion coefficient D can be studied as a function of average energy density of the system , Fig. 3.8b. We note our method, near infinite temperature ( = 0), matches independent calculations of the diffusion [162]. The ability to correctly simulate Floquet heating and diffusive behaviors in large-scale quantum systems enables us to study more complex late-time hydrodynamics. We now attempt to combine these two aspects by investigating the half-system driving protocol, where the drive only adds energy to the right side of the system. As a result, the heating timescale τ ∗ is twice of that of the global drive, Fig. 3.8a. In this setup, the dynamics of the local energy density is much richer than in the globally driven case: due to the inhomogenous heating, energy must be transported across the spin chain for the system to approach the infinite temperature state, Fig. 3.1c. As such, both heating and transport are crucial to capturing the late-time equilibration of the system. Moreover, by changing the frequency of the drive, one directly modifies the heating rate exponentially without much change to transport, thus controlling the relative importance between the two. 3.3.3 Extracting diffusion coefficients of a spatially uniform, static Hamiltonian In this section we discuss the method used to extract the diffusion constant for a spatially uniform Hamiltonian H, which is a sum of local terms: H= L−1 X site i=0 hi . (3.28) 79 We present the numerical experiment performed using DMT and Krylov subspace methods, and how the diffusion constant can be extracted from the obtained data. 3.3.3.1 Numerical Experiment To properly probe diffusive behavior it is imperative that the system is perturbed around an equilibrium (i.e. thermal) state of H. More specifically, we want to initialize the system in a thermal state of H + ηHperturb , where the form of Hperturb controls the type of perturbation imposed, while η controls its strength. Since we are interested in studying the diffusion of energy when evolved H, we want the perturbations to correspond to the eigenmodes of the diffusion equation, spatial oscilations of the energy density. We then consider a family [k] of Hperturb which generate the k-th mode: [k] Hperturb = L−1 X site i=0 hi cos kiπ . L−1 (3.29) In DMT, the thermal state can be straighforwardly generated by performing imaginary time evolution on the infinite temperature state ρT =∞ ∝ I: ρβ = Z −1 exp {−β [H + ηHperturb ]} . (3.30) Because Krylov subspace methods can only treat pure states, it is impossible to directly compute expectations of the thermal state. Nevertheless, expectation values over the thermal density matrix ρβ can be obtained by averaging over initial states, which are then imaginary time evolved: D Nave hD E i h Ei 1 XD 1 X 1/2 1/2 1/2 1/2 tr Oρβ = ψi ρβ O ρβ ψi ≈ ψi ρβ O ρβ ψi . D i=1 Nave i=1 (3.31) Due to the large size D of the Hilbert space, we cannot perform the entire calculation. Instead we approximate it by averaging over Nave number of random initial states |ψi i: |ψi i ∝ D X i=1 ci |ii , ci normally distributed complex variables . (3.32) 80 Due to quantum typicality, such random states behave as infinite temperature states (for local operators) [181], and so the number of Nave need not be very large (we use Nave = 50). Once the initial state is generated, the system is time evolved with H, and the local energy hi is calculated as a function of time evolved. We observe that the initial spatial profile of the local energy quickly decays and the system becomes spatially uniform due to the diffusion of the energy density. 3.3.3.2 Extraction of the diffusion constant P Consider a system with some conserved quantity S = j sj such that sj are local operators. Moreover let ṡj = i[H, sj ] be also local (as is guaranteed in for a local Hamiltonian). We call sj local conserved quantities. In our case, S = H and sj = hj , the local energy. When H is spatially uniform and at equilibrium sj (t) will be constant for all sites (up to edge effects). As a result, we can measure the distance from equilibrium by sX P(t) = (hsj (t)i − s̄)2 (3.33) j where s̄ ≡ S/L is independent of time. The decay of this quantity provides a proxy for the diffusion coefficient: if the system is diffusive with diffusion coefficient D, then the decay rate of this quantity is given by the decay rate of the slowest non-zero diffusive mode. For a generic initial state, this corresponds to P(t) ∝ exp[−π 2 tD/L2 ] (3.34) for t  L2 /(4π 2 D) (the decay rate of the second-slowest mode). The diffusion coefficient is extracted by fitting this long-time behavior of P(t). Alternatively, we can probe that slowest mode directly, by exciting a particular diffusive mode and measuring its magnitude. This is most straightforwardly implemented by preparing the lowest diffusive mode k = 1, sj (t = 0) ≈ cos(jπ/L) + C as the initial state and measuring the amplitude of the corresponding Fourier mode: jπ 2X cos sj (t) . sq=π/L (t) = L j=0 L−1 L−1 (3.35) (a) (b) Amplitude -0.100 L = 100 -2 10 -0.105 25 50 75 Site Number 100 10 ε = -0.3 ε = -0.2 ε = -0.1 0.3 0.2 0.1 Lowest mode sq=π/L(t) -3 0 (c) Decay Rate Local Energy Density 81 Distance from equilibrium P(t) 0 20 40 Time (1/J) 60 0.0 80 0 π/100 2π/100 Wavevector q 3π/100 Figure 3.11: (a) The evolution of the first Fourier mode under Hstatic . (b) Decay of Fourier modes at large system size. β is chosen such that the averaged energy density is set to be ¯ = −0.1. (c) The decay rate of Fourier modes depends quadratically on the wavevector. The system size L = 100. 2 2 In this, the decay of sq=π/L (t) will be ∝ e−π tD/L , from where D can be extracted. We note that the profile of sj (t) can also be fitted, with a leastsquare method, to the lowest Fourier mode. Both methods yield the same results. Fig. 3.11a,b illustrate both methods, investigating P(t) and sq=π/L (t) for H = Hstatic . We see both the slowest mode and the sum over all modes decay exponentially with time; they have the same decay rate. Here we note that, at early times, we observe non-exponential behavior in the decay of sq=π/L (t) until a timescale ∼ 1/hx , the integrability breaking term of our system. It is at this timescale that we expect the interactions to induce the appropriate diffusive behavior. Moreover, we can study the decay of higher Fourier modes by using the same methodology. In Fig. 3.11(c), we observe the quadratic dependence of the decay rate on the wavevector, supporting that the dynamics of local energy density is diffusive in our system. 3.3.3.3 Accuracy of extracted diffusion coefficients The work of Kloss, Bar Lev and Reichman [119] and ongoing (unpublished) work of Leviatan et al. find that TDVP shows “false convergence”: it can converge very quickly in bond dimension—but to dynamics with an unphysical diffusion coefficient. We must therefore check that DMT shows the correct diffusive dynamics. In this section we compare the diffusion constant extracted between DMT and Krylov-space dynamics. 82 (a) 10-3 sq=π/L(t) (DMT) sq=π/L(t) (Krylov) 10-4 10-5 hx = 1.03 10-2 Amplitude Amplitude 10 (b) hx = 0.21 -2 sq=π/L(t) (DMT) sq=π/L(t) (Krylov) P(t) (DMT) P(t) (DMT) P(t) (Krylov) 0 5 10 Time (1/J) 15 10-3 P(t) (Krylov) 0 5 10 Time (1/J) 15 Figure 3.12: Decay of Fourier modes at small system size (L = 20) for (a) our model Hstatic near integrability, with the parameters used in the bulk of the paper, and for (b) our model far from integrability (b). In each The averaged energy density is set to be ¯ = −0.1. Consider our static Hamiltonian L−1 L X X z z x x Hstatic = [Jσi σi+1 + Jx σi σi+1 ] + hx σix . i=1 (3.36) i=1 In the main text we used {J, Jx , hx , hy , hz } = {1, 0.75, 0.21, 0.17, 0.13}, similar to previous work [150]. Because we are considering nearest neighbor interactions, the only integrability breaking term is hx , leading to a naı̈ve estimate for the scattering length of λ ∼ J/hx ≃ 5. As a result, observing diffusion at small system sizes is difficult. Fig. 3.12 (left) highlights this difficulty. Nevertheless, we observe good agreement between DMT and Krylov in the dynamics. We believe that DMT artificially increases a dephasing rate for the model’s quasiparticles; this explains the gradually increasing discrepancy between the DMT and Krylov simulations. To check how well DMT can capture diffusion, we increase hx = 1.03, decreasing the scattering lengthscale and making small system sizes more amenable to studies of diffusion. Indeed, Fig. 3.12 clearly demonstrates the agreement between the two methods, and as a result, the ability of DMT to probe the diffusive physics. 3.3.4 Classical diffusion equation With the previously extracted parameters τE∗ and D(), we can calculate the hydrodynamical evolution by Eq. 3.16, and compare it with the DMT simulation of the microscopic model. We leave η to be the only fitting parameter, constant across the entire evolution, but dependent on the frequency of the drive. Starting with an initial product state, we let the system reach local equilibrium by evolving it to time t0 = 200/J. Choosing the resulting energy 83 density profile as the initial state of Eq. 3.16, we observe excellent agreement of the remainder of the evolution, Fig. 3.8c. Using the same methodology, we reproduce the dynamics for different driving frequencies, corroborating our hydrodynamical description for the local energy density dynamics under the Floquet quantum model, Fig. 3.1c. Finally, we note that the fitting parameter η shows the expected negative dependence on the driving frequency ω; at higher frequency, Deff on the right side of the chain approaches Hstatic and the spatial inhomogeneity in Deff is reduced. 3.3.4.1 Derivation and approximation In general, the equations governing the heat transport in classical systems are ∂t (x, t) = ∂x j(x, t) + q(x, t) (3.37) j(x, t) ∝ ∂x T (x, t) . (3.38) The first equation is energy conservation, and the second equation reflects that a non-uniform temperature T (x, t) will lead to a heat current j(x, t). By driving the right half of the chain, the conversion from energy density (x, t) to temperature T (x, t), as well as the energy absorption q(x, t) varies explicitly in position along the chain. In particular, the lowest order (x, t) correction to T (x, t) yields a heat current j(x, t) ∝ ∂x T (x, t) ∝ ∂[(1+ηg(x))(x, t)], in which the small parameter η characterizes the correction of the conversion from  to T on the driven half and g(x) captures the spatial profile of the inhomogeneity. In our model, g(x) should be close to the step function Θ(x − L/2). The diffusion equation then reads ∂t  = D∂x2 [(1 + ηg(x))] − q(x, t) ≈ D∂x2 [(1 + ηg(x))] −  g(x), τ∗ (3.39) where we have used the fact that q(x, t) ≈ −g(x)/τ ∗ (for the driven part). We remark that a spatially varying Deff may also lead to another two modifications in the heat equation: a spatially dependent definition of , and spatially dependent diffusion constant D. A heat equation including all these modifications can be written as ∂t [(1 + λg(x))] = ∂x {[D + δDg(x)]∂x [(1 + ηg(x))]} −  g(x) , τ∗ (3.40) where λ, δD and η are all small numbers. We find that the dynamics in our 84 experiment is not sensitive to the inclusion of these terms, as they do not lead to qualitatively different terms in the equation. In fact, we find that the η term is the most meaningful for well describing the dynamics than λ and δD. At early times, before the heating occurs (τDeff < t < τ ∗ ), we already observe a non-homogenous spatial profile of energy density, due to the temperature inhomogeneity induced by η. Moreover, since the spatial profile g(x) is close to the step function Θ(x − L/2), a higher derivative of it will contribute to larger correction. Therefore, the η term, which has a second derivative, should be the most important correction. 3.3.4.2 Solving the Heat Equation Since no energy can flow out of the boundary of the system, we require 0 = j(t)|x=0,L ≈ ∂x [(1 + ηg(x))]|x=0,L . Considering that g(x) remains constant deeply inside the driven and undriven parts, we can simplify the boundary condition as ∂x |x=0,L = 0. This can be immediately achieved by considering the cosine series of the problem (x, t) = ∞ X fn (t) cos n=0 nπx . L (3.41) The differential equation then becomes: ∞ X n=0 cos ∞ h X nπx nπ i2 nπx ∂t fn (t) = −D fn (t) cos L L L n=0 ∞ h X nπx i ∂x2 g(x) cos + Dη fn (t) L n=0 ∞ nπx 1 X − ∗ g(x) cos fn (t). τ n=0 L (3.42) 85 Integrating both sides with against cos kπx for k ∈ N yields (the second term L on the right-hand side is integrated by parts):  2 kπ ∂t fk (t) = −fk (t)D L ( )  2 ∞ X kπ 1 2 fn (t) Dη + ∗ − L(1 + δk0 ) n=0 L τ Z L kπx nπx dx g(x) cos cos . × L L 0 (3.43) The resulting equations can be cast in a vectorial form as ∂t f~(t) = M f~(t) , (3.44) where f~ is the vector of the Fourier components and M describes the coupling between the modes in the right-hand side of Eq. (3.43). In practice, the magnitude of the Fourier modes decays very quickly with n, so we can consider only the first N = 40 and not incur significant error. The time evolution of f~ then becomes exactly f~(t) = exp {(t − t0 )M } f~(t0 ) . 3.3.4.3 (3.45) Dynamics of the energy density We now describe the procedure by which we can obtain the dynamics of the energy density at late times in the L = 100 system. The hydrodynamical description holds only for systems near a local equilibrium. As such to ensure we have a correct initial state, we choose some initial time t0 and Fourier transform the energy density profile at that time, thus obtaining fn (t0 ). We can now consider the evolution iteratively, i.e. to consider a series of times {tn } and evolve our state using Eq. (3.45) from tn−1 to tn . Here, the heating timescale τ ∗ and the diffusion coefficient D() are from the global heating rate and the diffusion under static Hamiltonian, respectively. For the energyindependent parameter η, we opitimize its value to minimize the discrepency between the two evolutions (DMT and hydrodynamical model). In particular, we charaterize the discrepency by the standard deviation averaged over all P pP 2 time slices, namely, tn x |(x, tn )DM T − (x, tn )Hydro | /L. 86 -0.10 Local Energy Density -0.15 -0.20 -0.25 -0.30 -0.35 0 ω= 5.5 25 50 75 100 0 Site Number ω= 6.0 25 50 75 100 0 Site Number ω= 6.5 25 50 75 100 0 Site Number ω= 7.0 25 50 75 100 Site Number Figure 3.13: Time evolution of the energy density profile starting with the lowest energy density state as the initial state. We observe great agreement with the DMT results. The extracted values of η are shown in Fig. 3.14. In Fig. 3.13, we compare the energy density profile from the simulation with that arising from Eq. (3.45). Using the lowest energy density state as the initial state, we can then apply the above procedure to obtain the energy density dynamics for later times across a large frequency range. We also observe that η has a negative dependence on frequency ω as expected (Fig. 3.14), since its value is determined by the higher order corrections to Deff in ω −1 , which goes smaller when the driving frequency increases. Moreover, by increasing bond dimension χ in DMT, we check the convergence of the energy density (Fig. 3.15). 0.08 0.06 0.04 5.5 6.0 ω 6.5 7.0 Figure 3.14: Extracted η decreases as the driving frequency increases. 3.3.5 3.4 Hydrodynamics in large spin chains Discussion In this chapter, we used DMT to study the emergence of hydrodynamics in the late-time evolution of large-scale Floquet systems. We began by demonstrating the validity and the range of applicability of this new method by comparing it against Krylov subspace methods, showing excellent agreement of the late- 87 Figure 3.15: Convergence of energy density. (a) Direct comparison between different bond dimension χ. (b) The relative error of local energy density δ = χ − χ=180 . We chose ω = 6 and averaged energy density ¯ = −0.2 as an illustration. For other choice of parameters, we observe similar trends. time thermalization dynamics. By working at large system size (L = 100), we independently characterize the heating and the diffusive properties of the dynamics of local energy density in our model. Using the half-system drive, we combine these two effects and describe the resulting late-time dynamics with a simple hydrodynamical model, Eq. (3.16). 