Exact Bosonization in All Dimensions: the Duality Between Fermionic and Bosonic Phases of Matter Citation Chen, Yu-An (2020) Exact Bosonization in All Dimensions: the Duality Between Fermionic and Bosonic Phases of Matter. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/593v-5r52. https://resolver.caltech.edu/CaltechTHESIS:06082020-083409184 Abstract We describe an n-dimensional (n≥2) analog of the Jordan-Wigner transformation, which maps an arbitrary fermionic system to Pauli matrices while preserving the locality of the Hamiltonian. When the space is simply-connected, this bosonization gives a duality between any fermionic system in arbitrary n spatial dimensions and a new class of (n-1)-form Z₂ gauge theories in n dimensions with a modified Gauss’s law. We describe several examples of 2d bosonization, including free fermions on square and honeycomb lattices and the Hubbard model, and 3d bosonization, including a solvable Z₂ lattice gauge theory with Dirac cones in the spectrum. This bosonization formalism has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the manifold, a key feature for defining fermions. A new formula for Stiefel-Whitney homology classes on lattices is derived. We also derive the Euclidean actions for the corresponding lattice gauge theories from the bosonization. The topological actions contain Chern-Simons terms for (2+1)D or Steenrod Square terms for general dimensions. Finally, we apply the bosonization to construct various bosonic or fermionic symmetry-protectedtopological (SPT) phases. It has been shown that supercohomology fermionic SPT phases are dual to bosonic higher-group SPT phases. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: bosonization; symmetry protected topological phases Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Kapustin, Anton N. Thesis Committee: Chen, Xie (chair) Kapustin, Anton N. Alicea, Jason F. Motrunich, Olexei I. Defense Date: 25 May 2020 Funders: Funding Agency Grant Number US Department of Energy, Office of Science, Office of High Energy Physics DE-SC0011632 Record Number: CaltechTHESIS:06082020-083409184 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06082020-083409184 DOI: 10.7907/593v-5r52 Related URLs: URL URL Type Description https://doi.org/10.1016/j.aop. 2018.03.024 DOI Article adapted for ch. 2 https://doi.org/10.1103/PhysRevB.100.245127 DOI Article adapted for ch. 3 https://arxiv.org/abs/1911.00017 arXiv Article adapted for ch. 4 ORCID: Author ORCID Chen, Yu-An 0000-0002-8810-9355 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 13788 Collection: CaltechTHESIS Deposited By: Yu An Chen Deposited On: 09 Jun 2020 19:34 Last Modified: 01 Jul 2025 16:19 Thesis Files Preview PDF - Final Version See Usage Policy. 1MB Repository Staff Only: item control page

Exact bosonization in all dimensions: the duality between fermionic and bosonic phases of matter Thesis by Yu-An Chen In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2020 Defended May 25, 2020 ii © 2020 Yu-An Chen ORCID: 0000-0002-8810-9355 All rights reserved iii ACKNOWLEDGEMENTS My time at Caltech has been exciting and rewarding. It is a pleasure to work with so many talented and friendly people. First, I want to thank my advisor Prof. Anton Kapustin. His deep mathematical knowledge and clear physical intuition always fascinate me. I wouldn’t have been able to build my solid background in both physics and mathematics without his guidance. I thank him for teaching the way of thinking on dealing with the most abstract and hardest problems, especially as a pioneer in our field. To complete my Ph.D. thesis, I needed to explore and combine different fields. My ability was nurtured with his helpful advising. He also gave me a lot of freedom to study everything I am interested in. With his support, I was able to be an intern in the Google Quantum AI team, which developed the latest quantum computers. The best decision I have ever made at Caltech was to choose him as my advisor. I am also influenced by many other professors at Caltech. I am grateful to Prof. Xie Chen for her advice on teaching, research, and life. I was lucky to be a TA in her course. During the quarter, I learned how to organize a course and to give excellent lectures. She shared her opinion on my future directions as well, and this helped my career. She is very kind and makes me feel like she is my “elder sister”. I am inspired by Prof. Alexei Kitaev. By taking his classes and talking with him during lunch, I learned many things from him. I admire his vision of physics and his continually pioneering works. It is also my honor to have many conversations with Prof. Jason Alicea and Prof. Olexei Motrunich. They gave me my first guidance in the field of condensed matter theory. My Ph.D. life wouldn’t have been enjoyable without meeting many students and postdocs at Caltech. I first thank Hsiao-Yi Chen for many discussions and for helping me get through my dark first year. I thank group members Alex Turzillo and Minyoung You for collaboration. As the latest one to join this small group, I am greatly enlightened by them. I also thank Tyler Ellison and Nathanan Tantivasadakarn for the wonderful collaborating. The project represents the most important work I have completed in my last two years. Finally, I had stimulating discussions with many people, and I appreciate their help during my Ph.D.: Boqiang Shen, Ying-Hsuan Lin, Po-Shen Hsin, Bowen Yang, Zitao Wang, Baoyi Chen, Yunxuan Li, Kuan-Yu Lin, Hung-I Yang, Jayden Ooi, Junyu Liu, Wilbur Shirley, Pouria Dadras, Djordje iv Radicevic, Wenjie Ji, Yu-Ping Lin, and Su-Kuan Chu. Lastly, I thank my parents for their enormous support and love for me. They make me pursue my dream without any pressure. v ABSTRACT We describe an 𝑛-dimensional (𝑛 ≥ 2) analog of the Jordan-Wigner transformation, which maps an arbitrary fermionic system to Pauli matrices while preserving the locality of the Hamiltonian. When the space is simply-connected, this bosonization gives a duality between any fermionic system in arbitrary 𝑛 spatial dimensions and a new class of (𝑛 −1)-form Z2 gauge theories in 𝑛 dimensions with a modified Gauss’s law. We describe several examples of 2d bosonization, including free fermions on square and honeycomb lattices and the Hubbard model, and 3d bosonization, including a solvable Z2 lattice gauge theory with Dirac cones in the spectrum. This bosonization formalism has an explicit dependence on the second StiefelWhitney class and a choice of spin structure on the manifold, a key feature for defining fermions. A new formula for Stiefel-Whitney homology classes on lattices is derived. We also derive the Euclidean actions for the corresponding lattice gauge theories from the bosonization. The topological actions contain Chern-Simons terms for (2 + 1)D or Steenrod Square terms for general dimensions. Finally, we apply the bosonization to construct various bosonic or fermionic symmetry-protectedtopological (SPT) phases. It has been shown that supercohomology fermionic SPT phases are dual to bosonic higher-group SPT phases. Keywords: bosonization, lattice gauge theory, spin systems, SPT phases Thesis Supervisor: Anton Kapustin vi PUBLISHED CONTENT AND CONTRIBUTIONS [1] Yu-An Chen. Exact bosonization in arbitrary dimensions, 2019. URL https: //arxiv.org/abs/1911.00017. Yu-An Chen finished this paper alone. [2] Yu-An Chen and Anton Kapustin. Bosonization in three spatial dimensions and a 2-form gauge theory. Phys. Rev. B, 100:245127, Dec 2019. doi: 10. 1103/PhysRevB.100.245127. URL https://link.aps.org/doi/10.1103/ PhysRevB.100.245127. Yu-An Chen developed the explicit formula for 3d bosonization, calculated examples, derived the spacetime action, drew figures, and participated in the writing of the manuscript. [3] Yu-An Chen, Anton Kapustin, and Djordje Radicevic. Exact bosonization in two spatial dimensions and a new class of lattice gauge theories. Annals of Physics, 393:234 – 253, 2018. ISSN 0003-4916. doi: https://doi.org/10.1016/j.aop. 2018.03.024. URL http://www.sciencedirect.com/science/article/ pii/S0003491618300873. Yu-An Chen developed the explicit formula for 2d bosonization, derived the spacetime action, drew figures, and participated in the writing of the manuscript. [4] Yu-An Chen, Anton Kapustin, Alex Turzillo, and Minyoung You. Free and interacting short-range entangled phases of fermions: Beyond the tenfold way. Phys. Rev. B, 100:195128, Nov 2019. doi: 10.1103/PhysRevB.100.195128. URL https://link.aps.org/doi/10.1103/PhysRevB.100.195128. YuAn Chen calculated the indexes for low dimensional models and participated in the writing of the manuscript. vii TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii v vi vi vi 1 I Bosonization 8 Chapter II: Bosonization on two-dimensional lattices . . . . . . . . . . . . . 2.1 Square lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Euclidean 3D gauge theories and their fermionic duals . . . . . . . . Chapter III: Bosonization on three-dimensional lattices . . . . . . . . . . . . 3.1 Cubic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Euclidean 3+1D gauge theories with fermionic duals . . . . . . . . . 3.5 Gauging fermion parity . . . . . . . . . . . . . . . . . . . . . . . . Chapter IV: Exact bosonization in 𝑛 dimensions . . . . . . . . . . . . . . . . 4.1 Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Modified Gauss’s law and Euclidean action . . . . . . . . . . . . . . 9 9 12 17 23 27 27 29 34 36 42 45 45 47 II Constructions of SPT phases 52 Chapter V: (2+1)D SPT constructions . . . . . . . . . . . . . . . . . . . . . 5.1 Review of group cohomology bSPT constructions . . . . . . . . . . 5.2 Correspendence between bSPT and supercohomology fSPT . . . . . Chapter VI: (3+1)D constructions . . . . . . . . . . . . . . . . . . . . . . . 6.1 2-group, 2-group extension, and 2-gauge theory . . . . . . . . . . . 6.2 Auxiliary “2-group” bSPT . . . . . . . . . . . . . . . . . . . . . . . 6.3 Fermionic SPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter VII: Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Chains and cochains notations and (higher) cup products on a triangulation and a cubic lattice . . . . . . . . . . . . . . . . . . . . . . . Appendix B: A formula for Stiefel-Whitney homology classes . . . . . . . . . Appendix C: Identity for fermionic algebra . . . . . . . . . . . . . . . . . . . Appendix D: Gauging the symmetries . . . . . . . . . . . . . . . . . . . . . 53 53 54 59 59 62 66 69 70 75 78 81 viii Appendix E: Derivation of 𝜁 (first descendant of 𝜌) . . . . . . . . . . . . . . 84 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 1 Chapter 1 INTRODUCTION The first part of this thesis is bosonization in all dimensions. It is well known that the Jordan-Wigner transformation establishes an equivalence between the quantum Ising chain in a transverse magnetic field and a system of free spinless fermions on a 1d lattice. This equivalence is very useful and is the quickest way to solve the 2D Ising model.1 The use of the Jordan-Wigner transformation is not limited to the quantum Ising chain. It establishes a very general kinematic equivalence between 1d fermionic systems and 1d spin chains with a Z2 spin symmetry. One can regard it as a very special isomorphism between the algebra of fermionic observables with trivial fermion parity and the Z2 -even subalgebra of the algebra of observables in a spin chain. Its distinguishing feature is that it maps local observables2 with trivial fermion parity on the fermionic side to local observables that commute with the total spin parity Ö 𝑧 𝑧 (−1) 𝑆 = (−1) 𝑆𝑖 𝑖 on the bosonic side (𝑆𝑖𝑧 = 0, 1). Thus any local Hamiltonian for a 1d fermionic system can be mapped to a local spin chain Hamiltonian which preserves 𝑆 𝑧 modulo 2. In this sense, the Jordan-Wigner transformation is local. There are several ways to generalize the Jordan-Wigner transformation to 2d lattice systems. One obvious approach is to take a square lattice, represent it as a 1d system by picking a lattice path that snakes through the whole lattice and visits each site once, and apply the 1d Jordan-Wigner transformation. This leads to a bosonization map which maps some, but not all, local observables with a trivial fermion parity to local observables with trivial 𝑆 𝑧 . The lack of 2d locality causes problems since even very simple fermionic Hamiltonians are mapped to non-local spin Hamiltonians and vice versa. But there are interesting exceptions [1], the Kitaev honeycomb model [2] being among them [3–5]. Another approach to 2d bosonization is to use flux attachment [6, 7]: a fermion is represented by a boson interacting with a ChernSimons gauge field. Related ideas in the continuum have been the focus of much recent interest [8–19]. However, it is hard to make this precise on the lattice, due 1 2D here means “two Euclidean space-time dimensions.” 2 That is, observables that act nontrivially on a finite number of lattice sites. 2 to well-known difficulties with defining a lattice Chern-Simons theory. A popular strategy is to eliminate the Chern-Simons gauge field by solving its equations of motion, but this again leads to a non-local map. A very interesting example of exact 2d bosonization, or rather fermionization, was presented by A. Kitaev in his paper on the honeycomb model [2]. At first it appears quite special, but in fact, it provides a method for mapping an arbitrary system of Majorana fermions on a trivalent lattice to a system of bosonic spins on the same lattice. The spin Hilbert space is not a tensor product over all sites, but rather a subspace defined by a set of commuting constraints. There is one constraint for each face of the lattice, indicating that the dual bosonic system is a gauge theory. But it is a very unusual gauge theory since the gauge field is a composite of spins. The starting point of this paper is to describe a 2d analog of the Jordan-Wigner transformation, which obeys locality, and to give some examples of 2d bosonization. We will show that any 2d fermionic system on a lattice can be mapped to a system of bosons. On a topologically trivial space, this map is an equivalence, and every local fermionic Hamiltonian is mapped to a local spin Hamiltonian. The main novelty compared to the 1d case is that the bosonic system is a Z2 gauge theory. This means that the Hilbert space is not a tensor product of local Hilbert spaces, but a subspace in such a tensor product defined by a set of commuting local constraints. They can be interpreted as Gauss law constraints. Our bosonization procedure shares some similarities both with the flux-attachment approach and with Kitaev’s approach. It follows the same strategy as the fluxattachment approach but uses a lattice Z2 gauge field in place of a 𝑈 (1) ChernSimons gauge field. There is no problem writing down a Chern-Simons-like term for a Z2 gauge field. An additional benefit is that we do not need to introduce separate bosonic degrees of freedom to which the flux is attached: we make use only of degrees of freedom that are already present in the gauge field. Our bosonization procedure is completely general and local, just like in Kitaev’s approach, but there are a couple of differences too: (1) the fermions are complex rather than Majorana, so that the fermionic Hilbert space is naturally a tensor product over all sites; (2) the bosonic variables live on edges rather than on sites, and the gauge field is fundamental rather than composite. The connection between gauge symmetry and bosonization rests on the observation made in [20] that 2d bosonization should map fermionic systems to bosonic systems with a global 1-form Z2 symmetry and a suitable ’t Hooft anomaly. On a lattice, 3 global 1-form symmetries can exist only in gauge theories. The proposal of [20] was made concrete in [21] for topological systems (that is, spin-TQFTs), but it was implicit in that paper that the same strategy should apply for general fermionic systems on a lattice. In this thesis, we make this completely explicit. Namely, we show that on a simply-connected space one can isomorphically map the bosonic subalgebra of the algebra of local fermionic observables to the algebra of local gauge-invariant observables in a suitable Z2 lattice gauge theory. The bosonization map is not canonical and depends on some additional choices which depend on the type of lattice. We discuss two kinds of lattices in 𝑛-dimension: the 𝑛-dimensional cubic lattice and an arbitrary triangulation endowed with a branching structure. In either case, the fermions live on the 𝑛-cells, i.e., fermions at faces for 2d lattices. Gauss law constraints for gauge theories dual to fermionic systems are not standard. Their meaning becomes clearer if we discretize time and consider the corresponding Euclidean lattice partition function. It turns out that the unusual Gauss law arises ∫ from a Chern-Simons-like term 𝑖𝜋 𝐴 ∪ 𝛿 𝐴 in the spacetime action. These terms necessarily break invariance under the cubic symmetry (if one starts from a 2d square lattice). This approach can be generalized to 3d [22]. Every fermionic lattice system in 3d is dual to a Z2 2-form gauge theory with an unusual Gauss law. Here “2form gauge theory” means that the Z2 variables live on faces (2-simplices), while the parameters of the gauge symmetry live on edges (1-simplices). 