88 Chapter 4 HYDRODYNAMICS IN A DISORDERED LOW-FREQUENCY FLOQUET SYSTEM In Ch. 3, we considered the dynamics of a clean Floquet system. Since the work of Deutsch [54] and Srednicki [200], we have believed generic isolated quantum systems to satisfy the eigenstate thermalization hypothesis. We expect that for times greater than some local thermalization time, systems satisfying that hypothesis will be well-described by a hydrodynamical model. That is, that local observables will be given by a Gibbs state with spatially varying Lagrange multipliers for the model’s conserved density (temperature, chemical potential, etc.). Those Lagrange multipliers in turn evolve by a partial differential equation, generally (at lowest order) a diffusion equation. The key result of Ch. 3 was that one could extend this phenomenological description to include Floquet heating—as long as one was careful to draw the diffusion coefficient from the right (effective prethermal) Hamiltonian. But Basko, Aleiner, and Altschuler [19] (cf [159]) introduced a broad class of disordered systems—so-called many body localized systems—that do not satisfy the eigenstate thermalization hypothesis, and are not well-described by a diffusive hydrodynamical theory with any small number of local conserved quantities. Instead, these MBL systems have local conserved quantites: their Hamiltonians commute with local operators (l-bits or local integrals of motion [92, 193]). Consequently, the system retains a local memory of its initial state even at long times. Many disordered models display a transition: they satisfy the eigenstate thermalization at small disorder, but are localized at large disorder. The fruit-fly example is the random-field Heisenberg model [146, 160], but other examples— including the the second-neighbor hopping model of [159], a variant of which we study here—are also known to display both phases. A number of phenomenological renormalization group schemes exist to describe this transition[72, 172, 220]; all predict that on the thermalizing side of the transition, the system will display subdiffusive hydrodynamics due to rare insulating regions. But it is still not clear how far the Griffiths region extends. Do systems with MBL transitions display subdiffusion all the way to the clean limit? [108, 147] 89 find that it does; [144, 189, 246] find that it does not. We find one such model that does not; in fact, it displays a large diffusive regime. We take a robustly non-integrable model (a Floquet variant of the second-neighbor hopping model studied by Huse and Oganesyan [159]) and use DMT to probe its hydrodynamics. We find that this model displays a large diffusive region. In this region, we measure the diffusion constant and a characteristic ultraviolet length scale for the model, which we identify as a coherence length; curiously, this coherence length does not change with disorder. 4.1 Model We use a Floquet variant of the model of [159]. When written with fermion operators, that model is Xh wi ni + V (ni − 1/2)(ni+1 − 1/2) H= i i + (c†i ci+1 + h.c.) + (c†i ci+2 + h.c.) (4.1) . We, like Huse and Oganesyan, take V = 2. Upon Jordan-Wigner transformation the model becomes H= Xh i i ~ ·S ~ + [S + S z S − + h.c.]) + wi S z . 2(S i+1 i+2 i i (4.2) In each case the disorder fields wi are normally distributed about 0 with width W . The continuous-time model satisfies the ETH at W = 0, while Huse and Oganesyan find an ETH-MBL transition around W = 6. The model (4.2) has two local conserved quantities, namely spin (particle number) and energy density. The coupling between the two conserved quantities leads to important nonlinear corrections to the hydrodynamics [156, 202]. We avoid this complication by breaking energy conservation: we time evolve by a series of unitaries U (t) = (Ueven Uodd )t , t ∈ Z (4.3) and stroboscopically sample. Spatially random Floquet models have been used as generic toy models for quantum thermalization[33, 111, 219] for similar reasons: they provide minimally-structured models. The precise form we choose for Ueven,odd is motivated by computational convenience: essentially, we group nearest-neighbor sites and then perform a first-order even-odd Trotter decomposition of the continuous-time evolution 90 U (t) = e−iHt . More precisely, bL/4c Ueven = Y exp[−i δt h2j ] j=1 (4.4) bL/4c Uodd = Y exp[−i δt h2j+1 ] , j=1 where L is the system size and 1~ 1 (1) z z ~ hj = S 2j−1 · S2j + [W2j−1 S2j−1 + W2j S2j ] 2 2 1~ (2) + z − ~ hj = S 2j · S2j + [S2j−1 S2j S2j+1 + h.c.] 2 + z − + [S2j S2j+1 S2j+2 + h.c.] (1) (1) (4.5) (2) hj = α2j−1 hj + α2j hj+1 + hj with αl chosen to take care of the left and right boundaries:   1 l = 1 or l = L − 1 αl = 2 1 else . (4.6) δt is a parameter. We wish to choose δt large enough that a Magnus expansion far from convergence. If we choose δt too small then we may see an approximately local conserved Floqet energy density, which could lead to the nonlinear hydrodynamics we wish to avoid. The Floquet model has an MBL transition around disorder width W = 20 (see Fig. 4.1). 4.2 Method We use DMT, as described in Ch. 2, to simulate the time evolution of a random mixed initial state by the Floquet unitaries (4.3) . We take an initial state hX i 1 z ρ = exp µj S j Z j (4.7) with µj random in [−1, 1]. We choose this initial state because it is not far from the identity—which makes us more confident of DMT’s efficacy—while containing variability at all wavelengths. Gap ratio r 91 0.525 0.500 0.475 0.450 0.425 0.400 0 L=8 L = 10 L = 12 10 20 Disorder width W 30 Figure 4.1: Gap ratio for the Floquet model of Sec. 4.1 as a function of disorder width at half-filling. Error bars show standard deviation of per-realization average gap ratio divided by square root of number of realizations. Since the model in question (4.2) is three-local, we group each adjacent pair of physical sites (local onsite dimension 2) into a single effective site (local onsite dimension 4). The method then guarantees that no truncation will change any operator on six contiguous physical sites. This is a strong guarantee. In particular, truncation does not change hSjz i, h d5 d z Sj i, . . . , h 5 Sjz i . dt dt (4.8) Consider, then, DMT at some small, fixed bond dimension. Thanks to this guarantee, we expect that DMT should approximate the model’s dynamics well in the hydrodynamical regime, where higher derivatives are very small— even if it requires much higher bond dimension to faithfully approximate shortto intermediate-time dynamics. 4.2.1 Convergence testing One can crudely estimate truncation error by simulating the same system at two different bond dimensions (say χ0 and 2χ0 ) and comparing the expectation values of the observables at different bond dimensions. But as the Sjz (t) approach their (small) thermal values, this becomes less useful: we want to normalize the discrepancy by the site-to-site variation in one Sjz (t) or the other. (It is not sufficient to normalize by a typical value of Sjz (t), as this will just be given by the thermal value. Since we seek to measure deviations from this thermal value, we desire that our error be small compared to those deviations. Nor can we normalize by individual Sjz (t): as these cross zero, they will produce spurious divergences in the error.) We therefore define the relative 92 W = 0.0 W = 2.0 W = 4.0 W = 6.0 W = 8.0 W = 10.0 W = 12.0 z E32, 128(t) 0.20 0.15 0.10 0.05 0.00 0 100 200 300 400 500 t (gate applications) Figure 4.2: Relative convergence error (4.11) at a variety of disorder strengths, for system size L = 64. (We label only even disorder strengths, for compactness.) The thick line indicates W = 8.0, the largest disorder strength for which we judge the simulation to be converged. convergence error between bond two bond dimensions χ1 , χ2 Eχ1 ,χ2 [Sjz (t)] ≡ Sjz (t, χ = χ1 ) − Sjz (t, χ = χ2 )   . std Sjz (t, χ = χ2 ) (4.9) where we normalize by the standard deviation   std Sjz (t, χ = χ2 ) h1 X ≡ S z (t, χ = χ2 )2 L j j 2 i1/2 1X z Sj (t, χ = χ2 ) , − L j (4.10) which is a proxy for deviation from the thermal value and one quantity of physical interest. (We will consider the physics of this quantity in detail below.) In order to have a single measure, we take the RMS: Eχz 1 ,χ2 (t) ≡ X Eχ1 ,χ2 [Sjz (t)]2 1/2 . (4.11) We also call this the relative convergence error. The error increases at a rate roughly independent of disorder strength W until a peak number of gate applications tworst , after which the error decreases. tworst . It is tempting to interpret tworst as a local-thermalization time and the decay 93 of error after tworst as confirmation of our intuition that our method captures hydrodynamics accurately (even if it fails to capture short- or intermediatetime dynamics). We leave to future work the careful diagnostics required to be confident in this interpretation. z (t) . 0.1; we judge the method to be trustWe see that for W ≤ 8.0, E32,128 worthy for these disorder strengths. At first sight these results seem to contradict the previous chapter. There we compared DMT simulations of a Floquet system at high frequencies to quasiexact Krylov space simulations and found that DMT agrees very well with the Krylov space results at long times, but fails at short times (t ∼ 10/J, J a coupling). We, on the other hand, find that DMT converges well (at least for clean systems) at all times. The apparent discrepancy is due to the fact that the model of Ch. 3 is close to free-fermion integrability. Truncation with DMT is liable to change the nearly conserved free fermion occupation numbers; since these occupation numbers are important to the system’s short-time dynamics (e.g. the revivals clearly visible in Fig. 3.2). This chapter’s model, on the other hand, is far from any kind of integrability. 4.2.2 Comparison with exact simulations But in light of the work of Kloss et al. [119], it is not enough to check convergence. They applied TDVP as in [133] and found that it could display false convergence: that as one increases the bond dimension, one sometimes sees the TDVP simulations converge quickly—but to a value that disagrees with exact results. In this section we consider DMT on small systems (L ≤ 28). Because the system size is small, we can compare DMT to exact simulations using MPSs of large bond dimensions (cf. Sec. ?? We consider two initial states: a product of random S z eigenstates (chosen so the system is at half-filling) and a product state of rotated spins: |ψ(0)i = L−1 O j=0 [cos(πj/L) |↑i + sin(πj/L) |↓i] . (4.12) (The latter is chosen to mimic the slowest mode of the continuous diffusion P equation. Additionally, because it is not an eigenstate of j Sjz , its dynamics 94 should better mimic the large-system limit.) In each case we compare DMT to MPS simulations with bond dimension 2L/2 —large enough that the MPS is never truncated (except to remove values below a numerical noise floor). Fig. 4.3 shows the hydrodynamics extracted from product of random S z eigenstates, while Fig. 4.4 shows that extracted from the product of superpositions (4.12). At first sight, Fig. 4.3 appears to show the false convergence that has bedeviled efforts to use TDVP to extract diffusion coefficients. Bond dimensions χ = 64 and χ = 128 give very nearly the same value—but this value is not the correct (exact) value. But this difference between diffusion coefficients extracted from DMT and those extracted from exact dynamics is of the same order as the difference between diffusion coefficients extracted from the exact dynamics at different system sizes—which suggests that the physics DMT is failing to capture more than finite size effects. Turning to the second initial state, Eq. (4.12) and Fig. 4.4, we see more reasonable convergence properties—χ = 64, 128 differ by an amount on the order of 1%—and still different “exact” values. We believe that the discrepancies are a combination of finite-size effects and imprecisions in our protocol. 95 100 10−2 10−3 10−4 0 50 100 150 t (unitary applications)) 0.50 MPS exact DMT χ = 32 DMT χ = 64 DMT χ = 128 0.2 0.1 0.45 0.0 0.40 200 (DDMT,χ − DMPS)/DMPS Diffusion constant D P(t) 10−1 0.3 exact DMT, χ = 128 22 24 26 System size L 28 22 24 26 System size L 28 Diffusion constant D 100 P(t) 10−1 10−2 10 −3 10 −4 0 exact DMT, χ = 128 50 100 150 t (unitary applications)) 200 (DDMT,χ − DMPS)/DMPS Figure 4.3: Hydrodynamics starting from an S z product state randomly chosen at half-filling. Left: decay of the distance from equilibrium (3.33) for L = 26. Center: diffusion coefficient, as calculated exactly (with large bond dimension matrix product states) and with DMT at three bond dimensions. Right: error in the DMT diffusion coefficients. 0.500 MPS exact DMT χ = 32 DMT χ = 64 DMT χ = 128 0.15 0.475 0.10 0.450 0.425 0.05 22 24 System size L 26 22 24 System size L 26 Figure 4.4: Hydrodynamics starting from the initial state (4.12). Left: decay of the distance from equilibrium (3.33) for L = 26. Center: diffusion coefficient, as calculated exactly (with large bond dimension matrix product states) and with DMT at three bond dimensions. Right: error in the DMT diffusion coefficients. 4.2.3 Quantity of interest We directly measure sj (t) ≡ hSjz (t)i, but the time traces of onsite spins are not enlightening. Instead, we define a distance from equilibrium P≡ X j (sj − s̄)2 , (4.13) where s̄ is the average spin s̄ = 1X 1X sj (t) = sj (0) . L j L j (4.14) Writing the sj as a vector s, that distance from equilibrium is P = (s − s̄)T (s − s̄) . (4.15) 96 P is the total weight on all of the system’s decay modes: if u(α) is a mode with decay rate γ(α) , i.e. Au(α) = γ(α) u(α) 6= 0 , (4.16) then P≡ 4.2.4 X (sT u(α) )2 . (4.17) α Disorder averaging One can legitimately disorder-average any measurable quantity. The disorder average then characterizes the distribution of that quantity. This is not the only reasonable goal, however: one may wish to use the disorder average not to characterize the distribution at a given length, but to mimic a longer systems than one simulates, so the limit of many disorder reaizations becomes the thermodynamic limit. Using a disorder average in this way, to probe large system sizes, requires some kind of phenomenological renormalization group scheme. One must disorder average in a way that mimics this scheme. In our case, if we imagine connecting two disorder realizations with uniform resistances ρ1 , ρ2 , the resulting segment has resistivity ρ12 = 21 (ρ1 + ρ2 ). Since the diffusion coefficient is D ∝ ρ−1 , this means that we should average diffusion coefficients by D12 = [D1−1 + D2−1 ]−1 . Consider now the distance from equilibrium P in two regimes, divided by the Thouless time tTh ∼ L2 /D: 1. 1/J  t  tTh ∼ L2 /D, when P ∼ √1Dt . In this case P ∝ D−1/2 ∝ so we should disorder average by P̄ = h(P (r) )2 i1/2 . 2 √ ρ, (4.18) 2 2. tTh  t, when P ∼ e−(π D/2L )t . In this case P ∼ e−t/ρ , so we should disorder average h i