2-form gauge theories in 3+1D have local flux excitations, and the unusual Gauss law ensures that these excitations are fermions. This Gauss’s law can be described by the "Steenrod ∫ square" topological action 𝑖𝜋 𝐵 ∪ 𝐵 + 𝐵 ∪1 𝛿𝐵. The form of the modified Gauss law was first observed in [20]: a bosonization of fermionic systems in 𝑛 dimensions must have a global (𝑛 − 1)-form Z2 symmetry with a particular ’t Hooft anomaly. The standard Gauss law leads to a trivial ’t Hooft anomaly, so bosonization requires us to modify it in a particular way. We further extend the result to arbitrary 𝑛 dimensions. We show that every fermionic lattice system in 𝑛-dimension is dual to a Z2 (𝑛 − 1)-form gauge theory with a modified Gauss law. Our bosonization map is kinematic and local in the same sense as the Jordan-Wigner map: every local observable on the fermionic side, including the Hamiltonian density, is mapped to a local gauge-invariant observable on the Z2 gauge theory side. In the Euclidean picture, we show explicitly that our bosonization 4 map is equivalent to introducing the topological term in the action: ∫ 𝑆top = 𝑖𝜋 ( 𝐴𝑛−1 ∪𝑛−3 𝐴𝑛−1 + 𝐴𝑛−1 ∪𝑛−2 𝛿 𝐴𝑛−1 ), (1.1) 𝑌 where 𝐴𝑛−1 is (𝑛 − 1)-form gauge fields, a (𝑛 − 1)-cochain 𝐴𝑛−1 ∈ 𝐶 𝑛−1 (𝑌 , Z2 ), and 𝑌 is (𝑛 + 1)-dimensional spacetime manifold. When 𝐴𝑛−1 is closed (a cocycle), this term reduces to the Steenrod square operator [23]. This “Steenrod square” term appears in the construction of fermionic symmetry-protected-topological phases [24]. The second part of the thesis is about constructions of symmetry protected topological (SPT) phases. A major goal in condensed matter is the classification of phases given a certain symmetry. For quantum phases of matter, there are a class of gapped symmetric phases which have a unique ground state on a closed manifold known as symmetry protected topological (SPT) phases [25, 26]. In studying these phases, there have been two main programs that have evolved in tandem. The first is how to classify such phases given a symmetry group 𝐺. There are two main classes of phases. One is where the constituents are purely bosonic (bSPTs), and those that have physical fermions as excitations (fSPTs). For an onsite and unitary symmetry group, interacting bSPTs and fSPTs, are now believed to be classified by oriented cobordism [27] and spin cobordism [28] respectively. These classifications can also be obtained from a generalized cohomology theory [29–31]. The second is to construct Hamiltonians which realizes these phases. For bosonic SPTs, a successful class of models have been constructed using group cohomology[32]. Outside of this class, though they are predicted to exist using field theory arguments[30, 33] only a few phases “beyond” group cohomology have been constructed as exactly solvable lattice models[34–36]. For fermionic SPTs, a large class of models can be realized via free fermion models[37–39]. However, there are also certain interacting models that have no free-fermion counterpart[40]. In this thesis, we consider a large class of models that realize interacting fSPTs known as group supercohomology SPTs [41]. Using input data that characterize these models, we construct exactly solvable Hamiltonians that realize these phases and also construct their symmetric gapped boundaries, which exhibit topological order. The construction of exactly solvable Hamiltonians for fSPTs in (2+1)D [42, 43] and (3+1)D will be shown explicitly in this paper. The construction exploits a duality which relates the ground state of the fSPT to the ground state of a certain auxillary bSPT. In particular, gauging the Z2 (𝑛 − 2)-form symmetry of 5 this auxiliary bSPT (in 𝑛 spatial dimensions) gives rise to a 𝑛 − 1-form gauge theory with emergent fermions. We then use the boson-fermion duality developed in Part I to obtain the wavefunction and Hamiltonian for the fSPT in 𝑛 spatial dimensions. This picture is summarized in Figure 1.1. We remark that although there are previous works which construct exactly-solvable fSPT phases, some of which are outside supercohomology [21, 41, 44–49], our construction of fSPT phases gives an expression for the disentangler for the ground state wavefunction. Specifically, in the absence of symmetry, we are able to explicitly trivialize the fSPT wavefunction using a Finite Depth Quantum Circuit (FDQC) which commutes with the symmetry as a whole. Furthermore, the group structure (or stacking rule) of the fSPT phases can be obtained explicitly by a composition of disentangling unitaries. Having an explicit disentangler also allows us to construct a Hamiltonian realizing the phase where every eigenstate is itself an SPT, and therefore allowing the Hamiltonian to be many-body localized by introducing disorder. 𝑓 Figure 1.1: To construct a 𝐺 𝑓 = 𝐺 × Z2 supercohomology SPT model in 𝑛 spatial dimensions, we start with a model for a particular (𝑛 − 1)-group SPT phase determined by the supercohomology data (𝜌, 𝜈). Next, we gauge the Z2 (𝑛 − 2)-form symmetry of the (𝑛 − 1)-group to build the shadow model. We then condense the fermion in the shadow model, or apply the fermionization duality, to obtain a model for the supercohomology SPT phase corresponding to (𝜌, 𝜈). This thesis is organized as follows. In Part I, we construct the 2d bosonization map on the kinematic level, i.e. on the level of the algebras of observables in Section 2. We give several examples of bosonization for concrete fermionic systems, such as free fermions and the Hubbard model. Conversely, we describe the fermionization of the simplest lattice gauge theories with a non-standard Gauss law, which are dual to free fermionic theories and thus are integrable. The Euclidean partition 6 function for the gauge theories is derived in Section 2.4. In Section 3, we explicitly construct the 3d bosonization in both cubic lattice and general triangulation, which utilizes the mathematical tools in algebraic topology: higher cup products. As in the 2d case, many examples and the spacetime action are discussed. The complete generalization for the bosonization in all dimensions is derived in Section 4. The formalism comes from comparing the formula between 2d and 3d bosonization, which can be extended to their general form directly. In Part II, we begin by reviewing the construction of (2+1)D bSPTs [32] and supercohomology fSPTs [42] in Section 5. In Section 5.2, we construct an exactly solvable lattice Hamiltonian and the corresponding groundstate wavefunction of a fermionic SPT protected by an onsite finite unitary group 𝐺. We achieve this by using the supercohomology data to construct an auxiliary bSPT protected by a ˜ which is an extension of the 𝐺 symmetry by a Z2 symmetry. The symmetry 𝐺, fSPT is obtained by first gauging the Z2 symmetry followed by fermionization. The ground state wavefunction has an explicit disentangling circuit, which trivializes the SPT phase in the absence of symmetry. We use the circuit to derive the stacking rule for supercohomology phases. In Section 6, similar scheme applies to (3+1)D fSPTs. The only difference is that the auxiliary bSPT is now protected by a 2group symmetry[50], which is an extension of a 0-form 𝐺 symmetry by a 1-form Z2 symmetry. In Section 6.1, we review the basic properties for 2-group, 2-group extension, and 2-gauge theory. In Section 6.2, we use these concepts to construct a “2-group” bSPT. After gauging the 1-form symmetry, it becomes a Z2 gauge theory, which is fermionizable. In Section 6.3, the fSPT wavefunction and its Hamiltonian are derived. Notations and Coventions This subsection introduces the notations and conventions adopted for this thesis. The more details for algebraic topology and the definition of (higher) cup products is included in Appendix A. In this paper, we will always work with an arbitrary triangulation of a simply-connected 𝑛-dimensional manifold 𝑀𝑛 equipped with a branching structure (orientations on edges without forming a loop in any triangle). The vertices, edges, faces, and tetrahedra are denoted 𝑣, 𝑒, 𝑓 , 𝑡, respectively. The general 𝑑-simplex is denoted as Δ 𝑑 . We can label the vertices of Δ 𝑑 as 0, 1, 2, . . . , 𝑑 such that the directions of edges are from the small number to the larger number. We denote this 𝑑-simplex as Δ 𝑑 = h01 . . . 𝑑i. Its boundaries are (𝑑 − 1)-simplices h0, . . . , 𝑖ˆ, . . . , 𝑑i for 𝑖 = 0, 1, . . . , 𝑑, where 𝑖ˆ means 𝑖 is omitted. A formal sum of 7 𝑑-simplices modulo 2 forms an element of the chain 𝐶𝑑 (𝑀𝑛 , Z2 ). For every 𝑣, we define its dual 0-cochain 𝒗, which takes value 1 on 𝑣, and 0 otherwise, i.e. 𝒗(𝑣 0) = 𝛿𝑣,𝑣 0 . Similarly, e is an 1-cochain 𝒆(𝑒0) = 𝛿 𝑒,𝑒 0 , and so forth, i.e. 𝚫𝑑 being a 𝑑-cochain 𝚫𝑑 (Δ 0𝑑 ) = 𝛿Δ 𝑑 ,Δ 𝑑0 . All dual cochains will be denoted in ∫ bold. An evaluation of a cochain 𝒄 on a chain 𝑐0 will sometimes be denoted 𝑐 0 𝒄. When the integration range is not written, 𝒄 is assumed to be the top dimension and ∫ ∫ 𝒄 ≡ 𝑀 𝒄. A 𝑑-cochain 𝒄 𝑑 ∈ 𝐶 𝑑 (𝑀𝑛 , Z2 ) can be identified as Z2 fields living on 𝑛 each 𝑑-simplex Δ 𝑑 , with the value 𝒄 𝑑 (Δ 𝑑 ). The cup product ∪ of a 𝑝-cochain 𝛼 𝑝 and a 𝑞-cochain 𝛽𝑞 is a ( 𝑝 + 𝑞)-cochain defined as [𝛼 𝑝 ∪ 𝛽𝑞 ] (h0, 1, . . . , 𝑝 + 𝑞i) = 𝛼 𝑝 (h01 . . . 𝑝i) 𝛽𝑞 (h𝑝, 𝑝 + 1, . . . , 𝑝 + 𝑞i) = 𝛼 𝑝 (0 ∼ 𝑝) 𝛽𝑞 ( 𝑝 ∼ 𝑝 + 𝑞). (1.2) The definition of the higher cup product [20, 23] is [𝛼 𝑝 ∪𝑎 𝛽𝑞 ] (0, 1, · · · , 𝑝 + 𝑞 − 𝑎) = Õ 𝛼 𝑝 (0 ∼ 𝑖0 , 𝑖1 ∼ 𝑖2 , 𝑖3 ∼ 𝑖4 , · · · ) × 𝛽𝑞 (𝑖0 ∼ 𝑖1 , 𝑖2 ∼ 𝑖3 , · · · ), (1.3) 0≤𝑖0 <𝑖 1 <···<𝑖 𝑎 ≤ 𝑝+𝑞−𝑎 where 𝑖 ∼ 𝑗 represents the integers from 𝑖 to 𝑗, i.e. 𝑖, 𝑖 + 1, . . . , 𝑗, and {𝑖0 , 𝑖1 , . . . , 𝑖 𝑎 } are chosen such that the arguments of 𝛼 𝑝 and 𝛽𝑞 contain 𝑝 + 1 and 𝑞 + 1 numbers separately. The boundary operator is denoted by 𝜕. For an 𝑛-simplex Δ 𝑛 , 𝜕Δ 𝑛 consists of all boundary (𝑛 − 1)-simplices of Δ 𝑛 . The coboundary operator is denoted by 𝛿 (not to be confused with the Kronecker delta previously). On a 0-cochain 𝒗, 𝛿𝒗 is an 1-cochain acting on edges, and is 1 if 𝜕𝑒 contains 𝑣 and 0 otherwise: 𝛿𝒗(𝑒) = 𝒗(𝜕𝑒) = 𝛿𝑣,𝜕𝑒 . It is similar for simplices in any dimension. Finally, Δ 1𝑛 ⊃ Δ 2𝑛 0 or Δ 2𝑛 0 ⊂ Δ 1𝑛 means that the simplex Δ 1𝑛 contains Δ 2𝑛 0 as a subsimplex. A general rule of thumb is that the subset symbol always points to one higher dimension. Part I Bosonization 8 9 Chapter 2 BOSONIZATION ON TWO-DIMENSIONAL LATTICES 2.1 Square lattice We first introduce our bosonization method on an infinite 2d square lattice. Suppose that we have a model with fermions living at the centers of faces. Let us describe the generators and relations in the algebra of local observables with trivial fermion parity (the even fermionic algebra for short). On each face 𝑓 we have a single fermionic creation operator 𝑐 𝑓 and a single fermionic annihilation operator 𝑐†𝑓 with the usual anticommutation relations. The fermionic 𝑐† 𝑐 parity operator on face 𝑓 is 𝑃 𝑓 = (−1) 𝑓 𝑓 . It is a “Z2 operator” (i.e. it squares to 1). All operators 𝑃 𝑓 commute with each other. The even fermionic algebra is generated by these operators and the operators 𝑐†𝑓 𝑐 𝑓 0 , 𝑐 𝑓 𝑐 𝑓 0 , and their Hermitean conjugates, where 𝑓 and 𝑓 0 are two faces which share an edge. Overall, we get four generators for each edge and one generator for each face. In fact, one can make do with a single generator for each edge and a single generator for each face, provided we choose a consistent orientation of all faces and arbitrary orientations of all edges. We introduce Majorana fermions 𝛾 𝑓 = 𝑐 𝑓 + 𝑐†𝑓 , 𝛾 0𝑓 = (𝑐 𝑓 − 𝑐†𝑓 )/𝑖. (2.1) The algebra of Majorana fermions is {𝛾 𝑓 , 𝛾 𝑓 0 } = {𝛾 0𝑓 , 𝛾 0𝑓 0 } = 2𝛿 𝑓 , 𝑓 0 , {𝛾 𝑓 , 𝛾0𝑓 0 } = 0, (2.2) where {𝐴, 𝐵} = 𝐴𝐵 − 𝐵𝐴 is the anti-commutator. Then the operators 𝑃 𝑓 = −𝑖𝛾 𝑓 𝛾 0𝑓 and 𝑆 𝑒 = 𝑖𝛾 𝐿(𝑒) 𝛾 0𝑅(𝑒) are Z2 operators and generate the even algebra. Here 𝐿(𝑒) and 𝑅(𝑒) are faces to the left and to the right of the edge 𝑒 with respect to the chosen orientations.1 We will 1 That is, 𝐿(𝑒) is the face which induces the same orientation on 𝑒 as the given orientation of 𝑒, while 𝑅(𝑒) is the face which induces the opposite orientation. 10 7 8 c d 4 5 a 1 9 6 b 2 3 Figure 2.1: Bosonization on a square lattice requires constraints on vertices. refer to 𝑆 𝑒 as the hopping operator for edge 𝑒. It anticommutes with 𝑃 𝑓 if 𝑓 = 𝐿 (𝑒) or 𝑓 = 𝑅(𝑒) and commutes with all other 𝑃 𝑓 . Other relations depend on the choice of orientations. We will choose the usual (counterclockwise) orientation of the plane and point all horizontal edges to the east, and all vertical edges to the north; see Fig. 2.1. Then it is easy to see that 𝑆 𝑒 and 𝑆 𝑒 0 may fail to commute only if 𝑒 and 𝑒0 share a point and are perpendicular. If 𝑒 and 𝑒0 share a point and are perpendicular, then in the notation of Fig. 2.1 we have [𝑆56 , 𝑆58 ] = [𝑆25 , 𝑆45 ] = 0, {𝑆25 , 𝑆56 } = {𝑆58 , 𝑆45 } = 0. (2.3) In other words, 𝑆 𝑒 and 𝑆 𝑒 0 anticommute if 𝑒 and 𝑒0 issue from the same vertex and point either east and south, or north and west. They commute in all other cases. Additional relations emerge if we consider the product of four hopping operators corresponding to all edges issuing from a vertex. This corresponds to an operator taking a fermion full circle around the vertex. The resulting operator commutes with 𝑃 𝑓 for all 𝑓 and thus must be some function of these operators. Indeed, a short calculation shows that 𝑆58 𝑆56 𝑆25 𝑆45 = (𝑖𝛾𝑑 𝛾𝑐0 )(𝑖𝛾𝑐 𝛾𝑏0 )(𝑖𝛾𝑎 𝛾𝑏0 )(𝑖𝛾𝑑 𝛾𝑎0 ) = (𝑖𝛾𝑎0 𝛾𝑎 )(𝑖𝛾𝑐0 𝛾𝑐 ) (2.4) = 𝑃𝑎 𝑃𝑐 . It is clear intuitively and can be shown rigorously (see Appendix for a sketch of a proof) that these are all relations between our chosen generators if the lattice is infinite, or if it is finite but topologically trivial. 11 The dual description will consist of bosonic spins living on edges of the same lattice. 𝑦 The operators acting on each edge 𝑒 are Pauli matrices 𝜎𝑒𝑥 , 𝜎𝑒 , and 𝜎𝑒𝑧 . To reduce notation clutter, we denote them 𝑋𝑒 , 𝑌𝑒 , and 𝑍 𝑒 . " # " # " # 0 1 0 −𝑖 1 0 𝑋𝑒 = , 𝑌𝑒 = , 𝑍𝑒 = . (2.5) 1 0 𝑖 0 0 −1 This is the usual operator algebra of the toric code. We have two kinds of edges: edges oriented east and edges oriented north. If 𝑒 is oriented east (resp. north), let 𝑟 (𝑒) be the edge which points north (resp. east) and ends where 𝑒 begins. In the notation of Fig. 2.1, 𝑟 (𝑒 56 ) = 𝑒 25 , 𝑟 (𝑒 58 ) = 𝑒 45 . It will be useful to define the composite operator (2.6) 𝑈𝑒 = 𝑋𝑒 𝑍𝑟 (𝑒) . In the toric code language, 𝑈𝑒 is the operator moving the 𝜖-particle across edge 𝑒. We also define the “flux operator” at each face 𝑓 to be Ö 𝑊𝑓 = 𝑍𝑒 . (2.7) 𝑒⊂ 𝑓 Our bosonization map is defined as follows: 1. We identify the fermionic states |𝑃 𝑓 = 1i and |𝑃 𝑓 = −1i with bosonic states for which 𝑊 𝑓 = 1 and 𝑊 𝑓 = −1, respectively. This amounts to dualizing 𝑃 𝑓 = −𝑖𝛾 𝑓 𝛾 0𝑓 ←→ 𝑊 𝑓 . (2.8) 2. The fermionic hopping operator 𝑆 𝑒 is identified with 𝑈𝑒 defined above, 𝑆 𝑒 = 𝑖𝛾 𝐿 (𝑒) 𝛾 0𝑅(𝑒) ←→ 𝑈𝑒 . (2.9) All operator relations discussed above are preserved under this map. The only exception is the relation (2.4), which is absent on the bosonic side. Instead, the product 𝑆58 𝑆56 𝑆25 𝑆45 maps to Ö 𝑈58𝑈56𝑈25𝑈45 = 𝑊 𝑓𝑎 𝑋𝑒 . (2.10) 𝑒⊃𝑣 5 To get an algebra homomorphism, we must impose a constraint on the bosonic variables at vertex 5, namely Ö 𝑊 𝑓𝑐 𝑋𝑒 = 1. (2.11) 𝑒⊃𝑣 5 12 For a general vertex 𝑣, the constraint is 𝑊NE(𝑣) Ö 𝑋𝑒 = 1, (2.12) 𝑒⊃𝑣 where NE(𝑣) is the face northeast of 𝑣. We interpret this as a Gauss law for the bosonic system. The presence of the Gauss law means that we are dealing with a gauge theory. Since the constraint at each vertex is a Z2 operator, this is a Z2 gauge theory. The algebra of gauge-invariant observables on the bosonic side (i.e. the algebra of local observables commuting with all Gauss law constraints) is generated by operators 𝑈𝑒 and 𝑊 𝑓 , and there are no further relations between them apart from those which exist between 𝑆 𝑒 and 𝑃 𝑓 . Thus the above map is an isomorphism and defines a 2d version of the Jordan-Wigner transformation. The constraint (2.12) couples the electric charge at a vertex 𝑣 to the magnetic flux at face NE(𝑣). Thus our modified Gauss law implements charge-flux attachment, and it is not surprising that operators 𝑈𝑒 which move the flux behave as fermionic bilinears. Note also that the total fermion number operator2 𝐹= Õ1 2 𝑓 (1 + 𝑖𝛾 𝑓 𝛾 0𝑓 ) is mapped to the net magnetic flux Õ1 𝑓 2 (1 − 𝑊 𝑓 ). While the fermion number operator is ultra-local (it is a sum of operators each of which acts nontrivially only on fermions at a particular site), its bosonized version is not ultra-local. 2.2 Triangulation The bosonization method described above also works for any triangulation 𝑇 with a branching structure.3 The main idea of this approach was previously described in 2 Not to be confused with the fermion parity operator Î 𝑓 𝑃𝑓 . 3 A branching structure on a triangulation is an orientation for every edge such that for every face the oriented edges do not form an oriented loop. A branching structure specifies an ordering of vertices of every face: on each face 𝑓 there will be exactly one vertex (denoted 𝑓0 ) with two edges of 𝑓 oriented away from the vertex, one vertex ( 𝑓1 ) with one edge of 𝑓 entering it and one leaving it, and another vertex ( 𝑓2 ) with two edges of 𝑓 oriented towards it. 13 [21]. We will review this material first and then describe a general way to perform bosonization on a 2d triangulation. We assume again that we are given a global orientation and for an edge 𝑒 define 𝐿 (𝑒) and 𝑅(𝑒) to be the faces to the left and to the right of 𝑒, just as for the square lattice. On a face 𝑓 we have fermionic operators 𝑐 𝑓 , 𝑐†𝑓 , or equivalently a pair of Majorana fermions 𝛾 𝑓 , 𝛾0𝑓 . They are generators of a Clifford algebra. The fermion parity on face 𝑓 is 𝑃 𝑓 = −𝑖𝛾 𝑓 𝛾 0𝑓 . A Z2 fermionic hopping operator on an edge 𝑒 can again be defined by 𝑆 𝑒 = 𝑖𝛾 𝐿 (𝑒) 𝛾 0𝑅(𝑒) . The even fermionic algebra is generated by 𝑃 𝑓 and 𝑆 𝑒 for all faces and edges. The relations between them can also be described. Obviously, these operators are Z2 , and 𝑆 𝑒 anticommutes with 𝑃 𝑓 whenever 𝑒 ⊂ 𝑓 and commutes with it otherwise. The operators 𝑆 𝑒 and 𝑆 𝑒 0 sometimes commute and sometimes anticommute. To describe the commutation rule more precisely, it is convenient to use the cup product on Z2 1-cochains. Recall that a Z2 𝑝-cochain is a Z2 -valued function on 𝑝-simplices of the triangulation. In our case, 𝑝 can be 0, 1, or 2, corresponding to functions on vertices, edges, and faces respectively. The cup product of two 1-cochains is a 2-cochain defined as follows: (𝛼 ∪ 𝛽)(h012i) = 𝛼(h01i) 𝛽(h12i). Here 𝛼 and 𝛽 are arbitrary 1-cochains with values in Z2 , and 0, 1, and 2 are vertices of a face h012i, ordered in accordance with the branching structure. h01i (resp. h12i) is the edge from 0 to 1 (resp. from 1 to 2). The cup product is not commutative (or supercommutative, which is the same thing since we are working modulo 2). Let 𝒆 be the 1-cochain which takes value 1 on the edge 𝑒 and value 0 on all other edges. Then the commutation rule for 𝑆 𝑒 and 𝑆0𝑒 is 𝑆 𝑒 𝑆 = (−1) 𝑒0 ∫ 𝒆∪𝒆 0 +𝒆 0 ∪𝒆 𝑆𝑒0 𝑆𝑒 . (2.13) Here the integral of a 2-cochain is simply the sum of its values on all faces of the space manifold 𝑀: ∫ Õ 0 0 𝒆∪𝒆 +𝒆 ∪𝒆= (𝒆 ∪ 𝒆0 + 𝒆0 ∪ 𝒆). (2.14) 𝑓 ∈𝑀 In other words, if 𝑒 and 𝑒0 are distinct edges, 𝑆 𝑒 and 𝑆 𝑒 0 anticommute if 𝑒 and 𝑒0 belong the same face 𝑓 = h012i and their union contains edges h01i and h12i of that face, {𝑒, 𝑒0 } = {h01i, h12i}. They commute otherwise. 