(r) −1 −1 P = exp h ln P i . (4.19) In each case we denote by angle brackets a simple disorder average hAi = PNreals (r) 1 r=1 A . Most of our results derive from the longest time behavior— Nreals that is, the decay of the slowest hydrodynamic mode. We therefore use the disorder averaging protocol 2. 97 W = 3.0, protocol 4 100 101 L=16 L=32 L=64 L=128 L=256 L=512 100 10−1 −4 10 −4 10−6 10−6 0 250 500 t 750 1000 10−8 10−4 10−3 P̄4 10 L=16 L=32 L=64 L=128 L=256 L=512 √ ∝ 1/ t 10−2 P̄4 P (r) 10−2 (a) 10−5 10−7 10−2 t/L2 100 10−9 0 1 2 3 t/L2 Figure 4.5: Scaling collapse for the distance from equilibrium P(t) for 30 disorder realizations at L = 64, W = 3.0. Left: P (r) for each of the 30 realizations. Middle, right: disorder average per the protocol 2. The kink in the L = 512 curve is an artifact of that disorder averaging protocol. The dashed black line shows P1 ∝ √1t . Error bars show 5th and 95th percentiles in a non-parametric bootstrap scheme; they do not include the simulation error E32,128 . 10%. 4.3 Results We plot the distance from equilibrium P(t) for disorder width W = 3.0 in Fig. 4.5. There we see not only intermediate-time P4 ∼ t−0.5 scaling, but also a convincing finite-size scaling collapse t̃ = t/L2 , which is strongly indicative of diffusive behavior. This collapse captures not only the long-time limit but also the crossover from the intermediate-time power law. The scaling collapse does not work for L = 16. This is unsurprising. The scaling collapse should only work for system sizes L  lUV , where lUV is the UV cutoff of the diffusive hydrodynamics; because our Trotter gates are supported on four sites, one would expect the diffusive, hydrodynamical model to have a UV cutoff lUV & 4. In particular, the outermost four sites (two on the left and two on the right) are only affected by every other layer of gates; when L = 16, this is a significant portion of the system. We can extract effective whole-system diffusion coefficients by considering the long-time behavior of the P (r) . In the long-time limit lim P (r) ∝ e−2γ (r) t t→∞ (4.20) or d (r) P ∼ −2γ (r) t , (4.21) t→∞ dt where γ is the decay rate of the slowest eigenmode of the system’s hydrodynamics. We then identify the diffusion constant lim D(r) ≡ L2 γ (r) /2π 2 (4.22) 98 on the assumption that the slowest mode has characteristic wavevector π/L, even if it is not strictly the cosine one would find in the continuous diffusion equation with closed (homogeneous Neumann) boundary conditions and homogeneous diffusion coefficient. We then disorder-average1 D= "N reals X (r) −1 (D ) r=1 #−1 (4.23) Diffusion constant D 0.6 L = 16 L = 32 L = 64 L = 128 0.5 0.4 0.3 0.2 0.1 0 2 4 6 Disorder width W 8 Figure 4.6: Diffusion coefficients extracted from the distance from equilibrium (4.13). We perform a least-squares fit of each disorder realization’s distance from equilibrium log(P (r) (t)) vs t for 924 ≤ t ≤ 1024 for L = 64, 128 (or 150 ≤ t ≤ 250 for L = 16, 32), and then disorder average per Eq. (4.23). The difference in times is because for L = 16, 32, by t = 900 the system is close enough to equilibrium that we do not believe we can reliably see deviations from that equilibrium. Error bars show 5th and 95th percentiles in a nonparametric bootstrap scheme. Fig. 4.6 shows the disorder-averaged diffusion coefficients extracted from P4 at different disorder widths W . L = 16 shows slower diffusion than L = 32, 64, 128, but as noted above it does not properly reflect the long-system hydrodynamics. L = 128 shows somewhat faster diffusion at disorder widths W & 3.0; this is because for L & 128, W & 3.0 the higher decay modes have not entirely decayed away by the end of our simulation time, so the diffusion coefficient we extract is polluted by that faster decay. 1 Note that in the language of Le Doussal and Machta [129] this is the so-called “annealed” diffusion coefficient, corresponding to an immediate spatial average, as opposed to the “quenched” diffusion coefficient. Some care must be taken in comparing our results here to those like [118] that probe different length scales at different times. 99 100 Distance from equilibrium P W=0.0 W=1.0 W=2.0 W=3.0 W=4.0 W=5.0 W=6.0 W=7.0 W=8.0 10−2 10−4 10−6 10−1 100 101 102 Rescaled time D(W )t Figure 4.7: Another scaling collapse for the distance from equilibrium P(t) for 30 disorder realizations at L = 64. Error bars show 5th and 95th percentiles in a non-parametric bootstrap scheme. These results, while suggestive, are not sufficient to demonstrate that our data shows diffusiive behavior at all disorder widths 0 ≤ W ≤ 8.2 In Fig. 4.7 we plot the power P4 (t) against the rescaled time Dt for L = 64. All disorder widths 0 ≤ W ≤ 8 show a good scaling collapse on the shape of figure 4.6. Since the diffusion constant is a free parameter in the scaling and we determine the diffusion coefficient from exactly and only the data shown in Fig. 4.7, there is a certain circularity in the analysis that is absent from the scaling collapse of Fig. 4.6; nonetheless, the consistency in the overall shape of the curve demonstrates that the system is in fact diffusive for 0 ≤ W ≤ 8. 4.4 Discussion We have applied time evolution with DMT to a disordered low-frequency Floquet system. We first evaluated its performance in this case both by checking convergence in bond dimension and by comparing with exact simulations. We found that it gives moderately good results, with error of order 5% in the largest exactly-simulable clean systems. But if one seeks greater precision than that, DMT may not be able to offer that precision. And, more worryingly, time evolution with DMT does not appear to be self-diagnosing with respect to errors of that order. We believe that the physics DMT fails to capture consists of finite size effects, perhaps peculiar to this model, but much work remains to be done to understand those effects and convincingly argue 2 As the joke runs: we have thus far seen that at least one side of one sheep in Scotland is black. 100 that DMT will converge to better than this 5% precision in long systems. Additionally, work remains to be done to reconcile the convergence tests of Sec. 4.2.1 with the comparison with exact results of Sec. 4.2.2. The comparison with exact results show that a bond dimension χ = 32 gives a diffusion coefficient that differs from that of χ = 128 by ∼ 20% at L = 28, but the convergence tests at L = 64 show that local observables differ between χ = 32 and χ = 128 by . 5% in the clean case. This difference may be due to the different measures (diffusion coefficient vs. local observables), but more optimistically it could be due to improved performance of DMT at longer system sizes. Additionally, it is possible that DMT fails to capture Griffiths regions—but we see apparent convergence because the systems we use to check convergence happen not to have rare, anomalously slow regions. We found that—contra expectations based on [108, 147]—it displays a large regime where it is diffusive, not subdiffusive. Part II MBL-mobile: Many-body-localized engine 101 102 Chapter 5 MBL AND THERMODYNAMIC ENGINES In Ch. 4 we alluded to many body localization, but we did not treat a localized system. At most, many body localization could have caused regions of anomalously large resistance, leading to subdiffusion. But a tantalizing question is whether the unique properties of MBL could be put to use. So far, MBL was proposed to be used for robust quantum memories [158]. We believe, however, that the potential of MBL is much greater. MBL systems behave athermally, and athermality (lack of thermal equilibrium) facilitates thermodynamic tasks [8, 26, 43, 50, 55, 69, 75, 83, 91, 98, 229, 236]. When a cold bath is put in contact with a hot environment, for instance, work can be extracted from the heat flow. Could MBL’s athermality have thermodynamic applications? In this part we present such an application. We formulate, analyze, and numerically simulate an Otto engine cycle for a quantum many-body system that has an MBL phase. The engine contacts a hot bath and a narrow-band cold bath, as sketched in Fig. 5.1. This application unites the growing fields of quantum thermal machines [22, 28, 51, 65, 66, 99, 123, 130, 164, 199, 205, 232] and MBL [19, 92, 158–160, 191]. Our proposal could conceivably be explored in ultracold-atom [24, 37, 120, 148, 188], nitrogen-vacancy-center [125], trappedion [198], and possibly doped-semiconductor [124] experiments. Our engine relies on two properties that distinguish MBL from thermal systems: its spectral correlations [191, 196] and its localization. The spectralcorrelation properties enable us to build a mesoscale level-statistics engine. The localization enables us to link mesoscale engines together and create a large engine with extensive work output. Take an interacting finite spin chain as an example. Consider the statistics of the gaps between consecutive energy eigenvalues far from the energy band’s edges. A gap distribution P (δ) encodes the probability that any given gap has size δ. The MBL gap distribution enables small (and large) gaps to appear much more often than in ETH spectra [44]. This difference enables MBL to enhance our quantum many-body Otto cycle. Let us introduce the MBL and ETH distributions in greater detail. Let hδiE 103 Wb Figure 5.1: Schematic of MBL engine: We formulate an Otto engine cycle for a many-body quantum system that exhibits an MBL phase. We illustrate with a spin chain (green dots and black arrows). A random disorder potential (jagged red line) localizes the particles. Particles interact and hop between sites (horizontal red arrows). Consider strengthening the interactions and the hopping frequency. The system transitions from strong localization to a thermal phase or to weak localization. The engine thermalizes with a hot bath (flames) and with a cold bath (ice cube). The cold bath has a small bandwidth Wb , to take advantage of small energy gaps’ greater prevalence in the highly localized regime. denote the average gap at the energy E. MBL gaps approximately obey Poisson statistics [44, 159]: (E) PMBL (δ) ≈ 1 −δ/hδiE e . hδiE (5.1) (E) Any given gap has a decent chance of being small: As δ → 0, PMBL (δ) → hδi1 > E 0. Neighboring energies have finite probabilities of lying close together: MBL systems’ energies do not repel each other, unlike thermal systems’ energies. Thermalizing systems governed by real Hamiltonians obey the level statistics of random matrices drawn from the Gaussian orthogonal ensemble (GOE) [159]: (E) PGOE (δ) ≈ π δ − π4 δ 2 /hδi2E . 2 e 2 hδiE (5.2) (E) Unlike in MBL spectra, small gaps rarely appear: As δ → 0, PGOE (δ) → 0. MBL’s athermal gap statistics should be construed as a thermodynamic resource, we find, as athermal quantum states are [8, 26, 43, 50, 55, 69, 75, 83, 91, 98, 229, 236]. In particular, MBL’s athermal gap statistics improve our engine’s reliability: The amount Wtot of work extracted by our engine fluctuates relatively little from successful trial to successful trial. Athermal statistics 104 also lower the probability of worst-case trials, in which the engine outputs net negative work, Wtot < 0. Furthermore, MBL’s localization enables the engine to scale robustly: Mesoscale “subengines” can run in parallel without disturbing each other much, due to the localization inherent in MBL. Even in the thermodynamic limit, an MBL system behaves like an ensemble of finite, mesoscale quantum systems, due to its local level correlations [96, 196, 201]. Any local operator can probe only a discrete set of sharp energy levels, which emerge from its direct environment. This part is organized as follows. Ch. 6 describes the MBL-mobile in the adiabatic limit—we take the tuning to be infinitely slow. In Sec. 6.1, we introduce the basic idea with a qubit (two-level quantum system). In Sec. 6.2, we scale the engine up to a mesoscopic chain tuned between MBL and ETH. In Sec. 6.5, we calculate its work output and efficiency. We check these calculations numerically in Sec. 6.7 with numerical simulations of disordered spin chains. In Sec. 6.6, we argue that the mesoscopic segments can be combined into a macroscopic MBL system while operating in parallel. Ch. 7 describes the MBL-mobile at finite speed. We discuss limitations on the speed at which the engine can be run (and consequently the engine’s power). This leads us to a more careful consideration of diabatic corrections to the work output, communication amongst subengines, and the cold bath’s nature. We again test our analytic calculations in Sec. 7.2, In Ch. 8 we provide order-of-magnitude estimates for a localized semiconductor engine’s power and power density, and we compare our engine to some competitor engines. Background information, intuitive examples, and extensive calculations appear in [238]. 105 Chapter 6 THE MBL-MOBILE IN ITS ADIABATIC LIMIT We aim to formulate an MBL engine cycle for the thermodynamic limit. Our road to that goal runs through a finite-size, or mesoscale, MBL engine. In Sec. 6.1, we introduce the intuition behind the mesoscale engine via a qubit toy model. Then, we describe (Sec. 6.2) and quantitatively analyze (Sec. 6.5) the mesoscale MBL engine. Table 6.1 offers a spotter’s guide to notation. 6.1 Qubit toy model for the mesoscale engine At the MBL Otto engine’s core lies a qubit Otto engine whose energy eigenbasis transforms during the cycle [31, 112, 121, 122]. Consider a two-level system evolving under the time-varying Hamiltonian Hqubit (t) := (1 − αt )hσ x + αt h0 σ z . (6.1) σ x and σ z denote the Pauli x- and z-operators. αt denotes a parameter tuned between 0 and 1. Figure 6.1 illustrates the cycle. The engine begins in thermal equilibrium at a high temperature TH . During stroke 1, the engine is thermally isolated, and αt is tuned from 0 to 1. During stroke 2, the engine thermalizes to a temperature TC  TH . During stroke 3, the engine is thermally isolated, and αt returns from 1 to 0. During stroke 4, the engine resets by thermalizing with the hot bath. Let us make two simplifying assumptions (see [238, App. C] for a generalization): First, let TH = ∞ and TC = 0. Second, assume that the engine is tuned slowly enough to satisfy the quantum adiabatic theorem. We also choose1 h= δGOE , 2 h0 = δMBL , 2 and δGOE  δMBL . 1 The gaps’ labels are suggestive: A qubit, having only one gap, obeys neither GOE nor MBL gap statistics. But, when large, the qubit gap apes a typical GOE gap; and, when small, the qubit gap apes a useful MBL gap. This mimicry illustrates how the mesoscopic engine benefits from the greater prevalence of small gaps in MBL spectra than in GOE spectra. 106 Eigenenergies (Et(j)) (3) Et Wb 0 ⌧ 000 (2) Et ⌧ MBL GOE ⌧ 0 displ (1) Et ⌧ 00 0 1 Hamiltoniantuning parameter (αt) Figure 6.1: Qubit toy model for the MBL Otto cycle: A qubit models two “working levels” in the MBL Otto engine’s many-body spectrum. The energy (1) (2) (2) (1) eigenstates |Et i and |Et i span the “working subspace.” The gap Et −Et begins at size δGOE during a successful trial. The gap shrinks to δMBL , then returns to δGOE . In addition to changing the gap, each Hamiltonian tuning changes the eigenstates’ functional forms. The displacement δdispl is included for generality. The blue text marks the times t = 0, τ, . . . , τ 000 at which the strokes begin and end during a work-outputting trial. The spectator level (3) |Et i fails to impact the engine’s efficiency. The cold bath has too narrow a (3) bandwidth Wb to couple |Et i to any other level. If the engine begins any trial on the top green line, the engine remains on that line throughout the trial. Zero net work is outputted. Let us analyze the cycle’s energetics. The system begins with hHqubit (t)i = 0. . Stroke 1 preserves the T = ∞ state I/2. Stroke 2 drops the energy to − δMBL 2 δGOE The energy drops to − 2 during stroke 3. During stroke 4, the engine resets to zero average energy, absorbing heat hQ4 i = δGOE , on average. 2 The energy exchanged during the tunings (strokes 1 and 3) constitutes work [Eq. (1.137)], while the energy exchanged during the thermalizations (strokes 2 and 4) is heat [Eq. (1.138)]. The engine outputs the per-cycle power, or average work performed per cycle, hWtot i = 21 (δGOE − δMBL ). tot i MBL The efficiency is ηqubit = hW = 1 − δδGOE . This result is equivalent to the hQ4 i 107 Energy (Et) • Stroke 1 • Isentrope • W1 > 0 • Stroke 4 • Hot isochore • Q4 > 0 • Stroke 2 • Cold isochore • Heat -Q2 > 0 leaves the engine. • Stroke 3 • Isentrope • W3 > 0 ETH (“thermal”) MBL Hamiltonian [Hmeso(t)] Figure 6.2: Otto engine cycle for a mesoscale MBL system: Two energies in the many-body spectrum capture the cycle’s basic physics. The engine can be regarded as beginning each trial in an energy eigenstate drawn from a Gibbs distribution. The red dot represents the engine’s starting state in some trial of interest. During stroke 1, Hmeso (t) is tuned from “thermal” to MBL. During stroke 2, the engine thermalizes with a cold bath. Hmeso (t) returns from MBL to thermal during stroke 3. Stroke 4 resets the engine, which thermalizes with a hot bath. The tunings (strokes 1 and 3) map onto the thermodynamic Otto cycle’s isentropes. The thermalizations (strokes 2 and 4) map onto isochores. The engine outputs work W1 and W3 during the tunings and absorbs heat Q2 and Q4 during thermalizations. MBL gap statistics’ lack of level repulsion enhances the cycle: The engine “slides down” the lines that represent tunings, losing energy outputted as work. efficiency ηOtto of a thermodynamic Otto engine [Eq. (1.135)]. The gap ratio δMBL 1 . ηqubit equals also ηQHO [Eq. (1.136)] if the frequency plays the role of rγ−1 δGOE ratio ω/Ω is chosen to equal δMBL /δGOE . As argued in Sec. 6.6 (and 7.3), however, the qubit engine can scale to a large composite engine of densely packed qubit subengines operating in parallel. The dense packing is possible if the qubits are encoded in the MBL system’s localized degrees of freedom (l-bits, roughly speaking [93]). 6.2 Set-up for the mesoscale MBL engine The next step is an interacting finite-size system tuned between MBL and ETH phases. Envision a mesoscale engine as a one-dimensional (1D) system of N ≈ 108 Symbol N N E αt hδi Wb βH = 1/TH βC = 1/TC δ− v ξ> ξ< Xmacro g τcycle hδi(L) Significance Number of sites per mesoscale engine or per mesoscale subengine (in the macroscopic engine). Chosen, in the latter case, to equal ξ> . Dimensionality of one mesoscale (sub)engine’s Hilbert space. Unit of energy, average energy density per site. Hamiltonian parameter tuned from 0 (in the mesoscale engine’s ETH regime, or the macroscopic engine’s shallowly localized regime) to 1 (in the engine’s deeply MBL regime). Average gap in the energy spectrum of a length-N MBL system. Bandwidth of the cold bath. Small: Wb  hδi. Inverse temperature of the hot bath. Inverse temperature of the cold bath. Level-repulsion scale of a length-N MBL system. Minimal size reasonably attributable to any energy gap. Smallest gap size at which a Poissonian (5.1) approximates the MBL gap distribution well. Speed at which the Hamiltonian is tuned: v := E dαt t . Has dimensions of 1/time2 , in accordance with part of [46]. Localization length of macroscopic MBL engine when shallowly localized. Length of mesoscale subengine. Localization length of macroscopic MBL engine when deeply localized. Satisfies ξ< < ξ> . Characteristic X of the macroscopic MBL engine (e.g., X = N, hδi). Strength of coupling between engine and cold bath. Time required to implement one cycle. Average energy gap of a length-L MBL system. Table 6.1: Parameters of the mesoscopic and macroscopic MBL engines: Boltzmann’s and Planck’s constants are set to one: kB = ~ = 1. 10 sites. This engine will ultimately model one region in a thermodynamically large MBL engine. We will analyze the mesoscopic engine’s per-trial power hWtot i, the efficiency ηMBL , and work costs hWdiab i of undesirable diabatic transitions. 109 The mesoscopic engine evolves under the Hamiltonian Hmeso (t) := E [(1 − αt )HGOE + αt HMBL ] . Q(αt ) (6.2) The unit of energy, or average energy density per site, is denoted by E. The tuning parameter αt ∈ [0, 1]. When αt = 0, the system evolves under a (E) random Hamiltonian HGOE whose gaps δ are distributed according to PGOE (δ) [Eq. (5.2)]. When αt = 1, Hmeso (t) = HMBL , a Hamiltonian whose gaps are (E) distributed according to PMBL (δ) [Eq. (5.1)]. For a concrete example, take a random-field Heisenberg model in which the disorder strength is tuned. HGOE and HMBL have the same bond term, but the disorder strength varies in time. We simulate (a rescaled version of) this model in Sec. 6.7. The mesoscale engine’s cycle is analogous to the qubit cycle, including initialization at αt = 0, tuning of αt to one, thermalization with a temperatureTC bath, tuning of αt to zero, and thermalization [47, 63, 94, 132] with a temperature-TH bath. To highlight the role of level statistics in the cycle, we hold the average energy gap hδi constant.2 We do so using the renormalization factor Q(αt ).3 Section 6.7 details how we define Q(αt ) in numerical simulations. The key distinction between GOE level statistics (5.2) and Poisson (MBL) statistics (5.1) is that small gaps (and large gaps) appear more often in Poisson spectra. A toy model illuminates these level statistics’ physical origin: An MBL system can be modeled as a set of noninteracting quasilocal qubits [93]. Let gj denote the jth qubit’s gap. Two qubits, j and j 0 , may have nearly equal gaps: gj ≈ gj 0 . The difference |gj − gj 0 | equals a gap in the manybody energy spectrum. Tuning the Hamiltonian from MBL to ETH couples 2 hδi is defined as follows. The density of states at the energy E has the form µ(E) ≈ 2 2 (see Table 6.1 for the symbols’ meanings). Inverting µ(E) yields the local 1 average gap: hδiE := µ(E) . Inverting the average of µ(E) yields the average gap, √ N e−E /2N E 2πN E hδi := 3 √ 1 N 2 πN = R∞ = E. hµ(E)ienergies N dE µ2 (E) −∞ (6.3) Imagine removing Q(αt ) from Eq. (6.2). One could increase αt —could tune the Hamiltonian from ETH to MBL [191]—by strengthening a disorder potential. This strengthening would expand the energy band; tuning oppositely would compress the band. By expanding and compressing, in accordion fashion, and thermalizing, one could extract work. This engine would benefit little from properties of MBL, whose thermodynamic benefits we wish to highlight. So we “zero out” the accordion motion by fixing hδi through Q(αt ). For a brief discussion of the accordion-like engine, see Sec. 8.1.1. 110 the qubits together, producing matrix elements between the nearly degenerate states. These matrix elements force energies apart. To take advantage of the phases’ distinct level statistics, we use a cold bath that has a small bandwidth Wb . According to Sec. 6.1, net positive work is extracted from the qubit engine because δMBL < δGOE . The mesoscale analog of δGOE is ∼ hδi, the typical gap ascended during hot thermalization. The engine must not emit energy on this scale during cold thermalization. Limiting Wb ensures that cold thermalization relaxes the engine only across gaps δ ≤ Wb  hδi. Such anomalously small gaps appear more often in MBL energy spectra than in ETH spectra [71, 107, 161]. This level-statistics argument holds only within superselection sectors. Suppose, for example, that Hmeso (t) conserves particle number. The level-statistics arguments apply only if the particle number remains constant throughout the cycle [238, App. F]. Our numerical simulations (Sec. 6.7) take place at halffilling, in a subspace of dimensionality N of the order of magnitude of the N whole space’s dimensionality: N ∼ √2 N . We are now ready to begin analyzing the mesoscopic-engine Otto cycle. The
engine begins in the thermal state ρ(0) = e−βH HGOE /Z, wherein Z := Tr e−βH HGOE . The engine can be regarded as starting each trial in some energy eigenstate j drawn according to the Gibbs distribution (Fig. 6.2). During stroke 1, Hmeso (t) is tuned from HGOE to HMBL . We approximate the tuning as quantumadiabatic (diabatic corrections are left to Sec. 7.1). Stroke 2, cold thermalization, depends on the gap δj0 between the jth and (j − 1)th MBL levels. δj0 typically exceeds Wb . If it does, cold thermalization preserves the engine’s b energy, and the cycle outputs Wtot = 0. With probability ∼ W , the gap is hδi 0 small enough to thermalize: δj < Wb . In this case, cold thermalization drops the engine to level j −1. Stroke 3 brings the engine to level j −1 of HGOE . The gap δj between the (j − 1)th and jth HGOE levels is hδi  Wb , with the high probability ∼ 1 − (Wb / hδi)2 . Therefore the engine likely outputs Wtot > 0. Hot thermalization (stroke 4) returns the engine to ρ(0). 6.3 Notation and definitions: We focus on one mesoscopic engine S of N sites. The engine corresponds to a N Hilbert space of dimensionality N ∼ √2 N . The Hamiltonian, H(t) ≡ Hmeso (t), is tuned between HGOE , which obeys the ETH, and HMBL , which governs an 111 MBL system. ETH and MBL Hamiltonians typically have Gaussian DoSs4 µ(E) = √ N 2 2 e−E /(2N E ) , 2πN E (6.4) R∞ normalized to −∞ dE µ(E) = N . The unit of energy, or energy density per site, is E. We often extend energy integrals’ limits to ±∞, as the Gaussian peaks sharply about E = 0. N 1 = , and the average gap is hδi := R ∞ dE The local average gap is hδiE = µ(E) µ2 (E) −∞ √ 2 πN E [Eq. 6.3]. The average H gap, hδi, equals the average H GOE MBL gap, by N construction. hδi sets the scale for work and heat quantities. Hence we cast Q’s and W ’s as (number)(function of small parameters) hδi. The system begins the cycle in the state ρ(0) = e−βH HGOE /Z, wherein Z :=
Tr e−βH HGOE denotes the partition function. Wb denotes the cold bath’s bandwidth. We set ~ = kB = 1 . t H(t) is tuned at a speed v := E dα , wherein αt denotes the dimensionless dt tuning parameter. v has dimensions of energy2 , as in [195]. Though our v is not defined identically to the v in [195], we expect ours to behave similarly. 6.4 Quantitative analysis of the adiabatic mesoscale engine: some easy limits How well does the mesoscale Otto engine perform? We calculate average work hWtot i outputted per cycle and the efficiency ηMBL . We focus on the parameter regime in which the cold bath is very cold, the coldbath bandwidth Wb is very small, and the hot bath is very hot: TC  Wb  √ hδi, and N βH E  1. The mesoscale engine resembles a qubit engine whose state and gaps are averaged over. The gaps, δj and δj0 , obey the distributions (E) (E) PGOE (δj ) and PMBL (δj0 ) [Eqs. (5.2) and (5.1)]. Correlations between the HGOE and HMBL spectra can be neglected. In this section we make three simplifying assumptions: (i) The engine is as4 This is only approximately true, even in the thermodynamic limit. In the randomfield Heisenberg Hamiltonian (6.33) that we use for our numerical simulations, for example, the coupling term skews the the density of states: the mean of the DoS is tr H = 0, but the median is nonzero. Our numerics nonetheless match analytics performed assuming a Gaussian DoS (cf Sec. 6.7). This is because the Gaussian is sharply peaked around the mode: we can think of the skewness as an effective constant offset to the energy. Moreover, the DoS is essentially a parameter in our calculations: the details of some of our results are specific to this form for the DoS, but the physics does not depend on it. 112 sumed to be tuned quantum-adiabatically. (ii) The hot bath is at TH = ∞. Wb We neglect finite-temperature corrections, which scale as N (βH E)2 hδi 2 . (iii) (E) The gap distributions vary negligibly with energy: PGOE (δj ) ≈ PGOE (δj ), and (E) PMBL (δj0 ) ≈ PMBL (δj0 ), wherein hδiE ≈ hδi. We relax assumptions (ii) and (iii) in Sec. 6.4; relaxing assumption (i) must wait till Sec. 7.1. 6.4.0.1 Average work hWtot i per cycle: The key is whether the cold bath relaxes the engine downwards across the MBL-side gap δ 0 ≡ δj0 , distributed as PMBL (δ 0 ), during a given trial. If δ 0 < Wb , the engine has a probability 1/(1 + e−βC δ ) of thermalizing. So the overall probability of relaxation by the cold bath is ZWb 0 1 e−δ /hδi 0 1 (Wb − TC ln 2) , pcold ≈ dδ 0 ≈ −β δ hδi 1 + e C hδi (6.5) 0 0 wherein we neglected Wb / hδi by setting e−δ /hδi ≈ 1. Alternatively, the cold bath could excite the engine to a level a distance δ 0 above the initial level. Such an upward hop occurs with a probability ZWb 0 −δ 0 /hδi e−βC δ TC ln 2 0 e p̄cold ≈ dδ . 0 ≈ −β δ C hδi 1 + e hδi (6.6) 0 If the engine relaxed downward during stroke 2, then upon thermalizing with the hot bath during stroke 4, the engine gains heat hQi4 ≈ hδi, on average. If the engine thermalized upward during stroke 2, then the engine loses hδi during stroke 4, on average. Therefore, the cycle outputs average work hWtot i ≈ (pcold − p̄cold ) hδi + hQ2 i ≈ Wb − 2 ln 2 . βC (6.7) hQ2 i denotes the average heat absorbed by the engine during cold thermalization: ZWb 0 0 e−δ /hδi (Wb )2 0 δ hQ2 i ≈ − dδ ≈ − , (6.8) hδi 1 + e−βC δ0 2 hδi 0 113 which is  hQ4 i. This per-cycle power scales with the system size N as5 √ bandwidth Wb  hδi ∼ #effective ∼ E NN . energy eigenstates 6.4.0.2 Efficiency ηMBL : The efficiency is ηMBL = hWtot i hQ4 i + hQ2 i Wb = ≈1− . hQ4 i hQ4 i 2 hδi (6.9) Wb The imperfection is small, 2hδi  1, because the cold bath has a small bandwidth. This result mirrors the qubit-engine efficiency ηqubit .6 But our engine is a many-body system of N interacting sites. MBL will allow us to employ segments of the system as independent qubit-like subengines, despite interactions. In the absence of MBL, each subengine’s effective hδi = 0. With hδi vanishes the ability to extract hWtot i > 0. Whereas the efficiency is nearly perfect, an effective engine requires also optimized power. The MBL engine’s power will be limited by dynamical considerations, discussed in Ch. 7. 6.5 6.5.1 Quantitative analysis of the adiabatic mesoscale engine: a more detailed calculation Partial-swap model of thermalization Classical thermalization can be modeled with a probabilistic swap, or partial swap, or p-SWAP [186, 245]. Let a column vector ~v represent the state. The thermalization is broken into time steps. At each step, a doubly stochastic matrix Mp operates on ~v . The matrix’s fixed point is a Gibbs state ~g . Mp models a probabilistic swapping out of ~v for ~g : At each time step, the system’s state has a probability 1 − p of being preserved and a probability 5 The effective bandwidth is defined as follows. The many-body system has a Gaussian √ 2 2 N density of states: µ(E) ≈ √2πN e−E /2N E . The states within a standard deviation E N E of the mean have gaps obeying the Poisson or GOE level-spacing distributions, Eqs. (5.1) √ and (5.2). These states form the effective band, whose width scales as E N . 6 ηMBL is comparable also to ηQHO [Eq. (1.136)]. Imagine operating an ensemble of independent QHO engines. Let the jth QHO frequency be tuned between Ωj and ωj , distributed according to PGOE (Ωj ) and PMBL (ωj ). The average MBL-like gap ωj , conditioned RW RW on ωj ∈ [0, Wb ], is hωj i ∼ Wb1/hδi 0 b dωj ωj PMBL (ωj ) ≈ W1b 0 b dωj ωj = W2b . Averaging hωi Wb the efficiency over the QHO ensemble yields hηQHO i := 1 − hΩi ≈ 1 − 2hδi ≈ ηMBL . The mesoscale MBL engine operates at the ideal average efficiency of an ensemble of QHO engines. But MBL enables qubit-like engines to pack together densely in a large composite engine. 114 p ∈ [0, 1] of being replaced by ~g . This algorithm gives Mp the form Mp = (1 − p)I + p~g (1, 1). We illustrate with thermalization across  −β∆  two levels. Let 0 and ∆ label the e 1 levels, such that ~g = 1+e−β∆ , 1+e−β∆ : " −β∆ e 1 − p 1+e1−β∆ p 1+e −β∆ Mp = e−β∆ 1 1 − p 1+e p 1+e−β∆ −β∆ # . (6.10) The off-diagonal elements, or transition probabilities, obey detailed balance [40, 239]: PP (0→∆) = e−β∆ . (∆→0) Repeated application of Mp maps every state to ~g [239]: limn→∞ (Mp )n ~v = ~g . The parameter p reflects the system-bath-coupling strength. We choose p = 1: The system thermalizes completely at each time step. (If p 6= 1, a more sophisticated model may be needed for thermalization across > 2 levels.) 6.5.2 Average heat hQ2 i absorbed during stroke 2 Let j denote the HGOE level in which the engine begins the trial of interest. (j) We denote by Q2 the average heat absorbed during stroke 2, from the cold (j) bath. (Q2 will be negative and, provided that j is around the energy band’s center, independent of j.) The heat absorbed can be calculated easily from the following observation. Stroke 1 (adiabatic tuning) preserves the occupied level’s index. The level closest to j lies a distance δ away when stroke 3 begins. δ can have either sign, can lie above or below j. Heat is exchanged only if |δ| < Wb . Let us initially neglect the possibility that two nearby consecutive gaps are very small: |Ej±2 − Ej | ≤ Wb . We can write the average (over trials begun in level j) heat absorbed as (j) Q2 = ZWb −Wb dδ δ
e−βC δ (E) PMBL (δ) + O Wb3 / hδi2 . −β δ 1+e C (6.11) This equation assumes a Sommerfeld-expansion form, so (j) Q2 = −