14 2 5 3 4 6 1 0 Figure 2.2: A branching structure on a general triangulation. There is also a relation for each vertex 𝑣, analogous to (2.4), which reads Ö Ö 𝑆𝑒 ∼ 𝑃𝑓, 𝑒⊃𝑣 (2.15) 𝑓 ⊃𝐼 𝑣02 where 𝐼𝑣02 is the set of those faces 𝑓 = h012i for which 𝑣 is either 0 or 2. The product of hopping terms equal to the product of fermion parity on these faces up to a sign. Its explicit form will be derived later. To reproduce these relations in a bosonic model, we again introduce a spin variable for every edge and let 𝑋𝑒 , 𝑌𝑒 , 𝑍 𝑒 be the corresponding Pauli matrices. We let Ö 𝑊𝑓 = 𝑍𝑒 , (2.16) 𝑒⊂ 𝑓 as before. This operator measures flux through face 𝑓 . We anticipate that the bosonic model will be a gauge theory, and thus the algebra of gauge-invariant observables will be generated by 𝑊 𝑓 and Z2 operators 𝑈𝑒 which anticommute with 𝑊 𝑓 if 𝑒 ⊂ 𝑓 and commute with 𝑊 𝑓 otherwise. We also anticipate that in order for 𝑈𝑒 to behave as fermion hopping operators, we must implement charge-flux attachment. Our convention will be that if 𝑊 𝑓 = −1 for some face 𝑓 = h012i, then electric charge will be sitting at the vertex 0. Then the flux hopping operator will take the form Ö 𝒆( 𝑓 ) (2.17) 𝑈𝑒 = 𝑋𝑒 𝑍 𝑓01 12 , 𝑓 ∈{𝐿(𝑒), 𝑅(𝑒)} where 𝑓𝑖 𝑗 denotes the edge of 𝑓 = h𝑖 𝑗 𝑘i connecting vertices 𝑓𝑖 and 𝑓 𝑗 . Eq. (2.17) means that 𝑈𝑒 implements the motion of the magnetic flux across edge 𝑒 accompanied by the electric charge moving along edges 𝑓01 of those faces for which 𝑒 = 𝑓12 . For example, in Figure 2.2, we have 𝑈35 = 𝑋35 𝑍23 𝑍13 , 𝑈13 = 𝑋13 𝑍01 , and 𝑈03 = 𝑋03 . 15 We can simplify Eq. (2.17) using the cup product notation: Ö ∫ 𝒆 0∪𝒆 ). 𝑈𝑒 = 𝑋𝑒 ( 𝑍𝑒 0 (2.18) 𝑒0 With this form, we can easily check that the operators 𝑈𝑒 satisfy the same commutation relations as 𝑆 𝑒 . Ö ∫ 𝒆 0 ∪𝒆 Ö ∫ 𝒆 0 ∪𝒆 1 2 1 𝑈𝑒 1 𝑈𝑒 2 = 𝑋𝑒 1 ( 𝑍𝑒 0 ) 𝑋𝑒 2 ( 𝑍𝑒 0 2 ) 1 = (−1) 𝑒 10 ∫ 𝒆 2 ∪𝒆 1 +𝒆 1 ∪𝒆 2 2 𝑒 20 𝑋𝑒 2 ( Ö ∫ 2 𝑒 20 = (−1) ∫ 𝒆 2 ∪𝒆 1 +𝒆 1 ∪𝒆 2 𝒆 0 ∪𝒆 2 𝑍𝑒 0 2 ) 𝑋𝑒 1 ( Ö 𝑒 10 ∫ 𝒆 0 ∪𝒆 1 𝑍𝑒 0 1 ) (2.19) 1 𝑈𝑒 2 𝑈𝑒 1 . One can check that if we impose a Gauss law of the form Ö Ö 𝑋𝑒 = 𝑊𝑓 , (2.20) 𝑓 ∈𝐼 0 (𝑣) 𝑒⊃𝑣 where 𝐼 0 (𝑣) is the set of faces such that 𝑣 = 𝑣 0 for that face, then 𝑈𝑒 satisfy a relation very similar to that of 𝑆 𝑒 : Ö Ö 𝑈𝑒 ∼ 𝑊𝑓 . (2.21) 𝑓 ⊃𝐼 𝑣02 𝑒⊃𝑣 The equality holds up to a 𝑣-dependent sign. To be precise about the sign, around each vertex 𝑣, we formulate the fermionic identity (2.15) precisely: ∫ Ö ∫ 𝒗∪ 𝒇 + 𝒇 ∪𝒗 𝒗 = 1, (2.22) (−1) 𝑤2 𝑆 𝛿𝒗 𝑃𝑓 𝑓 ∫ Î 𝒗∪ 𝒇 + 𝒇 ∪𝒗 where 𝑓 𝑃 𝑓 = 𝑓 ⊃𝐼 𝑣02 𝑃 𝑓 by the definition of the cup product and 𝑆 𝛿𝒗 is the product of hopping terms on edges around 𝑣, similar to (2.15). To avoid sign ambiguity due to the ordering of the product, we defined the hopping term for 1-cochains. In general, for any 1-cochains 𝝀 and 𝝀0, Î 𝑆 ∫ 𝜆+𝜆 0 ≡ (−1) ∫ 𝝀∪𝝀 0 𝑆𝜆 0 𝑆𝜆 . ∫ (2.23) For example, 𝑆 𝑒1 +𝑒2 = (−1) 𝒆2 ∪𝒆1 𝑆 𝑒1 𝑆 𝑒2 = (−1) 𝒆1 ∪𝒆2 𝑆 𝑒2 𝑆 𝑒1 , which is independent from the ordering between 𝑒 1 and 𝑒 2 . The sign is related to 𝑤 2 ∈ 𝐶0 (𝑀2 , Z2 ), which is the 0-chain (a formal sum of vertices) which is Poincaré dual to the second 16 Stiefel–Whitney cohomology class 𝒘 2 (𝑀2 ). The explicit expression of 𝑤 2 is derived in Appendix B: Õ Õ h1i, (2.24) 𝑤2 = 𝑣+ 𝑓 =h012i∈−triangle 𝑣 which is the formal sum (mod∫ 2) of all vertices and vertex 1 for each “−”-oriented 𝒗 triangle h012i. The sign (−1) 𝑤2 is defined by ∫ 𝒗 = 𝒗(𝑤 2 ) = 𝑤2    1,  if 𝑤 2 contains 𝑣   0,  if 𝑤 2 doesn’t contain 𝑣 (2.25) The second Stiefel-Whitney class is the obstruction to a spin structure. The fermion can only be define on a manifold which admits spin structure 𝐸 ∈ 𝐶1 (𝑀2 , Z2 ) such that 𝜕𝐸 = 𝑤 2 .One can interpret the 1-chain 𝐸 as a lattice representation of a spin structure. Indeed, in the context of Riemannian geometry it is well-known that the 2nd Stiefel-Whitney class 𝒘 2 (𝑀) ∈ 𝐻 2 (𝑀, Z2 ) is an obstruction for defining a lift of the structure group of the tangent bundle from 𝑆𝑂 (𝑛) to 𝑆 𝑝𝑖𝑛(𝑛). Thus any trivialization of this class leads to a lift of the structure group to 𝑆 𝑝𝑖𝑛(𝑛) and enables one to define spinors. Since 𝐸 is a trivialization of the homology 0-cycle Poincaré-dual to 𝒘 2 (𝑀), a choice of 𝐸 is equivalent to a choice of a trivialization of 𝒘 2 (𝑀) and thus can be thought of as implicitly defining a spin structure. It is remarkable that although we are dealing with spinless fermions, a choice of spin structure is still required in order to construct the bosonization map. This fermionic identity (2.22) is proved in Appendix C, including higher dimensional ∫ versions. Under the mapping (−1) 𝐸 𝒆 𝑆 𝑒 → 𝑈𝑒 and 𝑃 𝑓 → 𝑊 𝑓 , it gives gauge constraints for bosonic operators: 𝐺 𝑣 = 𝑈𝛿𝒗 ∫ Ö 𝒗∪ 𝒇 + 𝒇 ∪𝒗 𝑊𝑓 𝑓 = Ö 𝑋𝑒 ( Ö ∫ 𝑍𝑒 0 𝛿𝒗∪𝒆 0 (2.26) ), 𝑒0 𝑒⊃𝑣 Î ∫ 𝒆 0 ∪𝜆 𝜆(𝑒) Î , which can be derived 𝑒 𝑋𝑒 𝑒 0 𝑍𝑒 0 where we have used the property 𝑈𝜆 = from the definition: ∫ 0 𝑈𝜆+𝜆 0 ≡ (−1) 𝜆∪𝜆 𝑈𝜆 0 𝑈𝜆 . (2.27) 17 Therefore, our complete bosonization map is Ö 𝑊𝑓 = 𝑍 𝑒 ←→ 𝑃 𝑓 = −𝑖𝛾 𝑓 𝛾 0𝑓 , 𝑈𝑒 = 𝑋𝑒 ( Ö 𝑒⊂ 𝑓 ∫ ∫ ∫ 𝒆 0 ∪𝒆 𝒆 𝐸 ) ←→ (−1) 𝑆 𝑒 = (−1) 𝐸 𝒆 𝑖𝛾 𝐿(𝑒) 𝛾 0𝑅(𝑒) , 𝑍𝑒 0 𝑒0 𝐺𝑣 = Ö 𝑋𝑒 ( 𝑒⊃𝑣 Ö ∫ 𝛿𝒗∪𝒆 0 𝑍𝑒 0 ∫ ) ←→ (−1) 𝑤2 𝑒0 𝒗 𝑆 𝛿𝒗 Ö ∫ 𝒗∪ 𝒇 + 𝒇 ∪𝒗 𝑃𝑓 (2.28) = 1, 𝑓 Ö 𝑊 𝑓 = 1 ←→ 𝑓 Ö 𝑃𝑓 𝑓 On the bosonic side, the gauge constraint 𝐺 𝑣 = 1 is required. On the fermionic side, total parity must be even. 2.3 Examples Spinless fermion on a square lattice As a first example of the bosonization map, consider the theory of complex fermions on a square lattice with nearest-neighbor hopping and an on-site chemical potential 𝜇. The Hamiltonian is Õ Õ 𝑐†𝑓 𝑐 𝑓 . (2.29) 𝐻=𝑡 (𝑐†𝐿(𝑒) 𝑐 𝑅(𝑒) + 𝑐†𝑅(𝑒) 𝑐 𝐿(𝑒) ) + 𝜇 𝑒 𝑓 To apply our bosonization procedure, we first express (2.29) in terms of Majorana operators, 𝑡 Õ 𝜇Õ (𝑖𝛾 𝐿(𝑒) 𝛾 0𝑅(𝑒) + 𝑖𝛾 𝑅(𝑒) 𝛾 0𝐿(𝑒) ) + 𝐻= (1 + 𝑖𝛾 𝑓 𝛾 0𝑓 ) 2 𝑒 2 𝑓  𝑡 Õ = 𝑖𝛾 𝐿(𝑒) 𝛾 0𝑅(𝑒) − 𝑖(𝛾 𝐿(𝑒) 𝛾 0𝑅(𝑒) )(−𝑖𝛾 𝐿(𝑒) 𝛾 0𝐿 (𝑒) )(−𝑖𝛾 𝑅(𝑒) 𝛾 0𝑅(𝑒) ) (2.30) 2 𝑒 𝜇Õ (1 + 𝑖𝛾 𝑓 𝛾 0𝑓 ). + 2 𝑓 The bosonized Hamiltonian that follows from (3.2) and (3.3) is a Z2 gauge theory with Hamiltonian 𝑡 Õ 𝜇Õ 𝑋𝑒 𝑍𝑟 (𝑒) (1 − 𝑊 𝐿(𝑒) 𝑊 𝑅(𝑒) ) + (1 − 𝑊 𝑓 ) (2.31) 𝐻= 2 𝑒 2 𝑓 and a gauge constraint ( Î 𝑒⊃𝑣 𝑋𝑒 )𝑊NE(𝑣) = 1 on each vertex. 18 7 4 1 𝑘 𝑙 𝑚 𝑗 2 𝑖 5 𝑛 8 𝑝 3 6 Figure 2.3: A branching structure on triangular lattice. Spinless fermion on a honeycomb lattice Next, consider fermions living on the faces of a triangular lattice (or on the vertices of a honeycomb lattice), shown on Fig. 2.3. We consider again the nearest neighbor hopping Hamiltonian Õ Õ 𝑐†𝑓 𝑐 𝑓 . (2.32) 𝐻=𝑡 (𝑐†𝐿(𝑒) 𝑐 𝑅(𝑒) + 𝑐†𝑅(𝑒) 𝑐 𝐿(𝑒) ) + 𝜇 𝑒 𝑓 The hopping operators map as ∫ 𝑆 𝑒 = 𝑖𝛾 𝐿 (𝑒) 𝛾 0𝑅(𝑒) ←→ (−1) 𝐸 𝒆 𝑈𝑒 (2.33) for a suitably chosen sign 𝐸. The sign is chosen so that the vertex relations between 𝑆 𝑒 and 𝑈𝑒 are identical. For the branching structure shown in Fig. 2.3 one can choose 𝐸 = 0 (𝐸 contains no edge at all), so that the bosonization map is simply 𝑆 𝑒 ←→ 𝑈𝑒 . (2.34) With this choice, for the explicitly denoted edges on Fig. 2.3 the operators 𝑈𝑒 defined by (2.17) are 𝑈58 = 𝑋58 , (2.35) 𝑈57 = 𝑋57 𝑍45 , 𝑈56 = 𝑋56 𝑍35 , and other edges are defined by translation. The bosonized Hamiltonian is 𝑡 Õ 𝜇Õ 𝐻= 𝑈𝑒 (1 − 𝑊 𝐿 (𝑒) 𝑊 𝑅(𝑒) ) + (1 − 𝑊 𝑓 ) 2 𝑒 2 𝑓 with gauge constraint ( each vertex. Î 𝑒⊃𝑣 𝑋𝑒 )𝑊NE(𝑣) 𝑊SE(𝑣) = 1 (i.e. ( Î (2.36) 𝑒⊃𝑣 5 𝑋𝑒 )𝑊𝑚 𝑊𝑛 = 1) on 19 It is well known that on the honeycomb lattice the Hamiltonian (2.32) gives rise to a dispersion law which has two Dirac points in the Brillouin zone. This is highly non-obvious for the equivalent gauge theory Hamiltonian (2.36). Any fermionic operator with vanishing net fermion parity can be written in terms of 𝑆 𝑒 and 𝑃 𝑓 and thus have a bosonic counterpart. We start from simple examples. To bosonize 𝑐†𝑘 𝑐 𝑙 from Fig. 2.3, we express it via Majorana operators as 1 (𝛾 𝑘 𝛾𝑙 + 𝛾 0𝑘 𝛾𝑙0 + 𝑖𝛾 𝑘 𝛾𝑙0 + 𝑖𝛾𝑙 𝛾 0𝑘 ), 4 and then map these Majorana operators in the usual way, 𝑐†𝑘 𝑐 𝑙 = (2.37) 𝛾 𝑘 𝛾𝑙 = (𝑖𝛾 𝑘 𝛾 0𝑗 )(𝑖𝛾𝑙 𝛾 0𝑗 ) ←→ 𝑈24𝑈45 , 𝛾 𝑘 𝛾𝑙0 = 𝑖(𝛾 𝑘 𝛾𝑙 )(−𝑖𝛾𝑙 𝛾𝑙0) ←→ 𝑖𝑈24𝑈45𝑊𝑙 , 𝛾𝑙 𝛾 0𝑘 = (−𝑖)(𝛾 𝑘 𝛾𝑙 )(−𝑖𝛾 𝑘 𝛾 0𝑘 ) ←→ −𝑖𝑈24𝑈45𝑊 𝑘 , (2.38) 𝛾 0𝑘 𝛾𝑙0 = (−𝑖)(𝛾𝑙 𝛾 0𝑘 )(−𝑖𝛾𝑙 𝛾𝑙0) ←→ −𝑈24𝑈45𝑊 𝑘 𝑊𝑙 . This way we obtain 1 𝑐†𝑘 𝑐 𝑙 = 𝑈24𝑈45 (1 + 𝑊 𝑘 )(1 − 𝑊𝑙 ). 