Wb2 π2 µ(E) + µ(E)(TC )2 + O [Wb ]3 / hδi2 + O µ(E)2 [TC ]3 . 2 6 (6.12) 115 The first correction accounts for our not considering two levels within Wb of level j. Next, we need to average this result over all initial states j, assuming the initial density operator, ρ(0) = e−βH HGOE /Z: hQ2 i := D hQ2 (E)i cold E therm. gaps  (6.13) ρ(0) Z ∞  (Wb )2 π 2 1 e−βH E 2 + (6.14) = − dE µ (E) 2 6 (βC )2 Z −∞ (    3 !  3 ! ) µ(E) Wb Wb −βC Wb +O + hδi O +O e . (6.15) hδi hδi βC We substitute in for the DoS from Eq. (6.4): N2 1 hQ2 i = 2πN E 2 Z  Z ∞ (Wb )2 π 2 1 2 2 + dE e−E /N E e−βH E + O(.) , − 2 2 6 (βC ) −∞ (6.16) wherein the correction terms are abbreviated. The integral evaluates to The partition function is Z= Z ∞ −∞ 2 dE µ(E)e−βH E = N eN (βH E) /2 . √ 2 πN E eN (βH E) /4 . (6.17) Substituting into Eq. (6.16) yields   (Wb )2 π 2 1 2 hQ2 i = − + e−N (βH E) /4 2 2 hδi 6 (βC ) hδi (  3 !   Wb µ(E) −βC Wb + O [µ(E) Wb ] e + hδi O hδi βC )  3 ! h i4  √ µ(E) +O +O N βH E . (6.18) βC 1 The prefactor was replaced with hδi via Eq. (6.3) in the main text. Equation (6.18) is compared with numerical simulations in Fig. 6.3. In the appropriate regime (wherein Wb  hδi and TC  Wb ), the analytics agree well with the numerics, to within finite-size effects. 116 In terms of small dimensionless parameters, " 1 hQ2 i = hδi − 2  Wb hδi 2 π2 1 + 6 (βC hδi)2 #  N 2 1 − (βH E) + O(.) . 4 (6.19) 1 The leading-order term is second-order. So is the βC correction; but (βC hδi) 2   2 Wb , by assumption. The βH correction is fourth-order—too small to inhδi clude. To lowest order, hQ2 i ≈ − 6.5.3 (Wb )2 . 2 hδi (6.20) Average heat hQ4 i absorbed during stroke 4 The hQ4 i calculation proceeds similarly to the hQ2 i calculation. When calculating hQ2 i, however, we neglected contributions from the engine’s coldthermalizing down two small gaps. Two successive gaps have a probability   ∼ Wb hδi 2 of being < Wb each. Thermalizing across each gap produces heat ≤ h i  3 Wb Wb . Each such pair therefore contributes negligibly to hQ2 i, as hδi O . hδi We cannot neglect these pairs when calculating hQ4 i. Each typical small gap widens, during stroke 3, to size ∼ hδi . These larger gaps are thermalized across stroke 4, contributing at the nonnegligible second order, as hduring i2  Wb ∼ hδi O to hQ4 i . Chains of ≥ 3 small MBL gaps contribute neglihδi gibly. The calculation is tedious, appears in [238, App. G 5], and yields hQ4 i ≈ Wb − Wb 2 ln 2 (Wb )2 + + 4 ln 2 . βC 2 hδi βC hδi (6.21) The leading-order terms are explained heuristically below Eq. (6.7) in the main text. The leading-order βC correction, − 2βlnC2 , shows that a warm cold bath lowers the heat required to reset the engine. Suppose that the cold bath is maximally cold: TC = 0. Consider any trial that S begins just above a working gap (an ETH gap δ > Wb that narrows to an MBL gap δ 0 < Wb ). Cold thermalization drops S deterministically to the lower level. During stroke 4, S must absorb 117 1.4 0.0025 1.2 0.0020 hQ2i/hδi 1.0 hQ2i/hδi 0.0015 0.8 0.0010 0.6 0.4 0.0005 0.2 0.0 0.0 0.0000 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 TC/hδi 1.2 Wb/hδi (b) hQ2 i vs. TC at TH = ∞ and Wb = (a) hQ2 i vs. Wb at TC = 0 and TH = ∞ 2−4 hδi √ hQ2i/E N 0.00002 0.00001 0.00000 −0.00001 −0.00002 0.0 0.5 1.0 √ βH E N 1.5 2.0 (c) hQ2 i vs. βH at TC = 0 and Wb = 2−4 hδi Figure 6.3: Magnitude | hQ2 i | of the average heat absorbed during cold thermalization (stroke 2) as a function of the cold-bath bandwidth Wb (6.3a), the cold-bath temperature TC (6.3b), and the hot-bath temperature TH = 1/βH (6.3c): The blue lines represent the magnitude of the analytical prediction (6.18); in the bottom figure (showing βH dependence) we use the full analytical result from [238]. See Sec. 6.7 for other parameters and definitions. The analytics match the numerics’ shapes, and the agreement b is fairly close, in the appropriate limits (where W  1 and TC / hδi  1, in hδi the gray shaded regions). The analytics systematically underestimate hQ2 i at fixed Wb , due to the small level repulsion at finite N . The analytical prediction (6.18) substantially underestimates hQ2 i when the cold-bath bandwidth is large, Wb & hδi. Such disagreement is expected: The analytics rely on Wb 0  1, neglecting chains of small gaps: δj0 , δj+1 , · · · < Wb . Such chains hδi proliferate as Wb grows. A similar reason accounts for the curve’s crossing the origin in Fig. 6.3b: We analytically compute hQ2 i only to second order in TC / hδi. 118 Q4 > 0 to return to its start-of-trial state. Now, suppose that the cold bath is only cool: TC & 0. Cold thermalization might leave S in the upper level. S needs less heat, on average, to reset than if TC = 0. A finite TC therefore b detracts from hQ4 i. The +4 ln 2 βW offsets the detracting. However, the C hδi b  1. positive correction is smaller than the negative correction, as W hδi A similar argument concerns TH < ∞. But the βH correction is too small to 2 2 b) include in Eq. (6.21): hQ4 i ≈ Wb − 2βlnC2 + (W e−N (βH E) /4 . 2hδi Figure 6.4 shows Eq. (6.21), to lowest order in TC , as well as the βH dependence of hQ4 i. The analytical prediction is compared with numerical simulations. The agreement is close, up to finite-size effects, in the appropriate regime (TC  Wb  hδi). 6.5.4 Average per-cycle power hWtot i By the first law of thermodynamics, the net work outputted by the engine equals the net heat absorbed. Summing Eqs. (6.21) and (6.20) yields the per-trial power, or average work outputted per engine cycle: hWtot i = hQ2 i + hQ4 i ≈ Wb − 2 ln 2 Wb + 4 ln 2 . βC βC hδi (6.22) Theleading-order βH correction is negative and too small to include—of order 2 Wb hδi hδi N (βH E)2 . Equation (6.22) agrees well with the numerics in the appropriate limits (TC  Wb  hδi) and beyond, as shown in Fig. 6.5. The main text contains the primary analysis of Eq. (6.22). Here, we discuss the hQ2 i correction, limiting behaviors, and scaling. 2 The negative hQ2 i = − (Whδib ) detracts little from the leading term Wb of hQ4 i: (Wb )2 b  Wb , since W  1. The hQ2 i cuts down on the per-trial power little. hδi hδi The limiting behavior of Eq. (6.22) makes sense: Consider the limit as Wb → 0. The cold bath has too small a bandwidth to thermalize the engine, so the engine should output no work, on averge. Indeed, the first and third terms in Eq. (6.22) vanish, being proportional to Wb . The second term vanishes because βC → ∞ more quickly than Wb → 0 : The cold bath is very cold. √ Equation (6.22) scales with the system size N no more quickly than N /2N , √ by the assumption Wb  hδi ∼ N /2N . This scaling makes sense: The engine outputs work because the energy eigenvalues meander upward and downward in Fig. 6.2 as H(t) is tuned. In the thermodynamic limit, levels squeeze to- 119 0.06 2.5 0.05 2.0 hQ4i/hδi hQ4i/hδi 0.04 1.5 0.03 1.0 0.02 0.5 0.0 0.0 0.01 0.00 0.2 0.4 0.6 0.8 1.0 1.2 Wb/hδi (a) hQ4 i vs. Wb at TC = 0 and TH = ∞ 0.00 0.02 0.04 0.06 TC/hδi (b) hQ4 i vs. TC at TH = ∞ and Wb = 2−4 hδi 0.00026 √ hQ4i/E N 0.00025 0.00024 0.00023 0.00022 0.0 0.5 1.0 √ βH E N 1.5 2.0 (c) hQ4 i vs. βH at TC = 0 and Wb = 2−4 hδi Figure 6.4: Average heat hQ4 i absorbed during hot thermalization (stroke 4) as a function of the cold-bath bandwidth Wb , the coldbath temperature TC , and the hot-bath temperature TH = 1/βH : The blue lines represent the analytical prediction (6.21), to lowest order in TC , with the βH dependence of hQ4 i, too small a correction to include in Eq. (6.21): 2 2 b) hQ4 i ≈ Wb − 2βlnC2 + (W e−N (βH E) /4 . In the bottom figure (showing βH 2hδi dependence) we again use the full analytical result from [238]. See Sec. 6.7 for other parameters and definitions. The analytics’ shapes agree with the numerics’, and the fit is fairly close, in the appropriate limits (where e−βC Wb  b 1, βC1hδi  1, and W  1, in the gray shaded regions). The predictions hδi underestimate hQ4 i; see the Fig. 6.3 caption. Figure 6.4c suggests that the numerics deviate significantly from the analytics: The numerics appear to depend on βH via a linear term absent from the hQ4 i prediction. This seeming mismatch appears symptomatic of finite sample and system sizes. 120 1.4 0.06 1.2 0.05 hWtoti/hδi hWtoti/hδi 1.0 0.04 0.8 0.03 0.6 0.02 0.4 0.01 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.00 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 TC/hδi Wb/hδi hWtot i vs. TC at TH = ∞ and Wb = (a) hWtot i vs. Wb at TC = 0 and TH = (b) −4 hδi 2 ∞ 0.00026 √ hWtoti/E N 0.00025 0.00024 0.00023 0.00022 0.00021 0.0 0.5 1.0 √ βH E N 1.5 2.0 (c) hWtot i vs. βH at TC = 0 and Wb = 2−4 hδi Figure 6.5: Per-cycle power hWtot i as a function of the cold-bath bandwidth Wb , the cold-bath temperature TC , and the hot-bath temperature TH = 1/βH : The blue lines represent the analytical prediction b hWtot i ≈ Wb − 2βlnC2 : Eq. (6.22), to first order in W and in βC1hδi . See Sec. 6.7 hδi for other parameters and definitions. The analytics largely agree with the nub  1, and ThδiC  1 (in the gray shaded merics in the appropriate regime: W hδi region). Outside that regime, the analytics underestimate hWtot i; see Fig. 6.3 for an analysis. Figure 6.5c suggests that the numerics depend on βH via a linear term absent from the analytical prediction; see the caption of Fig. 6.4c. gether. Energy eigenvalues have little room in which to wander, and S outputs little work—hence our parallelization of fixed-length mesoscopic subengines in the thermodynamic limit (Sec. 6.6). 6.5.5 Efficiency ηMBL in the adiabatic approximation The efficiency is defined as ηMBL := hWtot i . hQin i (6.23) 121 The numerator is averaged separately from the denominator because averaging Wtot over runs of one mesoscopic engine is roughly equivalent to averaging over tot i simultaneous runs of parallel subengines in one macroscopic engine. hW may hQin i Wtot therefore be regarded as the Qin of one macroscopic-engine trial. The positive-heat-absorbing-stroke is stroke 4, in the average trial:  hQ2 i hQin i = hQ4 i = hWtot i − hQ2 i = hWtot i 1 − hWtot i  = hWtot i (1 + φ) , (6.24) wherein φ := − hQ2 i Wb ≈ . hWtot i 2 hδi (6.25) Substituting from Eq. (6.24) into Eq. (6.23) yields ηMBL ≈ hWtot i Wb ≈1−φ= 1− . hWtot i (1 + φ) 2 hδi (6.26) Using suboptimal baths diminishes the efficiency. Adding βC -dependent terms from Eq. (6.22) to hWtot i yields φ0 = Wb ln 2 Wb 1 + − 2 ln 2 . 2 hδi βC hδi hδi βC hδi (6.27) 2 Wb −N (βH E) /4 e , is too small to include. The correction The βH correction, 1 − 2hδi shares the sign of βH : A lukewarm hot bath lowers the efficiency. Expressions (6.26) and (6.27) are compared with results from numerical simulations in Fig. 6.6. The analytics agree with the numerics in the appropriate regime (TC  Wb  hδi). 6.6 MBL engine in the thermodynamic limit In Secs.6.2, 6.5, and 6.4 we analyzed a single mesoscale engine. The power output of that mesoscale engine was very small, and indeed decreased exponentially with the size of the engine: hWtot i ∼ Wb  hδi ∼ 2−N . Worse, the tuning speed v must shrink exponentially: Hmeso (t) is ideally tuned quantumadiabatically. The time per tuning stroke must far exceed hδi−1 . The mesoscale engine scales poorly, but properties of MBL offer a solution. 122 0.97 1.0 0.9 0.96 ηMBL ηMBL 0.8 0.7 0.94 0.6 0.93 0.5 0.4 0.0 0.95 0.00 0.2 0.4 0.6 0.8 1.0 0.02 0.04 0.06 TC/hδi 1.2 Wb/hδi (b) ηMBL vs. TC at TH = ∞ and Wb ≈ (a) ηMBL vs. Wb at TC = 0 and TH = ∞ 10−4 √N E ≈ 0.04 hδi 0.990 ηMBL 0.985 0.980 0.975 0.970 0.965 0.0 0.5 1.0 √ βH E N 1.5 2.0 (c) ηMBL vs. βH at TC = 0 and Wb = 2−4 hδi Figure 6.6: Efficiency ηMBL as a function of the cold-bath bandwidth Wb , the cold-bath temperature TC , and the hot-bath temperature TH = 1/βH : The blue lines represent the analytical predictions (6.26) and (6.27). Figure (6.6c) shows the leading-order βH dependence of ηMBL , a Wb −N (βH E)2 /4 correction too small to include in Eq. (6.27): 1 − 2hδi e . See Sec. 6.7 for other parameters and definitions. The analytics agree with √ the numerics Wb TC fairly well in the appropriate regime ( hδi  1, hδi  1, and N TH E  1). The analytics underestimate ηMBL ; see the Fig. 6.3 caption. 123 The MBL engine’s advantage lies in having a simple thermodynamic limit that does not compromise efficiency or power output. A thermodynamically large MBL Otto engine consists of mesoscale subengines that operate mostly independently. This independence hinges on local level correlations of the MBL phase [96, 196, 201]; cf Sec.1.1.4: Subsystems separated by a distance L evolve roughly independently until times exponential in L, due to the localization [158]. Consider perturbing an MBL system locally. In the Heisenberg picture, the perturbing operator spreads across a distance logarithmic in time [158]. (See also [109].) To see this, recall from Sec. 1.1.3 that the gaps between states differing on L sites are roughly δ ∼ E2−L e−L/ξ . (6.28) On a timescale t the system is indifferent to gaps less than t−1 , for (Et)−1 ∼ 2−L e−L/ξ (6.29) L(t) ∼ (ln 2 + ξ −1 )−1 ln(HScalet) . (6.30) or The longer the time t for which the perturbation lasts, the farther the influence spreads. First imagine tuning an MBL Hamiltonian infinitely slowly, to preclude diabatic transitions: t → ∞ . Even if the Hamiltonian consists of spatially local terms, the perturbation to each term spreads across the lattice. The global system cannot be subdivided into independent subengines. 7 The global system’s average gap vanishes in the thermodynamic limit: hδi → 0 . Since hWtot i ∼ Wb  hδi, the per-cycle power seems to vanish in the thermodynamic limit: Wb → 0. But now imagine tuning the Hamiltonian at a finite speed v. How big of a region can the tuning of a single local term affect? Dimensional analysis 7 Granted, subengines are coupled together even if the Hamiltonian is quenched infinitely quickly: Hsim (t) encodes a nearest-neighbor interaction, for example. That interaction might be regarded as coupling the edge of subengine k with the edge of subengine k + 1 . But subengines’ edges may be regarded as ill-defined. The sites definitively in subengine k, near subengine k’s center, should not couple to the sites near subengine `’s center, for any ` 6= k , if the subengines are to function mostly independently. Additionally, one may separate subenegines with “fallow” buffer zones, as we do. 124 suggests that the relevant time scale is t ∼ Ev . Local perturbations then affect a region of length ∼ L(E/v) ∼ ζ ln(E 2 /v). On a length scale L(E/v), global level correlations govern the engine’s performance less than local level correlations do, i.e., less than R(L(E/v), ω) does. This correlator registers level repulsion at a scale independent of N . Finite-speed tuning renders finite the average gap accessible to independent subengines, the hδi that would otherwise close in the thermodynamic limit. Each mesoscale subengine therefore outputs hWtot i > 0 . We can explain the gap’s finiteness differently: Suppose that the engine’s state starts some trial with weight on the jth energy level. The eigenenergies wiggle up and down during stroke 1. The jth energy may approach the (j − 1)th. Such close-together energies likely correspond to far-apart subengines. If the levels narrowly avoided crossing, particles would be rearranged across a large distance. Particles must not be, as subengines must function independently. So the engine must undergo a diabatic transition: the engine’s state must retain its configuration. The engine must behave as though the approaching energy level did not exist. Effectively removing the approaching level from the available spectrum effectively creates a gap in the spectrum. One can create such an effective gap (can promote such diabatic transitions) by tuning the Hamiltonian at a finite v. Let us apply this principle to a chain of N -site mesoscale engines separated by N -site buffers. The engine is cycled between a shallowly localized (HGOE like) Hamiltonian, which has a localization length ξ> , and a deeply localized (HMBL -like) Hamiltonian, which has ξ<  ξ> . The key element in the construction is that the cold bath acts through local operators confined to < N ∼ ξ> sites. This defines the subengines of the thermodynamic MBL Otto engine. Localization guarantees that “what happens in a subengine stays in a subengine”: subengines do not interfere much with each other’s operation. This subdivision boosts the engine’s power. A length-N mesoscale engine oper√ E N ates at the average per-cycle power hWtot imeso ∼ Wb  2N (Sec. 6.5). A subdivided length-Nmacro MBL engine outputs average work ∼ Nmacro hWtot imeso . 2N In contrast, if the length-Nmacro engine were not subdivided, it would output √ macro average work ∼ E2NNmacro , which vanishes in the thermodynamic limit. We can be more quantitative. 125 6.7 Numerical simulations We use numerical exact diagonalization to check analytical results. In Sec. 6.7.1, we describe the Hamiltonian used in our numerics. In Sec. 6.7.6, we study engine performance in the adiabatic limit (addressed analytically in Sec. 6.5). The code is available at https://github.com/christopherdavidwhite/MBLmobile. Call the times at which the strokes end t = τ, τ 0 , τ 00 , and τ 000 (see Fig. 6.1). For each of Nreals ≈ 1, 000 disorder realizations, we computed the whole density matrix ρ(t) at t = 0, τ, τ 0 , τ 00 , τ 000 . The engine’s time-t internal energy is E(t) = Tr(H(t)ρ(t)) . The quantities of interest are straightforwardly hW1 i = E(0) − E(τ ) , hQ2 i = E(τ 00 ) − E(τ 0 ) , hW3 i = E(τ 000 ) − E(τ 00 ) , and hQ4 i = E(0) − E(τ 000 ) . (6.31) (6.32) We disorder-averaged these quantities before dividing to compute the effii+hW3 i . ciency, ηMBL = 1 − hW1hQ 4i 6.7.1 Hamiltonian The engine can be implemented with a disordered Heisenberg model. A similar model’s MBL phase has been realized with cold atoms [188]. We numerically simulated a 1D mesoscale chain governed by a Hamiltonian " N −1 # N X X E Hsim (t) = ~σj · ~σj+1 + h(αt ) hj σjz ; Q(h(αt )) j=1 j=1 (6.33) this is a special case of the general mesoscopic Hamiltonian (6.2) described in Sec. 6.2. Equation (6.33) describes spins equivalent to interacting spinless fermions. Energies are expressed in units of E, the average per-site energy density. For γ = x, y, z, the γth Pauli operator that operates nontrivially on the jth site is denoted by σjγ . The Heisenberg interaction ~σj · ~σj+1 encodes nearest-neighbor hopping and repulsion. The tuning parameter αt ∈ [0, 1] determines the phase occupied by Hsim (t). The site-j disorder potential depends on a random variable hj distributed uniformly across [−1, 1]. The disorder strength h(αt ) varies as h(αt ) = αt hGOE + (1 − αt )hMBL . When αt = 0, the disorder is weak, h = hGOE , and the engine occupies the ETH phase. When αt = 1, the disorder is strong, h = hMBL  hGOE , and the engine occupies the MBL phase. 126 The normalization factor Q(h(αt )) preserves the width of the density of states (DoS) and so preserves hδi. Q(h(αt )) prevents the work extractable via change of bandwidth from polluting the work extracted with help from level statistics (see Sec. 8.1.1 for a discussion of work extraction from bandwidth change). Q(h(αt )) is defined and calculated below. The ETH-side field had a magnitude h(0) = 2.0, and the MBL-side field had a magnitude h(1) = 20.0. These h(αt ) values fall squarely on opposite sides of the MBL transition at h ≈ 7. 6.7.2 Scaling factor We wish to keep the DoS constant through the cycle. To fix µ(E), we rescale the Hamiltonian by a factor Q(h(αt )). We define Q2 (h(αt )) as the disorder average of the variance of the unrescaled DoS: Q2 (h(αt )) := = * * N 1 X 2 E N j=1 j ! N 1 X Ej N j=1 − 1 Tr(H̃ 2 (t)) − N !2 + (6.34) disorder !2 + 1 Tr(H̃(t)) N . (6.35) disorder The H̃(t) denotes an unrescaled variation on the random-field Heisenberg Hamiltonian H(t) of Eq. (6.33): H̃(t) := E "N −1 X j=1 ~σj · ~σj+1 + h(αt ) N X j=1 # hj σjz . (6.36) To compute Q2 (h(αt )), we rewrite the unrescaled Hamiltonian as " N −1 # −1 N X X X
N + − z z z H̃(t) = E 2 σj σj+1 + h.c. + σj σj+1 + h(αt ) hj σj . j=1 j=1 (6.37) j=1 We assume that N is even, and we work at half-filling. The N2 -particle subspace
N has dimensionality N = N/2 . Let us calculate some operator traces that we will invoke later. Let X := QN x j=1 σ denote the global spin-flip operator. For any operator A such that † X AX = −A,
Tr(A) = Tr X † AX = −Tr(A) . (6.38) 127 We have used the evenness of N , which implies the invariance of the half-filling subspace under X. Also, Tr(A) = 0. In particular, 0 = Tr(σjz ) = Tr(σjz σjz0 σjz00 ), if j 6= j 0 6= j 00 . Traces of products of even numbers of σ z factors require more thought: z Tr(σjz σj+1 ) = (# states j, j + 1 =↑↑) + (# states j, j + 1 =↓↓) − 2(# states j, j + 1 =↑↓)      N −2 N −2 N −2 = + −2 N/2 − 2 N/2 N/2 − 1 1 . = −N N −1 (6.39)  (6.40) Similarly,

− + + − Tr [σj+ σj− ][σj+1 σj+1 ] = Tr [σj− σj+ ][σj+1 σj+1 ] = (# states j, j + 1 =↑↓)   N −2 = N/2 − 1 N =N , 4(L − 1) (6.41) (6.42) and
z Tr σjz σj+1 σjz0 σjz0 +1 = (# states j, j + 1, j 0 , j 0 + 1 =↑↑↑↑)   4 + (# states j, j + 1, j 0 , j 0 + 1 =↑↑↓↓) 2 + (# states j, j + 1, j 0 , j 0 + 1 =↓↓↓↓)   4 − (# states j, j + 1, j 0 , j 0 + 1 =↑↑↑↓) 1   4 − (# states j, j + 1, j 0 , j 0 + 1 =↑↓↓↓) 1       N −4 N −4 N −4 = +6 + N/2 − 4 N/2 − 2 N/2     N −4 N −4 −6 −6 N/2 − 3 N/2 − 1 3 , =N (N − 1)(N − 3) (6.43) (6.44) wherein the first equality’s combinatorial factors come from permutations on 128 sites j, j + 1, j 0 , and j 0 + 1. Assembling these pieces, we find Tr(H̃(t)) = E we compute Tr(H̃ 2 (t)): PN −1 j=1
Tr σjz σjz = −EN . Next, " N −1 N −1 X X + − − + + − 2 2 (σj σj )(σj+1 σj+1 ) + 4 H̃ (t) = E 4 (σj− σj+ )(σj+1 σj+1 ) j + j N −1 X z σjz σj+1 σjz0 σjz0 +1 + h2 (αt ) j,j 0 =1 N X h2j + (traceless terms) j=1 " N −1 N −1 X X + − − + + − 2 =E 4 (σj σj )(σj+1 σj+1 ) + 4 (σj− σj+ )(σj+1 σj+1 ) j + N −1 X j=1 # (6.45) j I+ N −2 X z σjz σj+2 + + h(αt )2 (αt ) z σjz σj+1 σjz0 σjz0 +1 j=1 j 0 =j+2 j=1 N X N −3 N −1 X X # h2j + (traceless terms) . j=1 (6.46) We take the trace, using Eqs. (6.39), (6.41), and (6.43): # N X N − 2 + h2 h2j . Tr(H̃ 2 (t)) = N 3N − 1 + N −1 j=1 " We disorder-average by taking h2j 7→ D E 2 Tr(H (t)) disorder R1 0 (6.47) dhj h2j = 31 : " 2 = N 3N − 1 + h N −2 +N N −1 3 # . (6.48) Substituting into Eq. (6.34), we infer the rescaling factor’s square: Q2 (h(αt )) = 3N − 2 + N −2 h2 +N . N −1 3 (6.49) Our results are insensitive to the details of Q. The width of the DoS in one disorder realization will differ from the disorder average (6.49). Moreover, that difference will vary as we tune h(αt ), because the disorder affects only one term. The agreement between the analytics, in which µ(E) is assumed to remain constant in t, and the numerics is therefore comforting: the engine is robust against small variations in the rescaling. 129 6.7.3 Representing states and Hamiltonians We structured our software to facilitate a possible extension: The cold bath might be modeled more realistically, as coupling to the engine only locally. We represent the state of one mesoscopic MBL Otto engine with a density matrix ρ ∈ CN ×N , and the Hamiltonian with a matrix H ∈ CN ×N , relative to the basis {|s1 i , . . . , |sN i} = {|↑ . . . ↑i , . . . , |↓ . . . ↓i} of products of σ z eigenstates. We track the whole density matrix, rather than just the energy-diagonal elements, with an eye toward the coherent superpositions that q diabatic cor
N 2 2N . rections create. For an N -site chain at half-filling, N = N/2 ≃ πN 6.7.4 Strokes 1 and 3: tuning The (l, m) entry of the initial-state density matrix is ρ(0)lm = hsl | 1 −βH H(0) 1 X −βH Ej (0) e |sm i = e hsl i Ej (0) hEj (0)i sm . (6.50) Z Z j The jth eigenstate of H(0), associated with energy Ej (0), is denoted by |Ej (0)i. We approximate the time evolution from 0 to τ (during stroke 1) as adiabatic. The evolution therefore does not move weight between levels: ρ(τ )lm = 1 X −βH Ej (0) e hsl i Ej (τ ) hEj (τ )i sm . Z j (6.51) If we represented our density matrix relative to an instantaneous energy eigenbasis, simulating the time evolution would be trivial: We would reinterpret the diagonal matrix ρ as being diagonal, with the same elements in a new basis. However, we wish to represent ρ(t) relative to the σjz product basis. This representation enhances the code’s flexibility, facilitating the inclusion of diabatic evolutions and a more detailed model of cold thermalization. To represent ρ(t) relative to the σjz product basis, we note that ρ(τ )lm = X j hsl i Ej (τ ) hEj (0)| ρ(0) |Ej (0)i hEj (τ )i sm = [U (τ, 0)ρ(0)U (τ, 0)† ]lm . (6.52) We have defined a time-evolution matrix U (τ, 0) ∈ C by U (τ, 0)lm = P j hsl i Ej (τ ) hEj (0)i sm . This matrix is easily computed via exact diagonalization of H(0) and H(τ ). N ×N We can compute the density matrix ρ(τ 00 ) at the end of stroke 3 (the tuning from MBL to GOE) from the density matrix ρ(τ 0 ) at the end of stroke 2 130 (the cold-bath thermalization) similarly: ρ(τ 00 ) = U (τ 00 , τ 0 )ρ(τ 0 )U (τ 00 , τ 0 )† . The time-evolution matrix U (τ 00 , τ 0 ) ∈ CN ×N is given by U (τ 00 , τ 0 )lm = X j hsl i Ej (0) hEj (τ )i sm . (6.53) [Recall that H(τ 00 ) = H(0) and H(τ 0 ) = H(τ ).] 6.7.5 Stroke 2: Thermalization with the cold bath During stroke 2, the system thermalizes with a bandwidth-Wb cold bath. We make three assumptions. First, the bandwidth cutoff is hard: The bath can transfer only amounts < Wb of energy at a time. Therefore, the cold bath cannot move probability mass between adjacent levels separated by just one gap δ 0 > Wb . Second, the bath is Markovian. Third, the system thermalizes for a long time. The bath has time to move weight across sequences of small 0 gaps δj0 , δj+1 , . . . < Wb . We can implement thermalization as follows. First, we identify sequences of levels connected by small gaps. Second, we reapportion weight amongst the levels according to a Gibbs distribution. Suppose, for example, that the MBL Hamiltonian H(τ ) contains the following chain of six energies, E1 , . . . , E6 , separated from its surrounding levels by large gaps (Fig. 6.7): (E2 − E1 ), (E3 − E2 ) < Wb , (E5 − E4 ) < Wb , and (6.54) (E4 − E3 ), (E6 − E5 ) > Wb . We suppress the time arguments to simplify notation. Before thermalization, the density operator is diagonal with respect to the energy basis: ρ(τ ) = P j ρj |Ej ihEj | . The weight on level j is denoted by ρj . Thermalization maps ρ(τ ) 7→ ρ(τ 0 ) = ρ1 + ρ2 + ρ3 −β E 1 C e + e−βC E2 + e−βC E3   × e−βC E1 |E1 ihE1 | + e−βC E2 |E2 ihE2 | + e−βC E3 |E3 ihE3 |   ρ4 + ρ5 −βC E5 −βC E4 e |E ihE | + e |E ihE | + −β E4 4 4 5 5 e C + e−βC E5 + ρ6 |E6 ihE6 | . (6.55) 131 Energies E6 Wb E5 E4 E3 E2 E1 Figure 6.7: Energies of a cold-thermalized system: We illustrate our implementation of cold thermalization with this example chain of six energies. The cold bath has a bandwidth of size Wb , depicted in green. 6.7.6 Results We compare the analytical predictions of Sec. 6.5 to numerical simulations of a 12-site engine governed by the Hamiltonian (6.33). During strokes 1 and 3, the state was evolved as though the Hamiltonian were tuned adiabatically. We index the energies Ej (t) from least to greatest at each instant: Ej (t) < Ek (t) ∀j < k. Let ρj denote the state’s weight on eigenstate j of the initial Hamiltonian, whose αt = 0. The engine ends stroke 1 with weight ρj on eigenstate j of the post-tuning Hamiltonian, whose αt = 1. The main results appear in Fig. 6.8. Figure 6.8a shows the average work extracted per cycle, hWtot i. Figure 6.8b shows the efficiency, ηMBL . In these simulations, the baths had the extreme temperatures TH = ∞ and TC = 0. This limiting case elucidates the Wb -dependence of hWtot i and of ηMBL : Disregarding finite-temperature corrections, on a first pass, builds intuition. Finite-temperature numerics appear alongside finite-temperature analytical calculations above. Figure 6.8 shows how the per-cycle power and the efficiency depend on the cold- 132 1.4 1.0 1.2 0.9 1.0 hWtoti/hδi 0.8 ηMBL 0.8 0.6 0.6 0.4 0.5 0.2 0.0 0.0 0.7 0.2 0.4 0.6 0.8 1.0 1.2 0.4 0.0 0.2 0.4 0.6 Wb/hδi Wb/hδi (a) (b) 0.8 1.0 1.2 Figure 6.8: Average per-cycle power hWtot i (left) and efficiency ηMBL (right) as functions of the cold-bath bandwidth Wb : Each red dot represents an average over 1,000 disorder realizations of the random-field Heisenberg Hamiltonian (6.33). The slanted blue lines represent the analytical predictions (6.7) and (6.9) of Sec. 6.5. When Wb  hδi (in the gray shaded region), hWtot i and ηMBL vary linearly with Wb , as predicted. The error bars are smaller than the numerical-data points. bath bandwidth Wb . As expected, hWtot i ≈ Wb . The dependence’s linearity, and the unit proportionality factor, agree with Eq. (6.7). Also as expected, Wb . The the efficiency declines as the cold-bath bandwidth rises: ηMBL ≈ 1 − 2hδi linear dependence and the proportionality factor agree with Eq. (6.9). The gray columns in Fig. 6.8 highlight the regime in which the analytics were b  1. If the cold-bath bandwidth is small, Wb < hδi, the performed, where W hδi analytics-numerics agreement is close. But the numerics agree with the analytics even outside this regime. If Wb & hδi, the analytics slightly underestimate ηMBL : The simulated engine operates more efficiently than predicted. To predict the numerics’ overachievement, one would calculate higher-order corrections in Sec. 6.5: One would Taylor-approximate to higher powers, modeling subleading physical processes. Such processes include the engine’s dropping across a chain of three small gaps, δ10 , δ20 , δ30 < Wb , during cold thermalization. The error bars are smaller than the numerical-data points. Each error bar represents the error in the estimate of a mean (of hWtot i or of ηMBL := 1 − hWtot i ) over 1,000 disorder realizations. Each error bar extends a distance hQin i √ (sample standard deviation)/ # realizations above and below that mean. 133 Chapter 7 THE MBL-MOBILE AT FINITE SPEED In Ch. 6, we assumed that we could tune the engine in strokes 2 and 4 infinitely slowly, as far as the individual mesoscale engines were concerned—but then we also assumed that we could tune it quickly enough that information does not propagate between those individual mesoscale engines. Are these two assumptions broadly consistent? That is, does a regime exist that is both slow with respect to unwanted transitions in a single mesoscale engine and fast with respect to transitions across multiple mesoscale engines? More generally, an experimenter would want to run our cycle in a finite time: she is restricted by the coherence times of her systems, by a desire to go to lunch, and ultimately by the requirement that she finish, publish, and graduate. How does our engine perform at finite speed? The time scales of a macroscopic engine are crucial for the assessment of the MBL Otto engine. In this chapter we estimate the restrictions on the speed with which the Hamiltonian must be tuned to avoid diabatic transitions (Sec. 7.1). We compare these diabatic corrections to numerical simulations (Sec. 7.2 After that we compute the restrictions on the speed required to prevent intersubengine communication (Sec. 7.3); we find that we can indeed tune the system quickly enough to avoid communication, but slowly enough to avoid undue diabatic transitions. We finally estimate the time required for cold thermalization (stroke 2) in Sec. 7.4. 7.1 Diabatic corrections We have modeled the Hamiltonian tuning as quantum-adiabatic, but realistic t tuning speeds v := E dα are finite. To understand diabatic tuning’s effects, dt we distinguish the time-t density matrix ρ(t) from the corresponding diagonal ensemble, ρdiag (t) = X j |Ej (t)i εj hEj (t)| , εj = hEj (t)| ρ |Ej (t)i wherein (7.1) 134 Energies (Et) APT APT HETH LandauZener LandauZener Frac-LandauZener Frac-LandauZener HMBL Hamiltonian [Hmeso(t)] Figure 7.1: Three (times two) classes of diabatic transitions: Hops to arbitrary energy levels, modeled with general adiabatic perturbation theory (APT), plague the ETH regime. Landau-Zener transitions and fractionalLandau-Zener transitions plague the many-body-localized regime. P and |Ej (t)i is an instantaneous energy eigenbasis of Hmeso (t) = j |Ej (t)i Ej (t) hEj (t)|. The average energy depends on ρ(t) only through ρdiag (t). [More generally, the state’s off-diagonal elements dephase under the dynamics. ρdiag (t) is “slow” and captures most of the relevant physics [44].] In the adiabatic limit, εj (t) = εj (0). We seek to understand how this statement breaks down when the tuning proceeds at a finite speed v It is useful to think of “infinite-temperature thermalization” in the sense of this diagonal ensemble: Fast tuning may push the diagonal-ensemble weights εj (t) towards uniformity—even though the process is unitary and the entropy S = −ρ(t) ln ρ(t) remains constant—thanks to the off-diagonal elements. The effects of diabatic tuning appear in three distinct regimes, which we label “fractional-Landau-Zener,” “Landau-Zener,” and “APT” (Fig. 7.1). We estimate the average per-cycle work costs hWdiab i of diabatic jumps, guided by the numerics in Sec. 6.7. We focus on TH = ∞ and TC = 0, for simplicity. Since TH = ∞, diabatic hops cannot bring ρdiag (t) closer to I/2N —cannot change the average energy—during stroke 1. Hence we focus on stroke 3. 7.1.1 Fractional-Landau-Zener transitions At the beginning of stroke 3, nonequilbrium effects could excite the system back across the small gap to energy level j. The transition would cost work and would prevent the trial from outputting Wtot > 0. We dub this excitation a fractional-Landau-Zener (frac-LZ) transition. It could be suppressed by a sufficiently slow drive [46]. The effects, and the resultant bound on v, are simple to derive 135 Let the gap start stroke 3 at size δ and grow to a size ∆ > δ. Because the two energy levels begin close together, one cannot straightforwardly apply the Landau-Zener formula. One must use the fractional-Landau-Zener result of De Grandi and Polkovnikov [46], v 2 (δ− )2 pfrac-LZ (δ) ≈ 16  1 1 + δ 6 ∆6  ≈ v 2 (δ− )2 . 16δ 6 (7.2) δ− denotes the MBL level-repulsion scale, the characteristic matrix element introduced by a perturbation between eigenstates of an unperturbed Hamiltonian. We suppose that energy-level pairs with pfrac-LZ . 1 are returned to the infinite-temperature state from which the cold bath disturbed them. These pairs do not contribute to hWtot i. Pairs that contribute have pfrac-LZ  1, i.e., δ & (vδ− )1/3 . (7.3) If the rest of the stroke is adiabatic, the average work performed during the cycle is hWtot i ∼ hQ4 i − hQ2 i − (vδ− )1/3 , (7.4) which results immediately in the correction hWdiab,frac-LZ i ∼ (vδ− )1/3 . (7.5) This correction is negligible at speeds low enough that v 7.1.2 (Wb )3 . δ− (7.6) Landau-Zener transitions While the system is localized, the disturbances induced by the tuning dH(t) dt can propagate only only a short distance lv . The tuning effectively reduces the mesoscale engine to an effective length-lv subengine. To estimate lv , we compare the minimum gap of a length-lv subsystem to the speed v: E2−lv e−lv /ξ< ∼ √ v. (7.7) This minimum gap—the closest that two levels are likely to approach—is given by the smallest level-repulsion scale, δ− . δ− characterizes the deeply localized 136 system, whose ξ = ξ< . Consequently, ln(E 2 /v) . lv ∼  2 ln 2 + 1ξ (7.8) < Suppose that lv ≤ N , and consider a length-lv effective subengine. In the adiabatic limit, hWtot i does not depend on the engine’s size. (hWtot i depends only on the bath bandwidth Wb  hδi.) To estimate how a finite v changes hWtot i, we consider the gaps δ < Wb of the size-lv subengine. We divide the gaps into two classes: 1. Gaps connected by flipping l-bits on a region of diameter l < lv . The tuning is adiabatic with respect to these gaps, so they result in work output. 