4 (2.39) Next, consider the operator 𝑐𝑖† 𝑐 𝑙 = 14 (𝛾𝑖 𝛾𝑙 + 𝛾𝑖0 𝛾𝑙0 + 𝑖𝛾𝑖 𝛾𝑙0 + 𝑖𝛾𝑖 𝛾𝑙0). Its first term is 𝛾𝑖 𝛾𝑙 = (𝑖𝛾 𝑗 𝛾𝑖0)(𝑖𝛾𝑙 𝛾 0𝑗 )(−𝑖𝛾 𝑗 𝛾 0𝑗 )(−𝑖𝛾𝑖 𝛾𝑖0)) ←→ 𝑈25𝑈45𝑊 𝑗 𝑊𝑖 , and the other terms can be computed the same way, giving 1 𝑐𝑖† 𝑐 𝑙 = 𝑈25𝑈45𝑊 𝑗 𝑊𝑖 (1 + 𝑊𝑖 )(1 − 𝑊𝑙 ) 4 1 = 𝑈25𝑈45𝑊 𝑗 (1 + 𝑊𝑖 )(1 − 𝑊𝑙 ). 4 (2.40) (2.41) Generalizing from (2.39) and (2.41), the rule for bosonization of a fermion bilinear 𝑐†𝑎 𝑐 𝑏 can be stated as follows. First choose an arbitrary path from face 𝑎 to face 𝑏. Start with (1 + 𝑊𝑎 )(1 − 𝑊𝑏 )/4, and follow the path. When the path passes through a face 𝑓 by crossing two edges with different orientations, we need to multiply by 𝑊 𝑓 . Then, for each edge 𝑒 the path crosses, we multiply by 𝑈𝑒 . For example, following the path 𝑚 → 𝑙 → 𝑘 → 𝑖, we can write down 1 𝑐𝑖† 𝑐 𝑚 = 𝑈25𝑈45𝑈57𝑊 𝑗 (1 + 𝑊𝑖 )(1 − 𝑊𝑚 ). (2.42) 4 If we use another path 𝑚 → 𝑛 → 𝑝 → 𝑖, it becomes 1 (2.43) 𝑐𝑖† 𝑐 𝑚 = 𝑈35𝑈56𝑈58𝑊𝑛 (1 + 𝑊𝑖 )(1 − 𝑊𝑚 ). 4 The above two formulas only differ by a gauge transformation. 20 The Hubbard model on a square lattice The Hamiltonian of Hubbard model (with fermions on faces of a square lattice) is Õ Õ 𝐻=𝑡 (𝑐†𝐿(𝑒), 𝜎 𝑐 𝑅(𝑒), 𝜎 + ℎ.𝑐.) + 𝑈 𝑛 𝑓 ↑𝑛 𝑓 ↓, (2.44) 𝑒, 𝜎 𝑓 where 𝜎 ∈ {↑, ↓} and 𝑛 𝑓 = 𝑐†𝑓 𝑐 𝑓 . It can be viewed as two copies of the nearest neighbor hopping Hamiltonian with an interaction on each face. Similar to (2.36), the bosonized system is a Z2 ×Z2 gauge theory on the dual lattice with a Hamiltonian 𝐻= 𝑡 Õ 𝜎 𝜎 𝑈Õ 𝜎 𝜎 (1 − 𝑊 ↑𝑓 )(1 − 𝑊 ↓𝑓 ) 𝑋𝑒 𝑍𝑟 (𝑒) (1 − 𝑊 𝐿(𝑒) 𝑊 𝑅(𝑒) )+ 2 𝑒, 𝜎 4 𝑓 (2.45) Î 𝜎 𝜎 with gauge constraints ( 𝑒⊃𝑣 𝑋𝑒𝜎 )𝑊NE(𝑣) 𝑊SE(𝑣) = 1 for 𝜎 =↑, ↓ at each vertex. On each edge, there are two species of spins labeled by ↑ and ↓. Note that the 𝑆𝑈 (2) spin symmetry is not manifest in this bosonized description. There exists a version of our bosonization procedure where the 𝑆𝑈 (2) symmetry is manifest. In that description, one of the Z2 gauge fields is replaced with a bosonic spin which lives on the vertices of the dual lattice. The 𝑆𝑈 (2) symmetry acts only on this spin variable. Some soluble 2+1D lattice gauge theories We have seen a couple of examples where a simple theory of free fermions on a lattice can be rewritten as a rather complicated Z2 lattice gauge theory on the dual lattice. Conversely, one can start with some simple Z2 gauge theory and ask if it can be rewritten as a theory of free fermions. The standard 2+1D Z2 lattice gauge theory introduced by F. Wegner [51] can be written in the Hamiltonian form as follows [52]. There is a spin on every edge 𝑒, with Pauli matrices 𝑋𝑒 , 𝑌𝑒 , 𝑍 𝑒 . The physical Hilbert space is a subspace of the tensor product space defined by the Gauss law constraints Ö 𝑋𝑒 = 1, ∀𝑣. (2.46) 𝑒⊃𝑣 The Hamiltonian is 𝐻 = 𝑔2 Õ 𝑒 𝑋𝑒 + 1 Õ 𝑊𝑓 , 𝑔2 𝑓 (2.47) where 𝑊 𝑓 is given by Eq. (2.16), as usual. This theory is not integrable and is related by Kramers-Wannier duality to the 3D Ising model. 21 0 3 6 𝛿®3 𝛿®2 𝑏 𝑏 𝛿®1 𝑎 1 4 𝑎®1 7 𝑎®2 𝑏 2 5 Figure 2.4: Fermions√at the center of faces form a honeycomb lattice. The vectors√are √ √ 3 3 3 1 1 defined as 𝛿®1 = (0, − 3 ), 𝛿®2 = (− 2 , 6 ), 𝛿®3 = ( 2 , 6 ) and 𝑎®1 = 𝛿®2 − 𝛿®1 = (− 12 , 23 ), √ 𝑎®2 = 𝛿®3 − 𝛿®1 = ( 1 , 3 ). 2 2 To get a Z2 gauge theory which is dual to a fermionic theory, we need to replace Eq. (2.46) with the modified Gauss law (2.12) on a square lattice, or with (2.20) on a general triangulation. The second (potential) term in Eq. (2.47) is still gaugeinvariant, but the first (kinetic) term is not. To fix this problem we simply replace each 𝑋𝑒 with 𝑈𝑒 = 𝑋𝑒 𝑍𝑟 (𝑒) , which is gauge-invariant by construction, and let 𝐻 0 = 𝑔2 Õ 𝑈𝑒 + 𝑒 1 Õ 𝑊𝑓 . 𝑔2 𝑓 (2.48) Since 𝑊 𝑓 maps to −𝑖𝛾 𝑓 𝛾 0𝑓 , and 𝑈𝑒 maps to 𝑖𝛾 𝐿(𝑒) 𝛾 0𝑅(𝑒) , the fermionic dual of this gauge theory is a theory of free fermions. To analyze this fermionic theory in more detail, let us specialize to the case of a regular triangular lattice (Fig. 2.4), so that fermions live on the vertices of a regular honeycomb lattice. By bosonization map (2.33), the Hamiltonian (2.48) is (up to a constant) equivalent to Õ Õ 𝐻 0𝑓 = 𝑡 (𝑐 𝐿(𝑒) 𝑐 𝑅(𝑒) − 𝑐†𝐿(𝑒) 𝑐†𝑅(𝑒) + 𝑐†𝐿 (𝑒) 𝑐 𝑅(𝑒) + 𝑐†𝑅(𝑒) 𝑐 𝐿(𝑒) ) + 𝜇 𝑐†𝑓 𝑐 𝑓 , (2.49) 𝑒 𝑓 Í ® where 𝑡 = 𝑔 2 and 𝜇 = 2/𝑔 2 . After the usual Fourier transform 𝑐 𝑥® = √1 𝑘® 𝑒𝑖 𝑘·®𝑥 𝑐 𝑘® , 𝑁 the Hamiltonian becomes Õ Õ Õ † † 𝐻 0𝑓 = (Δ 𝑘 𝑐 𝑘,𝑎 𝑐 + h.c.) + (𝜖 𝑐 𝑐 + h.c.) + 𝜇 (𝑐†® 𝑐 𝑘,𝑎 ® ® ® ® + 𝑐 ® 𝑐 𝑘,𝑏 ® ), − 𝑘,𝑏 𝑘® ® 𝑘,𝑏 𝑘,𝑎 𝑘® 𝑘,𝑎 𝑘® ® ® ® ® 𝑘,𝑏 𝑘® ® ® ® ® ® ® ® ® where Δ 𝑘® = 𝑡 (𝑒 −𝑖 𝑘·𝛿1 − 𝑒 −𝑖 𝑘·𝛿2 − 𝑒 −𝑖 𝑘·𝛿3 ) and 𝜖 𝑘® = 𝑡 (𝑒𝑖 𝑘·𝛿1 + 𝑒𝑖 𝑘·𝛿2 + 𝑒𝑖 𝑘·𝛿3 ). We can 22 (a) For 𝜇/𝑡 = 2, the band gap closes at 𝑘® = (± 23𝜋 , 0), which form two Dirac cones. (b) Another viewpoint from the top. Two Dirac cones lie in the first Brillouin zone (the hexagon). Figure 2.5: (Color online) Band structure of 𝐻BdG (equivalent to 𝐻 0𝑓 ). write this using the Bogoliubov-de-Gennes (BdG) formalism as 𝐻 0𝑓 = 1Õ † ® ® Ψ® 𝐻BDG ( 𝑘)Ψ 𝑘 𝑘 2 (2.50) 𝑘® with   𝑐  0   𝜇 −Δ ∗® 𝜖 𝑘®  𝑘,𝑎 ®  𝑘    †  −Δ    0 −𝜖 𝑘®   𝑘® −𝜇 𝑐 − 𝑘,𝑏 ®  ® 𝐻BDG ( 𝑘) =  ∗ (2.51)  , Ψ𝑘® =  .  𝜖®  𝑐®  0 𝜇 −Δ ∗®  𝑘,𝑏  𝑘   𝑘  𝑐 †   −𝜖 ®∗ −Δ 𝑘® −𝜇   0  − 𝑘,𝑎  𝑘    ®  q q 2 2 ® The eigenvalues are 𝐸 ( 𝑘) = ± |Δ 𝑘® | + (|𝜖 𝑘® | + 𝜇) , ± |Δ 𝑘® | 2 + (|𝜖 𝑘® | − 𝜇) 2 . The gap closes at 𝑘 = (± 2𝜋 3 , 0) and 𝜇/𝑡 = 2 (𝑔 = 1). The spectrum is shown in Fig. 2.5. 23 2.4 Euclidean 3D gauge theories and their fermionic duals In this section, we will derive the Euclidean 3D actions for gauge theories dual to some fermionic systems. Before considering the nearest neighbor hopping Hamiltonian (2.30), let us first look at the simpler Majorana hopping Hamiltonian on the square lattice, Õ Õ 𝐻 = −𝐴 𝑖𝛾 𝐿(𝑒) 𝛾 0𝑅(𝑒) − 𝐵 (−𝑖𝛾 𝑓 𝛾 0𝑓 ), (2.52) 𝑒 𝑓 whose bosonic dual is a gauge theory with a Hamiltonian Õ Õ 𝐻 = −𝐴 𝑋𝑒 𝑍𝑟 (𝑒) − 𝐵 𝑊𝑓 . 𝑒 (2.53) 𝑓 Without loss of generality, we can assume 𝐴 > 0. The Gauss law constraint is ! Ö Ö 𝐺𝑣 ≡ 𝑋𝑒 𝑍 𝑒 0 = 1. (2.54) 𝑒 0 ⊂NE(𝑣) 𝑒⊃𝑣 The partition function is Z = Tr 𝑒 −𝛽𝐻 = Tr 𝑇 𝑀 , (2.55) where 𝑇 is the transfer matrix defined as ! Ö 𝛿𝐺 0𝑣 ,1 𝑒 −𝛿𝜏𝐻 . 𝑇= (2.56) 𝑣 The prime on 𝐺 𝑣 means that it acts to bra on the left, which will be clear in later calculations. The first factor projects to the gauge-invariant sector of the Hilbert space. We can rewrite it using a Z2 Lagrange multiplier field 𝜆 𝑣 as 1−𝑍𝑒 0 1−𝜆 𝑣 Í 1−𝜆 𝑣 Í 1−𝑋𝑒 1 Õ 1 0 0 (−1) 2 𝑒 ⊃𝑣 2 (−1) 2 𝑒 0 ⊂NE(𝑣) 2 . (2.57) 𝛿𝐺 𝑣 ,1 = (1 + 𝐺 𝑣 ) = 2 2 𝜆 =±1 𝑣 Let us define |𝑚(𝜏)i = |{𝑆 𝑒 }i as the configuration of spins (in the 𝑍 𝑒 basis). To evaluate the matrix element h𝑚0 (𝜏 + 𝛿𝜏)|𝑇 |𝑚(𝜏)i, we insert the “decomposition of unity” in terms of a full basis of 𝑋𝑒 (momentum) eigenstates, using the identity 1−𝑆 𝑥 𝑧0 𝑧 1 Õ 𝑓 (𝑆 𝑥 , 𝑆 𝑧 )(−1) 4 (2−𝑆 −𝑆 ) , (2.58) h𝑆 𝑧0 | 𝑓 (𝜎 𝑥 , 𝜎 𝑧 )|𝑆 𝑧 i = 2 𝑆 𝑥 =±1 where we assume 𝜎 𝑥 is always left to 𝜎 𝑧 in 𝑓 (𝜎 𝑥 , 𝜎 𝑧 ). The matrix element is h𝑚0 (𝜏 + 𝛿𝜏)|𝑇 |𝑚(𝜏)i # " Õ Ö Î 1−𝜆 𝑣 Í 𝑧0 𝑧 ∝ (−1) 4 𝑒 0 ⊂NE(𝑣) (1−𝑆𝑒 0 ) 𝑒 𝐵𝛿𝜏 𝑒 0 ⊂NE(𝑣) 𝑆𝑒 0 × {𝜆 𝑣 } 𝑣 Ö Õ    1−𝑆𝑒𝑥 𝑥 𝑆𝑧 𝑧 Í 𝑧0 𝐴𝛿𝜏𝑆 0 [2−𝑆 −𝑆 + (1−𝜆 )] 0 𝑒 𝑒 𝑒 𝑣 𝑣 ⊂𝑒 𝑟 (𝑒)  . ×  (−1) 4 𝑒   𝑒 𝑆𝑒𝑥 =±1    (2.59) 24 The next order of business is to integrate out the intermediate momentum fields, i.e. to perform the sum over the 𝑆 𝑥 in the second bracket. This bracket equals Î 𝐴𝛿𝜏𝑆𝑟𝑧 (𝑒) −𝐴𝛿𝜏𝑆𝑟𝑧 (𝑒) 𝑧 0 𝑧 Î +𝑒 𝑆 𝑒 𝑆 𝑒 𝑣 0 ⊂𝑒 𝜆 𝑣 0 ). To simplify it, we need to consider two 𝑒 (𝑒 cases: 𝑆𝑟𝑧 (𝑒) = 1 and 𝑆𝑟𝑧 (𝑒) = −1. First, for 𝑆𝑟𝑧 (𝑒) = 1, we can simply write 𝑒 𝐴𝛿𝜏𝑆𝑟𝑧 (𝑒) +𝑒 −𝐴𝛿𝜏𝑆𝑟𝑧 (𝑒) 𝑧 0 𝑧 𝑆𝑒 𝑆𝑒 Ö 𝜆 𝑣 0 = 𝑒 𝐴𝛿𝜏 + 𝑒 −𝐴𝛿𝜏 𝑆 𝑒𝑧 0 𝑆 𝑒𝑧 𝑣 0 ⊂𝑒 Ö 𝜆𝑣 0 𝑣 0 ⊂𝑒 = 𝐶𝑒 𝐽𝑆 𝑒𝑧 0 𝑆 𝑒𝑧 Î 𝑣 0 ⊂𝑒 𝜆 𝑣 0 (2.60) , where 𝐶 2 = 2 sinh(2𝐴𝛿𝜏) and tanh 𝐽 = 𝑒 −2𝐴𝛿𝜏 . For the other case 𝑆𝑟𝑧 (𝑒) = −1, 𝑒 𝐴𝛿𝜏𝑆𝑟𝑧 (𝑒) +𝑒 −𝐴𝛿𝜏𝑆𝑟𝑧 (𝑒) 𝑧 0 𝑧 𝑆𝑒 𝑆𝑒 Ö 𝜆𝑣 0 𝑣 0 ⊂𝑒 = 𝐶𝑒 𝐽𝑆 𝑒𝑧 0 𝑆 𝑒𝑧 Î 𝑣 0 ⊂𝑒 𝜆 𝑣 0 (−1) (2.61) 𝑧 Í 𝑧0 1 2 [2−𝑆 𝑒 −𝑆 𝑒 + 𝑣 0 ⊂𝑒 (1−𝜆 𝑣 0 )] . We can combine (2.60) and (2.61) into the single equation Ö 𝐴𝛿𝜏𝑆𝑟𝑧 (𝑒) −𝐴𝛿𝜏𝑆𝑟𝑧 (𝑒) 𝑧 0 𝑧 𝑒 +𝑒 𝜆𝑣 0 𝑆𝑒 𝑆𝑒 𝑣 0 ⊂𝑒 = 𝐶𝑒 Î 𝐽𝑆 𝑒𝑧 0 𝑆 𝑒𝑧 𝑣 0 ⊂𝑒 𝜆 𝑣 0 (−1) (2.62) 𝑧0 𝑧 Í 𝑧 1 4 [2−𝑆 𝑒 −𝑆 𝑒 + 𝑣 0 ⊂𝑒 (1−𝜆 𝑣 0 )] (1−𝑆𝑟 (𝑒) ) . We can now substitute (2.62) back to (3.51) and write the matrix element in the suggestive form h𝑚0 (𝜏 + 𝛿𝜏)|𝑇 |𝑚(𝜏)i ÕÖ Î Í 𝑧 𝑧0 𝑧 Î 𝑧0 1 1 ∝ 𝑒 𝐾 𝑒 0 ⊂NE(𝑣) 𝑆𝑒 0 +𝐽𝑆𝑒 𝑆𝑒 𝑣 0 ⊂𝑒 𝜆 𝑣 0 (−1) 2 (1−𝜆 𝑣 ) 𝑒 0 ⊂NE(𝑣) 2 (1−𝑆𝑒 0 ) {𝜆 𝑣 } 𝑣,𝑒 1 × (−1) 2 (1−𝑆𝑟𝑧 (𝑒) ) [ 21 (1−𝑆 𝑒𝑧0 )+ 21 (1−𝑆 𝑒𝑧 )+ Í 1 𝑣 0 ⊂𝑒 2 (1−𝜆 𝑣 0 ) (2.63) ], where 𝐾 ≡ 𝐵𝛿𝜏. We can interpret 𝜆 𝑣 as gauge fields on temporal links. Therefore, the first, exponential term can be thought of as the exponential of the anisotropic Wegner action [51, 52] Õ Ö 𝐽𝑓 𝑆𝑒 , (2.64) 𝑓 𝑒⊃ 𝑓 where 𝐽 𝑓 is different for spatial and temporal faces of the 3d lattice. The rest can be thought of as a topological factor in the partition function which gives the correct anomaly factors of −1 for fermionic statistics. Let 𝑎𝑖 ∈ 𝐶 0 (𝐿, Z2 ) be the 0-cochain on the 𝑖-th layer with value 21 (1 − 𝜆 𝑣 ) on vertex 𝑣. We regard it as the Z2 gauge field on temporal links between the 𝑖-th and (𝑖 + 1)-th layers. Let 𝛼𝑖 ∈ 𝐶 1 (𝐿, Z2 ) be the 25 1-form on the 𝑖-th layer that represents the values of 𝑆 𝑒𝑧 on the 𝑖-th layer. Then we can express the “topological” factors in the last line of (2.63) as Í Ö 1 1−𝜆 𝑣 Í 𝑧0 1 (1−𝑆𝑟𝑧 (𝑒) ) [ 21 (1−𝑆 𝑒𝑧0 )+ 12 (1−𝑆 𝑒𝑧 )+ 𝑣 0 ⊂𝑒 12 (1−𝜆 𝑣 0 )] (−1) 2 𝑒 0 ⊂NE(𝑣) 2 (1−𝑆𝑒 0 ) (−1) 2 𝑣, 𝑒 ≡ (−1) 𝑎𝑖 (Δ (𝛿𝛼𝑖+1 )) (−1) 𝛼𝑖 (Δ (𝛼𝑖+1 +𝛼𝑖 +𝛿𝑎𝑖 )) , (2.65) where Δ (𝑥) is the Poincaré dual of 𝑥 at relative position (− 12 , − 12 ) (i.e. Δ (𝛿NE(𝑣) ) = 𝑣 and Δ (𝛿 𝑒 ) = 𝑟 (𝑒)). This expression is invariant (up to boundary terms) under the gauge transformation 𝑎𝑖 → 𝑎𝑖 + 𝑓𝑖 + 𝑓𝑖+1 and 𝛼𝑖 → 𝛼𝑖 + 𝛿 𝑓𝑖 . If we put the Hamiltonian (2.52) on a general triangulation instead of a square lattice, its bosonic dual is Õ Õ 𝐻 = −𝐴 𝑈𝑒 − 𝐵 𝑊𝑓 . (2.66) 𝑒 𝑓 Its partition function is 𝑍= Õ 𝑒 −𝑆topo 𝑒 𝐾 Í Í 𝑧 𝑧 𝑓𝜏 𝑆 𝑓𝜏 𝑓𝑠 𝑆 𝑓𝑠 +𝐽 (2.67) , 𝑆𝑧 , 𝜆 where 𝑓𝑠 and 𝑓𝜏 are faces of spatial and temporal types, 𝑆 𝑧𝑓 ≡ Í ∫ Î ∫ 𝑒 −𝑆topo = (−1) 𝑖 [ 𝑎𝑖 ∪𝛿𝛼𝑖+1 + 𝛼𝑖 ∪(𝛼𝑖 +𝛼𝑖+1 +𝛿𝑎𝑖 )] . 𝑧 𝑒⊂ 𝑓 𝑆 𝑒 , and (2.68) Notice that (2.68) is analogous to Chern-Simon action on a general triangulation of 3d manifold ∫ 𝑆CS = 𝑖𝜋 𝑎 ∪ 𝛿𝑎. (2.69) This kind of topological term results in charge-flux attachment and generates fermionic degrees of freedom. Now, let us go back to the usual fermionic hopping Hamiltonian on a general triangulation Õ Õ 𝑐†𝑓 𝑐 𝑓 , (2.70) 𝐻 = −2𝐴 (𝑐†𝐿 (𝑒) 𝑐 𝑅(𝑒) + 𝑐†𝑅(𝑒) 𝑐 𝐿 (𝑒) ) + 2𝐵 𝑒 𝑓 and its bosonic dual (up to some constant) Õ Õ 𝐻 = −𝐴 𝑈𝑒 (1 − 𝑊 𝐿(𝑒) 𝑊 𝑅(𝑒) ) − 𝐵 𝑊𝑓 𝑒 (2.71) 𝑓 ∫ Î Î 𝛿 𝑣 ∪𝛿 𝑓 with gauge constraints on vertices ( 𝑒⊃𝑣 𝑋𝑒 )( 𝑓 𝑊 𝑓 ) = 1. The only difference from the Majorana hopping Hamiltonian is the factor (1 − 𝑊 𝐿 (𝑒) 𝑊 𝑅(𝑒) ). With 26 some careful calculations, one can show the partition function is Õ Í Î Í Î 𝑧 𝑧 𝑍= 𝑒 −𝑆topo 𝑒 𝐾 𝑓𝑠 𝑒 ⊂ 𝑓𝑠 𝑆𝑒 +𝐽 𝑓𝜏 𝑒 0 ⊂ 𝑓𝜏 𝑆𝑒 0 𝑆𝑧 , 𝜆 × 𝑒− Î 𝑧 Î 𝑧 𝐽 −ln 2 Í 𝑒𝑠 [1+( 𝑒 ⊂𝐿 (𝑒𝑠 ) 𝑆 𝑒 ) ( 𝑒 ⊂𝑅 (𝑒𝑠 ) 𝑆 𝑒 )] 2 × 𝑒 −𝑙 Í 𝑒𝑠 [1+( Î 𝑧 Î 𝑧 𝑧 0 𝑧 Î 𝑣 ⊂𝑒𝑠 𝜆 𝑣 ) 𝑒 ⊂𝐿 (𝑒𝑠 ) 𝑆 𝑒 ) ( 𝑒 ⊂𝑅 (𝑒𝑠 ) 𝑆 𝑒 )] (1−𝑆 𝑒𝑠 𝑆 𝑒𝑠 , (2.72) where 𝑙 is taken to be infinity and 𝑒 𝑠 is a edge on a spatial slice. Taking 𝑙 → ∞ imposes additional gauge constraints to the previous lattice gauge theory (2.67), and the topological term is not affected. 27 Chapter 3 BOSONIZATION ON THREE-DIMENSIONAL LATTICES 3.1 Cubic lattices Next, we introduce our bosonization method on an infinite 3d cubic lattice. Suppose that we have a model with fermions living at the centers of cubes. Let us describe the generators and relations in the algebra of local observables with trivial fermion parity (the even fermionic algebra for short). On each cube 𝑡 we have a single fermionic creation operator 𝑐 𝑡 and a single fermionic annihilation operator 𝑐†𝑡 with the usual anticommutation relations. The fermionic † parity operator on cube 𝑡 is 𝑃𝑡 = (−1) 𝑐𝑡 𝑐𝑡 . All operators 𝑃𝑡 commute with each other. We work in the Majorana basis 𝛾𝑡 = 𝑐 𝑡 + 𝑐†𝑡 , 𝛾𝑡0 = (𝑐 𝑡 − 𝑐†𝑡 )/𝑖. (3.1) The even fermionic algebra is generated by 𝑖𝛾 𝐿( 𝑓 ) 𝛾 0𝑅( 𝑓 ) and −𝑖𝛾𝑡 𝛾𝑡0 where each face is assigned an orientation from cube 𝐿 ( 𝑓 ) to cube 𝑅( 𝑓 ). To illustrate the definition of these operators, we draw the dual lattice of the original lattice. In Fig. 3.1, fermions live on vertices and the orientations of each dual edge (face of the original lattice) are taken to be along +𝑥, +𝑦, and +𝑧 directions. The Majorana hopping operator is defined by 𝑆 𝑓 = 𝑖𝛾 𝐿( 𝑓 ) 𝛾 0𝑅( 𝑓 ) , where 𝐿 ( 𝑓 ) and 𝑅( 𝑓 ) are source and sink (starting and ending points) of dual edge 𝑓 in the dual lattice. 𝑆 𝑓𝑖 and 𝑆 𝑓 𝑗 anti-commute only when both dual edges 𝑓𝑖 and 𝑓 𝑗 start from the same point or both end at the same point. The dual bosonic system has Z2 spins living on faces of the original lattice (or equivalently, on edges of the dual lattice). To define bosonic hopping operators 𝑈 𝑓 , we need to choose a framing for each edge of the dual lattice, i.e. a small shift of each dual edge along some orthogonal direction. We also assume that when projected on some generic plane (such as the plane of the page) a shifted dual edge intersects all dual edges transversally. For example, in Fig. 3.1 such a framing is indicated by red, green, and blue lines (for dual edges along 𝑥, 𝑦 and 𝑧 directions, respectively), and the shift of the dual edge 1 intersects dual edges 3 and 4 1. Now 1 There are many choices of framing, and accordingly many versions of the bosonization map. By construction, they are related by automorphisms of the algebra of observables. 28 Figure 3.1: (Color online) For edges in the dual lattice, the "framing" is defined by green, red, and blue edges, which is a small shift of duel edges [53]. Given a dual edge 𝑓 , the operator 𝑈 𝑓 is defined as 𝑋 𝑓 times 𝑍 𝑓 0 for those 𝑓 0 which intersect the framing of 𝑓 when projected to the plane (i.e. 𝑈 𝑓1 = 𝑋1 𝑍3 𝑍4 , 𝑈 𝑓2 = 𝑋2 𝑍7 𝑍8 , and 𝑈 𝑓3 = 𝑋3 𝑍5 𝑍6 ). we define 𝑈 𝑓 as a product of 𝑋 𝑓 with all 𝑍 𝑓 0 such that 𝑓 0 intersects the framing of 𝑓 when projected to the plane of the page. For example, the hopping operator for the dual edge 1 is 𝑈1 = 𝑋1 𝑍3 𝑍4 . Notice that 𝑈1 , 𝑈3 , and 𝑈4 anti-commute with each other and 𝑈3 , 𝑈5 , and 𝑈6 anti-commute with each other, while 𝑈2 and 𝑈3 commute, and 𝑈1 and 𝑈8 commute. One can check that 𝑆 𝑓 and 𝑈 𝑓 have the same commutation relations. Therefore, the bosonization map in 3D can be defined as follows: Î 1. For any cube 𝑡 let 𝑊𝑡 ≡ 𝑓 ⊂𝑡 𝑍 𝑓 . We identify the fermionic states |𝑃𝑡 = 1i and |𝑃𝑡 = −1i with bosonic states for which 𝑊𝑡 = 1 and 𝑊𝑡 = −1, respectively. Thus 𝑃𝑡 = −𝑖𝛾𝑡 𝛾𝑡0 ←→ 𝑊𝑡 . (3.2) 2. The fermionic hopping operator 𝑆 𝑓 is identified with 𝑈 𝑓 defined above: 𝑆 𝑓 = 𝑖𝛾 𝐿( 𝑓 ) 𝛾 0𝑅( 𝑓 ) ←→ 𝑈 𝑓 . (3.3) As in 2d, the bosonic operators satisfy some constraints. In Fig. 3.2, we calculate the product of 𝑆 𝑓 around the red square on the dual lattice: 29 Figure 3.2: (Color online) The framing of the hopping term defined previously is indicated by the green square, while the gauge constraint involves the 𝑍 operators in the opposite framing (blue dashed square). 𝑆 𝑓1 𝑆 𝑓2 𝑆 𝑓3 𝑆 𝑓4 =(𝑖𝛾𝑑 𝛾𝑐0 )(𝑖𝛾𝑏 𝛾𝑐0 )(𝑖𝛾𝑎 𝛾𝑏0 )(𝑖𝛾𝑎 𝛾𝑑0 ) = − (−𝑖𝛾𝑏 𝛾𝑏0 )(−𝑖𝛾𝑑 𝛾𝑑0 ) (3.4) = − 𝑃𝑏 𝑃𝑑 ←→ −𝑊𝑏 𝑊𝑑 . Its bosonic dual defined by (3.3) is the product of the corresponding operators 𝑈 𝑓 . Their definition involves a framing of the red square given by the green square: 𝑈 𝑓1 𝑈 𝑓2 𝑈 𝑓3 𝑈 𝑓4 =(𝑋1 𝑍2 𝑍6 )(𝑋2 𝑍12 𝑍13 )(𝑋3 𝑍11 𝑍14 )(𝑋4 𝑍3 𝑍5 ) = − 𝑋1 𝑋2 𝑋3 𝑋4 𝑍5 𝑍6 (𝑍2 𝑍3 𝑍11 𝑍12 𝑍13 𝑍14 ) (3.5) = − 𝑋1 𝑋2 𝑋3 𝑋4 𝑍5 𝑍6𝑊𝑏 . Comparing (3.4) and (3.5), we get the constraint 1 =𝑋1 𝑋2 𝑋3 𝑋4 𝑍5 𝑍6𝑊𝑑 (3.6) =𝑋1 𝑋2 𝑋3 𝑋4 𝑍1 𝑍4 𝑍5 𝑍6 𝑍7 𝑍8 𝑍9 𝑍10 The operators 𝑍’s are the edges crossed by dashed square in Fig. 3.2. The framing for gauge constraints is opposite to the framing used to define hopping operators. We have a gauge constraint for each face of dual lattice. In terms of the original lattice, there is one gauge constraint for each edge. All these constraints commute and thus define a Z2 2-form gauge theory with an unusual Gauss law. 3.2 Triangulation The bosonization method described above also works for any triangulation. For an arbitrary triangulation 𝑇 of a 3d manifold 𝑀3 , we choose a branching structure. 30 3 0 1 2 Figure 3.3: (Color online) A branching structure on a tetrahedron. The orientation of each face is determined by the right-hand rule. We defined this as the “+” tetrahedron, the directions of faces 012 and 023 are inward (blue) while the directions of faces 123 and 013 are outward (red). The directions of faces are reversed in the “−” tetrahedron (mirror image of this tetrahedron). A branching structure is a choice of an orientation on each edge such that there is no oriented loop on any triangle. One simple way is to label vertices by different numbers and assign the direction of an edge from the vertex with smaller number to the vertex with larger number (see Fig. 3.3). Each tetrahedron has two inward faces and two outward faces (by right-hand rule). We place fermions at the centers of tetrahedra. Each tetrahedron 𝑡 contains Majorana operators 𝛾𝑡 and 𝛾𝑡0. We define the fermionic hopping operator on each face 𝑓 as 𝑆 𝑓 = 𝑖𝛾 𝐿( 𝑓 ) 𝛾 0𝑅( 𝑓 ) , (3.7) where 𝐿 ( 𝑓 )/𝑅( 𝑓 ) is the tetrahedron with 𝑓 as a outward/inward face. Notice that 𝑆 𝑓 and 𝑆 𝑓 0 anti-commute only when 𝑓 and 𝑓 0 share a tetrahedron with both 𝑓 and 𝑓 0 inward or outward. To express this property mathematically, we introduce (higher) cup product used in algebraic topology. The definition and properties of the (higher) cup products are described in Section A. If 𝛽1 and 𝛽2 are 2-cochains, then 𝛽1 ∪1 𝛽2 (0123) = 𝛽1 (023) 𝛽2 (012) + 𝛽1 (013) 𝛽2 (123). (3.8) Therefore, the commutation relations can be expressed as 𝑆𝑓𝑆 𝑓0 = (−1) ∫ 𝒇 ∪1 𝒇 0 + 𝒇 0 ∪1 𝒇 𝑆 𝑓 0𝑆 𝑓 , (3.9) where the notation 𝒇 ∈ 𝐶 2 (𝑇, Z2 ) denotes the 2-cochain with value 1 on face 𝑓 and 0 otherwise, and the integral represents the sum over all tetrahedra. The even fermionic algebra is generated by the operators 𝑆 𝑓 for all faces and the fermionic parity operators 𝑃𝑡 for all tetrahedra. 