2. Gaps connected by flipping l-bits on a region of diameter l = lv . The tuning is resonant with these gaps and so thermalizes them, in the sense of the diagonal ensemble: The tuning makes the instantaneous-energyeigenvector weights εj uniform, on average. Type-1 gaps form a v-independent O(1) fraction θ of the length-lv subengine’s short-length-scale gaps.1 Type-2 gaps therefore make up a fraction 1 − θ. Hence Landau-Zener physics leads to a v-independent O(1) diabatic correction (1 − θ)Wb to hWtot i, provided that v is high enough that lv < N . 7.1.3 APT transitions When the system is in the ETH phase (or has correlation length ξ ∼ N ), typical minimum gaps (points of closest approach) are still given by the levelrepulsion scale, which is now hδi. Hence one expects the tuning to be adiabatic if v  hδi2 . (7.9) This criterion is at least as stringent (depending on the system size) as the requirement (7.6) that fractional Landau-Zener transitions occur rarely. Both 1 We can estimate θ crudely. For a given diameter-lv subset, each gap connected by a diameter-(lv − 1) operator can be made into a diameter-lv gap: One flips the last (lv )th l-bit, for θ ∼ 1/2. This estimate neglects several combinatorial matters. A more detailed analysis would account for the two different diameter-(lv − 1) regions of a given length-lv subengine, gaps connected by l-bit flips in the intersections of those subengines, the number of possible diameter-lv subengines of an N -site system, etc. 137 1.0 hWtoti/Wb 0.8 0.6 0.4 0.2 0.0 0.0 0.5 1.0 (vδ−)1/3/Wb 1.5 2.0 Figure 7.2: Average per-cycle work as a function of tuning speed: We numerically simulated 995 disorder realizations of the random-field Heisenberg Hamiltonian (6.33) for a system of N = 8 sites (red dots). The results are compared to the analytical estimate (6.7) for the adiabatic work output (blue line) and an empirical straight-line fit Wtot = W0 − W1 (vδ− )1/3 /Wb (black line). Errors p in the estimate of the mean, computed as (sample standard deviation)/ (# realizations), lead to error bars smaller than the numerical-data points. fractional Landau-Zener transitions and APT transitions bound the cycle time τcycle less stringently than thermalization with the cold bath; hence a more detailed analysis of APT transitions would be gratuitous. Such an analysis would rely on the general adiabatic perturbation theory of De Grandi and Polkovnikov [46]; hence the moniker “APT transitions.” 7.2 Numerical simulations We simulate the engine as in Ch. 6 (in particular Sec. 6.7)—with the exception that now in the work strokes, strokes 1 and 3, Hsim (t) was tuned at finite speed v. We numerically study the preclusion of communication between mesoscale subengines (addressed analytically in Sec. 7.3) only insofar as these results follow from diabatic corrections: Limitations on computational power restricted the system size to 12 sites. Computational limitations restricted the engine to 8 sites. (These simulations quickly become slow to run because our upper bounds on v scale as powers of hδi ∼ 2−N .) 138 7.2.1 Simulating finite-time tuning We simulate a stepwise tuning, taking αt = δt bt/δtc . T (7.10) GOE δt denotes a time-step size, and T ∝ hMBL −h denotes the time for which one v tuning stroke lasts. This protocol is more violent than the protocols treated analytically: v is assumed to remain finite in the diabatic analytics. In the numerics, we tune by sudden jumps (for reasons of numerical convenience). We work at TH = ∞ and TC = 0—again, to capture the essential physics without the complication of finite-temperature corrections. To implement this stepwise tuning, we take α(t) = δt bt/δtc , T (7.11) where δt denotes a time-step size and T ∝ (hMBL − hGOE )/v denotes the total tuning time, and we compute a time-evolution unitary for the whole stroke by chaining together the unitaries for each time step. For stroke 1, U (τ, 0; v, δt) = e−iH(τ −δt)δt e−iH(τ −2δt)δt . . . e−iH(0)δt , (7.12) with the number of time steps set by the speed. We use the time step δt = 0.405 hδi, but our results are not sensitive to time step’s size. In judging the engine’s effectiveness at a finite v, we must estimate the levelrepulsion scale δ− . We do this by diagonalizing 106 disorder realizations at the relevant disorder width, h = 20, for N = 8 sites. A histogram of the gaps is plotted in Fig. 7.3. We then visually estimate the point at which the distribution turns over. Our results are not sensitive to this value. Figure 7.2 shows the average work output, hWtot i, as a function of v. Despite the simulated protocol’s violence, both a fractional-Landau-Zener correction Wfrac-LZ ∼ (vδ− )3 , explained in Sec. 7.1.1, and a v-independent O(1) LandauZener correction, explained in Sec. 7.1.2, are visible. We believe that the adiabatic numerics (v = 0 red dot) differ from the analytics (blue line) due to finite-size effects: For small systems away from the spectrum’s center, the average gap estimated from the density of states can vary appreciably over one gap. These numerics confirm the analytics and signal the MBL Otto engine’s robustness with respect to changes in the tuning protocol. 139 Number of gaps 60000 40000 20000 0 0.00 0.02 0.04 0.06 hδi/δ 0.08 Figure 7.3: Level-spacing distribution for 106 disorder realizations of the random-field Heisenberg model at disorder width h = 20 and system size N = 8 (blue line). The vertical black line shows the estimate of the levelrepulsion parameter δ− . 7.2.2 Enlightenment from numerical methods Sec. 7.2 discusses numerical simulations of a finite-speed MBL engine. We simulate not the notionally smooth tuning of Sec. 6.2, but rather a stepwise tuning with step size δt ≃ 0.405 hδi−1 . Faithfully simulating the smooth-tuning protocol is substantially more difficult than simulating the stepwise protocol. In either case, the tuning time grows exponentially with system size, because it is set by (an estimate of) the smallest gap. But in the stepwise protocol, we fix the step size in terms of the average gap, so the step size grows with the the tuning time. Faithfully simulating the smooth tuning protocol, on the other hand, requires step size constant with system size. The highest frequency ωmax in the problem is of order the many-body bandwidth ∼ Q1 N ∼ 1, and even with higher-order methods we require step size δt ∼ ωmax : to see this heuristically, note that a fourth-order polynomial can approximate at most two periods of a sine. And yet the stepwise protocol with long timestep δt ≃ 0.405 hδi−1 captures the essential physics. Why is this? What about that physics is amenable to capture by such a crude method2 ? A timestep δt = 0.405 hδi−1 can capture physics on energy scales smaller than δt−1 ≃ 2 hδi. The results of Fig. 7.2 are 2 Given our reliance on the numerics to catch persuasive but subtly incorrect arguments, this could be could merely signal that we made a mistake that happened to line up with the ways in which the stepwise protocol differs from the smooth protocol. But let us proceed on the assumption that this is not the case. 140 not sensitive to δt, suggesting that diabatic corrections occur on energy scales small This is (weak) evidence for our implicit supposition in Sec. ss:diabaticcorrection that far-away levels (that is, levels that are not nearest in energy). It is also evidence that the fractional Landau-Zener and Landau-Zener transitions of Sec’s 7.1.1 and 7.1.2—which take place on an energy scale δ− —are indeed the dominant contribution. To claim this definitively would require a much more careful analysis of convergence in δt than we have performed. Looking beyond convergence analysis, this is a place where the well-developed methods of numerical solution of ODEs may have physical insight to offer. In particular, adaptive stepsize methods would directly probe the energy scale required to capture the system’s essential physics. The challenge in applying these methods would be constructing a good quantitative measure of how accurately an individual step has been approximated. 7.3 Precluding communication between subengines To maintain the MBL engine’s advantage, we must approximately isolate subengines. The subengines’ (near) independence implies a lower bound on the tuning speed v: The price paid for scalability is the impossibility of adiabaticity. Suppose that Hmacro (t) were tuned infinitely slowly. Information would have time to propagate from one subengine to every other. The slow spread of information through MBL [109] lower-bounds v. This consideration, however, does not turn out to be the most restrictive constraint on the cycle time. Therefore, we address it only qualitatively. As explained in Sec. 7.1.2, v determines the effective size of an MBL subengine. Ideally, v is large enough to prevent adiabatic transitions between configurations extended beyond the mesoscale N . For each stage of the engine’s operation, v should exceed the speed given in Eq. (7.7) for the localization length ξ of a length-(N + 1) chain: v  [δ− (N + 1, ξ)]2 ∼ E 2 2−2(N +1) e−2(N +1)/ξ . (7.13) (We have made explicit the dependence of the level-repulsion scale δ− on the mesoscale-engine size N and on the localization length ξ.) This scale quickly decays as the system is taken through stroke 1. This implies 3 b) [from that the speed should interpolate between [δ− (N + 1, ξ> )]2 and δ−(W (N,ξ< ) Ineq. (7.6)]. 141 7.4 Lower bound on the cycle time from cold thermalization Thermalization with the cold bath (stroke 2) bounds τcycle more stringently than the Hamiltonian tunings do. The reasons are (i) the slowness with which MBL thermalizes and (ii) the restriction Wb  hδi on the cold-bath bandwidth. We elaborate after introducing our cold-thermalization model (see [238, App. I] for details). We envision the cold bath as a bosonic system that couples to the engine locally, as via the Hamiltonian Hint = g Z Wb /ξ> dω −Wb /ξ> NX macro   c†j cj+1 + h.c. j=1 × δ(h0|cj Hmacro (τ )c†j+1 |0i − ω) . bω + b†ω
(7.14) The coupling strength is denoted by g. cj and c†j denote the annihilation and creation of a fermion at site j. Hmacro (t) denotes the Hamiltonian that would govern the engine at time t in the bath’s absence. Cold thermalization lasts from t = τ to t = τ 0 (Fig. 6.1). bω and b†ω represent the annihilation and creation of a frequency-ω boson in the bath. The Dirac delta function is denoted by δ(.). The bath couples locally, e.g., to pairs of nearest-neighbor spins. This locality prevents subengines from interacting with each other much through the bath. The bath can, e.g., flip spin j upward while flipping spin j + 1 downward. These flips likely change a subengine’s energy by an amount E. The bath can effectively absorb only energy quanta of size ≤ Wb from any subengine. The cap is set by the bath’s speed of sound [114], which follows from microscopic parameters in the bath’s Hamiltonian [134]. The rest of the energy emitted during the spin flips, |E − Wb |, is distributed across the subengine as the intrinsic subengine Hamiltonian flips more spins. Let τth denote the time required for stroke 2. We estimate τth from Fermi’s Golden Rule, Γf i = 2π |hf |V |ii|2 µbath . ~ (7.15) Cold thermalization transitions the engine from an energy level |ii to a level |f i. The bath has a density of states µbath ∼ 1/Wb . We estimate the matrix-element size |hf |V |ii| as follows. Cold thermaliza- 142 tion transfers energy Eif ∼ Wb from the subengine to the bath. Wb is very small. Hence the energy change rearranges particles across a large distance L  ξ = ξ< , due to local level correlations. V nontrivially transforms just a few subengine sites. Such a local operator rearranges particles across a large distance L at a rate that scales as Ee−L/ξ 2−L ∼ δ− . Whereas E sets the scale of the level repulsion δ− , g sets the scale of |hf |V |ii|. The correlation length ξ = ξ< during cold thermalization. We approximate L with the subengine length ξ> . Hence |hf |V |ii| ∼ gδE− . We substitute into Eq. (7.15). The transition rate Γf i = τ1th . Inverting yields τcycle ∼ τth ∼ Wb  E gδ− 2 . (7.16) To bound τcycle , we must bound the coupling g. The interaction is assumed to be Markovian: Information leaked from the engine dissipates throughout the bath quickly. Bath correlation functions must decay much more quickly than the coupling transfers energy. If τbath denotes the correlation-decay time, τbath < g1 . The small-bandwidth bath’s τbath ∼ 1/Wb , so g < Wb . This inequality, with Ineq. (7.16), implies τcycle = τth > E2 10 2ξ> /ξ< 3ξ> ∼ e 2 . 2 Wb (δ− ) E (7.17) . The final expression follows if Wb ∼ hδi 10 Like Markovianity, higher-order processes bound τth . Such processes transfer energy E > Wb between the engine and the cold bath. Because they require n ∼ E/Wb excitations in the bath, they will be suppressed by a factor ∼ g E/Wb . To understand these processes in more detail, write Htot = Hmacro (t) + Hbath + Hint for the Hamiltonian that governs the engine-and-bath composite. Htot generates the time-evolution operator U (t) := e−iHtot t . Consider Taylor-expanding U (t). The `th term is suppressed in g ` , contains 2` fermion operators cj and c†j 0 , and contains ` boson operators bω and b†ω0 . This term encodes the absorption, by the bath, of ` energy quanta of sizes ≤ Wb . The subengine gives the bath a total amount ∼ `Wb of heat. The subengine should not lose so much heat. Hence higher-order processes should occur much more slowly than the 143 rate-g processes: τhigh−ord.  τth . (7.18) Let us construct an expression for the left-hand side. Which processes most urgently require suppressing? Processes that change the subengine’s energy by & hδi. Figure 6.2 illustrates why. If the right-hand leg has length & hδi, the right-hand leg could be longer than the left-hand leg. If it were, the trial would yield net negative work, Wtot < 0. The bath would absorb energy hδi hδi packets of energy ∼ Wb each. Hence the from a subengine by absorbing ∼ W b hδi bath would appear to need to flip ∼ L = W spins to absorb energy ∼ hδi. (We b switch from fermion language to spin language for convenience.) However, the length-L spin subchain has a discrete effective energy spectrum. The spectrum might lack a level associated with the amount (initial energy) − hδi of energy. hδi If so, the bath must flip more than W spins—local level correlations suggest b o n hδi , ξ> . Energy is rearranged across the ∼ ξ> spins. Hence L = max W b distance L at a rate ∝ g L . 144 Chapter 8 RACING THE MBL-MOBILE In Chs 7, we analyzed the MBL engine mostly in terms of scaling properties. But what are the constant factors in front of those scalings? In particular, what would the power density of an experimental realization look like? And how does our engine compare to other quantum Otto engines? 8.1 Comparisons with competitor Otto engines We compare the MBL Otto engine to four competitors: a bandwidth engine, a variant of the MBL engine that is tuned between two disorder strengths, an engine of quantum dots, and an Anderson-localized engine. We argue that the MBL Otto engine is more robust against perturbations than the bandwidth, Anderson, and quantum-dot engines. We also argue that our MBL engine is more reliable than the equal-disorder-strength engine: Our MBL engine’s Wtot varies less from trial to trial and suppresses worst-case trials, in which Wtot < 0. This paper’s arguments go through almost unchanged for an Anderson-localized medium. Such a medium would lack robustness against interactions, though: Even if the interactions do not delocalize the medium— which would destroy the engine—they would turn the Anderson engine into an MBL engine. One can view our MBL engine as an easy generalization of the Anderson engine. 8.1.1 Comparison with bandwidth engine Imagine eliminating the scaling factor Q(h(αt )) from the Hamiltonian (6.33). The energy band is compressed and expanded as the disorder strength h(αt ) is ramped down and up. The whole band, rather than a gap, contracts and widens as in Fig. 6.2, between a size ∼ ENmacro h(α0 ) and a size ∼ ENmacro h(α1 )  ENmacro h(α0 ). The engine can remain in one phase throughout the cycle. The cycle does not benefit from the “athermality” of local level correlations. Furthermore, this accordion-like motion requires no change of the energy eigenbasis’s form. Tuning may proceed quantum-adiabatically: v ≈ 0. The ideal engine suffers no diabatic jumps, losing hWdiab imacro = 0. 145 But this engine is impractical: Consider any perturbation V that fails to commute with the ideal Hamiltonian H(t): [V, H(t)] 6= 0. Stray fields, for example, can taint an environment. As another example, consider cold atoms in an optical lattice. The disorder strength is ideally Eh(αt ). One can strengthen the disorder by strengthening the lattice potential Ulattice . Similarly, one can raise the hopping frequency (ideally E) by raising the pressure p. Strengthening t) = h(αt ) Ulattice and p while achieving the ideal disorder-to-hopping ratio Eh(α E requires fine control. If the ratio changes from h(αt ), the Hamiltonian H(t) acquires a perturbation V that fails to commute with other terms. This V can cause diabatic jumps that cost work hWdiab imacro . Can the bandwidth engine not withstand several hops—say, through 0.02Nmacro levels? No, because the ground state pulls away from the rest of the spectrum as Nmacro grows. Suppose, for simplicity, that TC = 0 and TH = ∞. The bandwidth engine starts stroke 1 in ρ(0) = I/Nmacro . Diabatic hops preserve ρ(t) during stroke 1, on average: The engine as likely hops upward as drops. Cold thermalization drops the engine to the ground state (plus an exponentially small dusting of higher-level states). The ground-state energy is generically extensive. Hence the engine absorbs hQ2 imacro ∼ −Nmacro , on average. Suppose that, during stroke 3, the engine jumps up through 2% of the levels. The engine ends about two standard deviations below the spectrum’s cen√ ter, with average energy ∼ Nmacro . While returning to TH = 0 during the √ average stroke 4, the bandwidth engine absorbs hQ4 imacro ∼ Nmacro . The av√ erage outputted work hWtot imacro = hQ4 imacro + hQ2 imacro ∼ Nmacro − Nmacro . As Nmacro grows, hWtot imacro dips farther below zero. A few diabatic jumps threaten the bandwidth engine’s ability to output hWtot i > 0. The bandwidth engine’s v must decline as Nmacro grows also because the typical E shrinks. The smaller the gaps, the greater whole-system gap hδimacro ∼ Nmacro the likelihood that a given v induces hops. As hδimacro → 0, v must → 0. The MBL Otto cycle proceeds more quickly, due to subengines’ parallelization. 8.1.2 Comparison with MBL engine tuned between same-strength disorder realizations Take our MBL Otto cycle, and vary not the disorder strength, but the disorder realization during each cycle. The disorder strength h(αt ) in Eq. (6.33) would remain  1 and constant in t, while the random variables hj would change. Let S̃ denote this constant-h(αt ) engine, and let S denote the MBL engine. 146 S̃ takes less advantage of MBL’s “athermality,” as S̃ is not tuned between level-repelling and level-repulsion-free regimes. Yet S̃ outputs the amount hWtot i of work outputted by S per cycle, on average. Because Wb is small, cold thermalization drops S̃ across only small gaps δ 0  hδi. S̃ traverses a trapezoid, as in Fig. 6.2, in each trial. However, the MBL engine has two advantages: greater reliability and fewer worst-case (negativework-outputted) trials. Both the left-hand gap δ and the right-hand gap δ 0 traversed by S̃ are Poissondistributed. Poisson-distributed gaps more likely assume extreme values than (E) (E) GOE-distributed gaps: PMBL (δ) > PGOE (δ) if δ ∼ 0 or δ  hδi [44]. The left-hand gap δ traversed by S is GOE-distributed. Hence the Wtot outputted by S̃ more likely assumes extreme values than the Wtot outputted by S. The greater reliability of S may suit S better to “one-shot statistical mechanics” [8, 25, 42, 43, 52, 55, 74, 75, 83, 91, 97, 210]. In one-shot theory, predictability of the work Wtot extractable in any given trial serves as a resource. Additionally, S suffers fewer worst-case trials than S̃. We define as worst-case a trial in which the engine outputs net negative work, Wtot < 0. Consider again Fig. 6.2. Consider a similar figure that depicts the trapezoid traversed by S̃ in some trial. The left-hand gap, δ, is distributed as the right-hand gap, (E) δ 0 , is, according to PMBL (δ). Hence δ has a decent chance of being smaller than δ 0 : δ < δ 0 . S̃ would output Wtot < 0 in such a trial. Suppose, for simplicity, that TH = ∞ and TC = 0. The probability that any given S trial outputs Wtot < 0 is pworst ≈ (Prob. that the left-hand gap < the right-hand gap) (8.1) × (Prob. that the right-hand gap is small enough to be cold-thermalized) Wb . (8.2) ≈ (Prob. that the left-hand gap < Wb ) × hδi (E) The initial factor is modeled by the area of a region under the PGOE (δ) curve. The region stretches from δ = 0 to δ = Wb . We approximate the region as a Wb − π4 (Wb )2 /hδi2 Wb triangle of length Wb and height π2 hδi ∼ hδi 2 e 2 , [δ ≈ Wb , Eq. (5.2),  2 Wb 1 π Wb b and hδi  1]. The triangle has an area of 2 · Wb · 2 hδi2 ∼ W . Substituting hδi 147 into Eq. (8.2) yields pworst ∼  Wb hδi 3 . (8.3) Let p̃worst denote the probability that any given S̃ trial outputs Wtot < 0. p̃worst shares the form of Eq. (8.2). The initial factor approximates to the area of a (E) region under the PMBL (δ) curve. The region extends from δ = 0 to δ = Wb . (E) 1 . Combining the The region resembles a rectangle of height PMBL (0) ≈ hδi Wb rectangle’s area, hδi , with Eq. (8.2) yields p̃worst ∼  Wb hδi 2 . (8.4) b Since W  1, pworst  p̃worst .1 hδi 8.1.3 Quantum-dot engine MBL has been modeled with quasilocal bits [34, 93]. A string of ideally independent bits or qubits, such as quantum dots, forms a natural competitor. A qubit Otto engine’s gap is shrunk, widened, and shrunk [11, 61, 67, 68, 88]. In addition to the order-of-magnitude analysis below, we make two points about implementations’ practicality. First, the MBL potential’s generic nature offers an advantage. MBL requires a random disorder potential {h(αt )hj }, e.g., a “dirty sample,” a defect-riddled crystal. This “generic” potential contrasts with the pristine background required by quantum dots. Imposing random MBL disorder is expected to be simpler. On the other hand, a quantum-dot engine does not necessarily need a small-bandwidth cold bath, Wb  hδi. 8.1.4 Anderson-localized engine Anderson localization follows from removing the interactions from MBL. One could implement our Otto cycle with an Anderson insulator because Anderson Hamiltonians exhibit Poissonian level statistics (5.1). But strokes 1 and 3 would require the switching off and on of interactions. Tuning the interaction, as well as the disorder-to-interaction ratio, requires more effort than tuning just the latter. 1 The discrepancy is exaggerated if the exponent in Eq. (8.3) rises, if the left-hand S Hamiltonian is modeled with a Gaussian ensemble other than the GOE. The Gaussian unitary ensemble (GUE) contains an exponent of 4; the Gaussian symplectic ensemble (GSE), an exponent of 6. Different ensembles model different symmetries. 148 Also, particles typically interact in many-body systems. MBL particles interact; Anderson-localized particles do not. Hence one might eventually expect less difficulty in engineering MBL engines than in engineering Andersonlocalized engines. 8.2 Order-of-magnitude estimates Localized engine: Localization has been achieved in solid-state systems.2 Consider silicon doped with phosphorus [124]. A distance of ∼ 10 nm may separate phosphorus impurities. Let our engine cycle’s shallowly localized regime have a localization length of ξ> ∼ 10 sites, or 100 nm. The workoutputting degrees of freedom will be electronic. The localized states will correspond to energies E ∼ 1 eV. Each subengine’s half-filling Hilbert space
has dimensionality N√ = 10 ∼ 102 . Hence each subengine has an effective 5 ∼ 10 meV. The cold-bath bandwidth must average gap hδi ∼ E NN ∼ 110eV 2 satisfy hδi  Wb . We set Wb to be an order of magnitude down from hδi: Wb ∼ 1 meV ∼ 10 K. The cold-bath bandwidth approximates the work outputted by one subengine per cycle:3 hWtot i ∼ Wb ∼ 1 meV [Eq. (6.7)]. What volume does a localized subengine fill? Suppose that the engine is threedimensional (3D).4 A little room should separate the subengines. Classicalcontrol equipment requires more room. Also, the subengine needs space to connect to the baths. We therefore associate each subengine with a volume of V ≈ (100 nm)3 . The last element needed is the cycle time, τcycle . We choose for δ− to be a little smaller than Wb —of the same order: δ− ∼ Wb ∼ 1 meV. In the extreme case 2 (10−15 eV s)(1 eV)2 ~E 2 allowed by Ineq. (7.17), τcycle ∼ Wb~E ∼ (W ∼ 1 µs. 3 ∼ (δ− )2 (1 meV)3 b) b The localized engine therefore operates with a power P ∼ τW ∼ 11meV ≈ µs cycle −16 10 W. Interestingly, this P is one order of magnitude greater than a flagellar motor’s [27] power, according to our estimates. 2 This localization is single-particle, or Anderson [12], rather than many-body. Sec. 8.1.4 extends the MBL Otto engine to an Anderson-localized Otto engine. 3 The use of semiconductors would require corrections to our results. (Dipolar interactions would couple the impurities’ spins. Energy eigenfunctions would decay as power laws with distance.) But we aim for just a rough estimate. 4 Until now, we have supposed that the engine is 1D. Anderson localization, which has been realized in semiconductors, exists in all dimensionalities. Yet whether MBL exists in dimensionalities D > 1 remains an open question. Some evidence suggests that MBL exists in D ≥ 2 [24, 37, 125]. But attributing a 3D volume to the engine facilitates comparisons with competitors. We imagine 10-nm-long 1D strings of sites. Strings are arrayed in a plane, separated by 10 nm. Planes are stacked atop each other, separated by another 10 nm. 149 We can assess the engine by calculating not only its power, but also its power 10−16 W 3 density. The localized engine packs a punch at VP ∼ (10 −7 m)3 = 100 kW/m . Car engine: The quintessential Otto engine powers cars. A typical car engine outputs P ∼ 100 horsepower ∼ 100 kW . A car’s power density is kW P ∼ 100 = 1 MW/ m3 (wherein L represents liters). The car engine’s VP V 100 L exceeds the MBL engine’s by only an order of magnitude, according to these rough estimates. Array of quantum dots: MBL has been modeled with quasilocal bits [34, 93]. A string of ideally independent bits or qubits, such as quantum dots, forms a natural competitor. A qubit Otto engine’s gap is shrunk, widened, and shrunk [11, 61, 67, 68, 88]. A realization could consist of double quantum dots [165, 166]. The scales in [165, 166] suggest that a quantum-dot engine could output an amount Wtot ∼ 10 meV of work per cycle per dot. We approximate the cycle time τcycle with the spin relaxation time: τcycle ∼ 1 µs. (The energy eigenbasis need not rotate, unlike for the MBL engine. Hence diabatic hops do not lower-bound the tot ∼ 101meV ∼ 10−15 W. ideal-quantum-dot τcycle .) The power would be P ∼ τWcycle µs The quantum-dot engine’s power exceeds the MBL engine’s by an order of magnitude. However, the quantum dots must be separated widely. Otherwise, they will interact, as an ETH system. (See [122] for disadvantages of interactions in another quantum thermal machine. Spin-spin couplings cause “quantum friction,” limiting the temperatures to which a refrigerator can cool.) We compensate by attributing a volume V ∼ (1 µm)3 to each dot. The power density becomes VP ∼ 1 kW/m3 , two orders of magnitude less than the localized engine’s. Localization naturally implies near independence of the subengines. 8.3 Outlook The realization of thermodynamic cycles with quantum many-body systems was proposed very recently [32, 62, 99, 130, 149, 153, 164, 211]. MBL offers a natural platform, due to its “athermality” and to athermality’s resourcefulness in thermodynamics. We designed an Otto engine that benefits from the discrepancy between many-body-localized and “thermal” level statistics. The engine illustrates how MBL can be used for thermodynamic advantage. Realizing the engine may provide a near-term challenge for existing experimental set-ups. Possible platforms include cold atoms [24, 37, 120, 148, 150 188]; nitrogen-vacancy centers [125]; ion traps [198]; and doped semiconductors [124], for which we provided order-of-magnitude estimates. Realizations will require platform-dependent corrections due to, e.g., variable-range hopping induced by particle-phonon interactions. As another example, semiconductors’ impurities suffer from dipolar interactions. The interactions extend particles’ wave functions from decaying exponentially across space to decaying as power laws. Reversing the engine may pump heat from the cold bath to the hot, lowering the cold bath’s temperature. Low temperatures facilitate quantum computation and low-temperature experiments. An MBL engine cycle might therefore facilitate state preparation and coherence preservation in quantum many-body experiments: A quantum many-body engine would cool quantum many-body systems. We have defined as work the energy outputted during Hamiltonian tunings. Some battery must store this energy. We have refrained from specifying the battery’s physical form, using an implicit battery model. An equivalent explicit battery model could depend on the experimental platform. Quantumthermodynamics batteries have been modeled abstractly with ladder-like Hamiltonians [197]. An oscillator battery for our engine could manifest as the mode of an electromagnetic field in cavity quantum electrodynamics. MBL is expected to have thermodynamic applications beyond this Otto engine. A localized ratchet, for example, could leverage information to transform heat into work. Additionally, the paucity of transport in MBL may have technological applications beyond thermodynamics. Dielectrics, for example, prevent particles from flowing in undesirable directions. But dielectrics break down in strong fields. To survive, a dielectric must insulate well—as does MBL. In addition to suggesting applications of MBL, this work identifies an opportunity within quantum thermodynamics. Athermal quantum states (e.g., ρ 6= e−H/T /Z) are usually regarded as resources in quantum thermodynamics [26, 43, 50, 69, 75, 77, 83, 91, 98, 140, 141, 229, 236, 237]. Not only athermal states, we have argued, but also athermal energy-level statistics, offer thermodynamic advantages. Generalizing the quantum-thermodynamics definition of “resource” may expand the set of goals that thermodynamic agents can achieve. Optimization offers another theoretical opportunity. We have shown that the engine works, but better protocols could be designed. For example, we pre- 151 scribe nearly quantum-adiabatic tunings. Shortcuts to adiabaticity (STA) avoid both diabatic transitions and exponentially slow tunings [2, 35, 51, 53, 122, 209]. STA have been used to reduce other quantum engines’ cycle times [2, 51, 53]. STA might be applied to the many-body Otto cycle, after being incorporated into MBL generally. 152 BIBLIOGRAPHY [1] ITensor Library (version 2.0.11) http://itensor.org. [2] O. Abah and E. Lutz. Performance of superadiabatic quantum machines. ArXiv e-prints, November 2016. [3] O. Abah, J. Roßnagel, G. Jacob, S. Deffner, F. Schmidt-Kaler, K. Singer, and E. Lutz. Single-ion heat engine at maximum power. Phys. Rev. Lett., 109:203006, Nov 2012. 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Physical Review Letters, 75(19):3537–3540, November 1995. INDEX AKLT Hamiltonian, x Anderson localization, 2 avalanche picture for MBL, 4 Gottesman-Knill theorem, xi bond dimension, 13 l-bit, 2, 4 canonical mixed, x canonical form left, 17 mixed, 19 right, 18 convergence false convergence, 93 convergence testing, 140 many-body localization , see MBLxi matrix product operator , see MPO 21 matrix product state, see MPSix, see MPS11 MBL, xi MPO, 21 MPS, ix, 11 density matrix renormalization group, see DMRG density matrix truncation, see DMT DMRG, ix DMT, xii orthogonality center, 19 eigenstate thermalization hypothesis, xi ETH, see eigenstate thermalization hypothesis extended Church-Turing thesis, xi resonant region, 4 isometry, 12 prethermalization timescale, 61, 64 prethermalization Hamiltonian, 59 Schmidt decomposition, 11 singular value decomposition, see SVD SPT, see symmetry protected topological phase subdiffusion, xi SVD, 11 symmetry protected topological phase, x Floquet drive, 59, 61 heating, 59 timescale, 61–64 free fermion model, 93 173