31 The dual bosonic variables are Z2 spins which live on faces of the triangulation. As before, the flux operator Ö 𝑊𝑡 = 𝑋𝑓 𝑓 ⊃𝑡 corresponds to 𝑃𝑡 under the bosonization map. Next we need to find bosonic operators 𝑈 𝑓 which have the same commutation relation as fermionic operators 𝑆 𝑓 . We should define 𝑈 𝑓 as 𝑋 𝑓 times 𝑍 𝑓 0 for some faces 𝑓 0 which share a tetrahedron with 𝑓 and have the same orientation with respect to the tetrahedron. One way to define 𝑈 𝑓 is Ö Ö ∫ 𝒇 0∪ 𝒇 1 𝒇 (𝑡 012 ) 𝒇 (𝑡123 ) . (3.10) 𝑍𝑡023 𝑍𝑡013 = 𝑋𝑓 𝑍𝑓0 𝑈𝑓 = 𝑋𝑓 𝑓0 𝑡∈{𝐿( 𝑓 ),𝑅( 𝑓 )} 𝑈 𝑓 satisfy the commutation relation 𝑈 𝑓 𝑈 𝑓 0 = (−1) ∫ 𝒇 ∪1 𝒇 0 + 𝒇 0 ∪1 𝒇 𝑈 𝑓 0𝑈 𝑓 (3.11) which is the same as (3.9). The final step is to determine the constraints on bosonic variables. There is one Î such constraint for each edge 𝑒. In the product 𝑓 ⊃𝑒 𝑆 𝑓 , the only surviving terms are −𝑖𝛾𝑡 𝛾𝑡0 with one face going inward and one face going outward of 𝑡. Therefore, the product can be written as Ö Ö 𝑆𝑓 ∼ 𝑃𝑡 , (3.12) 𝑓 ⊃𝑒 𝑡|𝑒=𝑡 01 ,𝑡 03 ,𝑡12 ,𝑡23 where ∼ means that it is equal up to a sign, which will be treated carefully in the next paragraph. This produce the gauge constraints for bosonic operators: Ö Ö 𝑈𝑓 ∼ 𝑊𝑡 . (3.13) 𝑓 ⊃𝑒 𝑡|𝑒=𝑡01 ,𝑡03 ,𝑡12 ,𝑡23 For a tetrahedron 𝑡 containing an edge 𝑒 with adjacent faces 𝑓1 and 𝑓2 , consider the following product which gives 𝑊𝑡 for 𝑒 = 𝑡01 , 𝑡03 , 𝑡12 , 𝑡23 and 1 otherwise: Ö ( 𝒇 + 𝒇 )∪ 𝒇 0+ 𝒇 0∪ ( 𝒇 + 𝒇 ) 1 1 2 𝑍𝑓01 2 1 𝑍 𝑓1 𝑍 𝑓2 𝑓 0 ⊂𝑡 =   𝑊𝑡 ,  if 𝑒 = 𝑡01 , 𝑡03 , 𝑡12 , 𝑡23   1, otherwise  Substituting this into (3.13), we have Ö Ö Ö ∫ 𝒇 0∪ 𝒇 + 𝒇 ∪ 𝒇 0 Ö ∫ 𝒇 0∪ 𝛿𝒆+𝛿𝒆∪ 𝒇 0 1 1 1 1 𝑈𝑓 ∼ 𝑍𝑓0 = 𝑍𝑓0 . 𝑓 ⊃𝑒 𝑓 ⊃𝑒 𝑓 0 𝑓0 (3.14) (3.15) 32 On the other hand, the product Î Ö Ö 𝑓 ⊃𝑒 𝑈𝑓 ∼ Ö 𝑋𝑓 𝑓 ⊃𝑒 𝑈 𝑓 is ∫ 𝑍𝑓0 𝒇 0 ∪1 𝒇 =( Ö 𝑓0 𝑓 ⊃𝑒 𝑋𝑓 ) Ö ∫ 𝑍𝑓0 𝒇 0 ∪1 𝛿𝒆 . (3.16) 𝑓0 𝑓 ⊃𝑒 Identifying (3.15) and (3.16) gives Ö Ö ∫ 𝛿𝒆∪ 𝒇 0 1 = 1. ( 𝑋𝑓 ) 𝑍𝑓0 (3.17) 𝑓0 𝑓 ⊃𝑒 This is the modified Gauss law (gauge constraint) on each edge 𝑒. One can check that constraints for different edges 𝑒 1 and 𝑒 2 commute since ∫ (𝛿𝒆 1 ∪1 𝛿𝒆 2 + 𝛿𝒆 2 ∪1 𝛿𝒆 1 ) = ∫ (𝒆 1 ∪ 𝛿𝒆 2 + 𝛿𝒆 2 ∪ 𝒆 1 + 𝒆 2 ∪ 𝛿𝒆 1 + 𝛿𝒆 1 ∪ 𝒆 2 ) = 0, (3.18) = where we have used the property ∫ ∫ 𝛿𝒆 1 ∪1 𝛿𝒆 2 = (𝒆 1 ∪ 𝛿𝒆 2 + 𝛿𝒆 2 ∪ 𝒆 1 ). To be more precise about the signs in (3.15) and (3.16), we give the definition of 𝑆 𝛽 for a 2-cochain 𝛽 ∈ 𝐶 2 (𝑇, Z2 ): 𝑆 𝛽 𝑆 𝛽 0 = (−1) ∫ 𝛽 0 ∪1 𝛽 (3.19) 𝑆 𝛽+𝛽 0 . We can also define 𝑈 𝛽 in the same way. It can be checked that 𝑈𝛽 = Ö 𝛽( 𝑓 ) 𝑋𝑓 Ö ∫ 𝑍𝑓0 𝒇 0 ∪1 𝛽 . (3.20) 𝑓0 𝑓 ∫ Î Î 𝒇 0 ∪1 𝛿𝒆 For example, we have 𝑈𝛿𝑒 = ( 𝑓 ⊃𝑒 𝑋 𝑓 ) 𝑓 0 𝑍 𝑓 0 . The explicit formula for the identity in fermionic evn algebra is ∫ Ö ∫ 𝒆∪ 𝒕+𝒕∪ 𝒆 𝒆 1 1 𝑤2 (−1) 𝑆 𝛿𝒆 𝑃𝑡 =1 (3.21) 𝑡 which is derived in Appendix C. The 1-chain 𝑤 2 consists of all edges of the triangulation, together with the (02) edge for all “+” tetrahedra and the (13) edge for all “−” tetrahedra: Õ Õ Õ 𝑤2 = 𝑒+ 𝑡02 + 𝑡13 . (3.22) 𝑒 𝑡∈+tetrahedra 𝑡∈−tetrahedra This is exactly the 1-chain which is Poincaré-dual to the second Stiefel-Whitney class. It is a 1-cycle modulo 2 (that is, its boundary is trivial when regarded as 33 a 0-chain with coefficients in Z2 ). If the topological space corresponding to the triangulation is simply-connected, then any 1-cycle is a boundary of some 2-cycle 2. Thus we can define ∫ 𝑆 𝐸𝛽 = (−1) 𝐸 𝛽 𝑆 𝛽 , (3.23) where 𝐸 is a 2-chain such that 𝜕𝐸 = 𝑤 2 . Such a 𝐸 is not unique, but any two choices differ by a 2-cycle. We see that if we identify 𝑆 𝐸𝑓 and 𝑈 𝑓 , then the bosonic variables must satisfy a gauge constraint (3.17). The 3d bosonization map can be summarized as follows: Ö 𝑊𝑡 = 𝑍 𝑓 ←→ 𝑃𝑡 = −𝑖𝛾𝑡 𝛾𝑡0, 𝑓 ⊂𝑡 𝑈𝑓 = 𝑋𝑓 ( Ö ∫ 𝑍𝑓0 𝒇 0 ∪1 𝒇 ∫ ∫ ) ←→ (−1) 𝐸 𝒇 𝑆 𝑓 = (−1) 𝐸 𝒇 𝑖𝛾 𝐿 ( 𝑓 ) 𝛾 0𝑅( 𝑓 ) , 𝑓0 𝐺𝑒 = Ö 𝑓 ⊃𝑒 𝑋𝑓 ( Ö ∫ 𝑍𝑓0 𝛿𝒆∪1 𝒇 0 ∫ ) ←→ (−1) 𝑤2 𝑓0 𝒆 𝑆 𝛿𝒆 Ö ∫ 𝑃𝑡 𝒆∪1 𝒕+𝒕∪1 𝒆 (3.24) = 1, 𝑡 Ö 𝑊𝑡 = 1 ←→ Ö 𝑡 𝑃𝑡 , 𝑡 where 𝑤 2 is defined in (3.22) and 𝐸 satisfies 𝜕𝐸 = 𝑤 2 . The modified Gauss law looks complicated, but it can be written down more concisely if we describe the spin configurations by a 2-cochain 𝐵 ∈ 𝐶 2 (𝑇, Z2 ). Our convention is that 𝐵( 𝑓 ) = 1 if 𝑍 𝑓 = −1 and 𝐵( 𝑓 ) = 0 if 𝑍 𝑓 = 1. Thus the unconstrained Hilbert space is spanned by vectors |𝐵i for all 𝐵. A 2-form gauge transformation has a 1-cochain Λ as a parameter and acts by 𝐵 ↦→ 𝐵 + 𝛿Λ. For a general Λ, the Gauss law constraint is given by ∫ ∫ 𝛿Λ∪1 𝒇 0 ª ©Ö ª ©Ö 𝑋𝑓 ® ­ 𝑍𝑓0 ­ ® (−1) Λ∪𝛿Λ = 1. 0 « 𝑓 ∈𝛿Λ ¬ « 𝑓 ¬ (3.25) 2 Actually, one can show that the second Stiefel-Whitney class of any oriented 3-manifold is trivial, and therefore the above 1-cycle is a boundary even in the non-simply-connected case. Our bosonization procedure works also for non-simply-connected spaces. The only difference is that apart from local Gauss law constraints one also needs to impose non-local constraints, one for each non-trivial class in 𝐻1 (𝑋, Z2 ). For example, for a 3-torus one would need to impose three constraints, one for each direction 𝑥, 𝑦, 𝑧. These constraints on the bosonic side are needed to reproduce the freedom to impose either periodic or anti-periodic boundary conditions on the fermions. But in this paper we limit ourselves to topologically-trivial (simply-connected) spaces. 34 ∫ This formula is proved by 𝒆 ∪ 𝛿𝒆 = 0 and induction: ∫ ∫ Ö Ö 𝛿Λ ∪ 𝒇 0 Ö Ö 𝛿Λ ∪ 𝒇 0 1 1 1 2 1 2 Λ1 ∪𝛿Λ1 ( 𝑋 𝑓1 )( 𝑍𝑓0 )(−1) ( 𝑋 𝑓2 )( 𝑍𝑓0 )(−1) Λ2 ∪𝛿Λ2 𝑓1 ∈𝛿Λ1 𝑓10 1 𝑓2 ∈𝛿Λ2 𝑓20 2 ∫ ∫ © Ö ª ©Ö 𝛿(Λ1 +Λ2 )∪1 𝒇 0 ª 𝑋𝑓 ® ­ 𝑍𝑓0 =­ ® (−1) Λ1 ∪𝛿Λ1 +Λ2 ∪𝛿Λ2 (−1) 𝛿Λ1 ∪1 𝛿Λ2 0 « 𝑓 ∈𝛿(Λ1 +Λ2 ) ¬ « 𝑓 ¬ ∫ © Ö ª ©Ö 𝛿(Λ1 +Λ2 )∪1 𝒇 0 ª 𝑍𝑓0 𝑋𝑓 ® ­ =­ ® (−1) (Λ1 +Λ2 )∪𝛿(Λ1 +Λ2 ) , 0 « 𝑓 ∈𝛿(Λ1 +Λ2 ) ¬ « 𝑓 ¬ (3.26) ∫ ∫ where we use the identity 𝛿Λ1 ∪1 𝛿Λ2 = Λ1 ∪ 𝛿Λ2 + 𝛿Λ2 ∪ Λ1 in the last equality. Eq. (3.25) can be concisely written as ∫ ©Ö ª 𝑋 𝑓 ® (−1) Λ∪𝛿Λ+𝛿Λ∪1 𝐵 = 1, ­ « 𝑓 ∈𝛿Λ ¬ (3.27) where 𝐵 is an arbitrary 2-cochain with values in Z2 . Consider now the following 2-form gauge theory defined on a general triangulated 4D manifold 𝑌 : ∫ Õ 𝑆(𝐵) = |𝛿𝐵(𝑡)| + 𝑖𝜋 (𝐵 ∪ 𝐵 + 𝐵 ∪1 𝛿𝐵). (3.28) 𝑌 𝑡 Here 𝐵 ∈ 𝐶 2 (𝑌 , Z2 ) is a Z2 field living at each face, and the gauge symmetry acts by 𝐵 → 𝐵 + 𝛿Λ for an arbitrary 1-cochain Λ. The second term is the Steenrod square topological action [23], which is used in [24] to construct fermionic topological phases. The action is gauge-invariant up to a boundary term: ∫ 𝑆(𝐵 + 𝛿Λ) − 𝑆(𝐵) = (Λ ∪ 𝛿Λ + 𝛿Λ ∪1 𝐵). (3.29) 𝜕𝑌 This boundary term determines the Gauss law for the wave-function Ψ(𝐵) on the spatial slice 𝑋 = 𝜕𝑌 : Ψ(𝐵 + 𝛿Λ) = (−1) 𝜔(Λ,𝐵) Ψ(𝐵), (3.30) ∫ where 𝜔(Λ, 𝐵) = 𝑋 (Λ ∪ 𝛿Λ + 𝛿Λ ∪1 𝐵). The Gauss law is the same as the gauge constraint (3.27). In Section 3.4, we use this observation to construct a 4D lattice action for particular Hamiltonian gauge theories with the modified Gauss law. 3.3 Examples In this section, we describe examples for the bosonization map in (3+1)D. 35 Soluble 3+1D lattice gauge theories Î The standard Gauss law for a 2-form Z2 gauge theory is 𝑓 ⊃𝑒 𝑋 𝑓 = 1. Such a bosonic gauge theory is dual to a theory of bosonic spins living on the vertices of the dual lattice. In particular, the quantum Ising model can be described by a Z2 2-form gauge theory with the Hamiltonian Õ 1 Õ 𝑋𝑓 + 2 𝐻 𝐼𝑠𝑖𝑛𝑔 = 𝑔 2 𝑊𝑡 . (3.31) 𝑔 𝑡 𝑓 This model is not soluble. If we impose the modified Gauss law (3.6) instead, the simplest analogous gaugeinvariant Hamiltonian is Õ 1 Õ 𝑈𝑓 + 2 𝐻𝑏 = 𝑔2 𝑊𝑡 . (3.32) 𝑔 𝑡 𝑓 The first and second term can be thought of as the kinetic and potential energies, respectively. This is dual to the fermionic Hamiltonian Õ 𝑐 𝐿( 𝑓 ) 𝑐 𝑅( 𝑓 ) − 𝑐†𝐿( 𝑓 ) 𝑐†𝑅( 𝑓 ) 𝐻 𝑓 =𝑡 𝑓 + 𝑐†𝐿 ( 𝑓 ) 𝑐 𝑅( 𝑓 ) + 𝑐†𝑅( 𝑓 ) 𝑐 𝐿( 𝑓 )  +𝜇 Õ 𝑐†𝑡 𝑐 𝑡 , (3.33) 𝑡 where 𝑡 = 𝑔 2 and 𝜇 = 𝑔22 . The fermionic Hamiltonian is free and thus soluble. By Í ® Fourier transform 𝑐 𝑥® = √1 𝑘® 𝑒𝑖 𝑘·®𝑥 𝑐 𝑘® , the fermionic Hamiltonian becomes 𝑁 𝐻𝑓 = Õ 𝜖 𝑘® 𝑐†® 𝑐 𝑘® + Õ 𝑘 (Δ 𝑘® 𝑐 𝑘® 𝑐 − 𝑘® + h.c.) (3.34) 𝑘® 𝑘® with 𝜖 𝑘® = 𝜇 + 2𝑡 (cos 𝑘 𝑥 + cos 𝑘 𝑦 + cos 𝑘 𝑧 ) and Δ 𝑘® = 𝑡 (𝑒 −𝑖𝑘 𝑥 + 𝑒 −𝑖𝑘 𝑦 + 𝑒 −𝑖𝑘 𝑧 ). The Hamiltonian (3.34) can be written in the Bogoliubov-de-Gennes (BdG) formalism as 1Õ † ® ® 𝐻𝑓 = Ψ® 𝐻BDG ( 𝑘)Ψ (3.35) 𝑘 𝑘 2 𝑘® with " ® = 𝐻BDG ( 𝑘) 𝜖 𝑘® −Δ ∗® −Δ 𝑘® −𝜖 𝑘® 𝑘 # " , # 𝑐 𝑘® Ψ𝑘® = † . 𝑐 ® (3.36) −𝑘 The spectrum is 𝐸2 = 𝑡 2 (3 + 2 cos(𝑘 𝑥 − 𝑘 𝑦 ) + 2 cos(𝑘 𝑥 − 𝑘 𝑧 ) + 2 cos(𝑘 𝑦 − 𝑘 𝑧 )) (3.37) + [𝜇 + 2𝑡 (cos 𝑘 𝑥 + cos 𝑘 𝑦 + cos 𝑘 𝑧 )] 2 . 4𝜋 Notice that for 𝜇 = 0 the gap closes for 𝑘® = (𝑞, 𝑞 + 2𝜋 3 , 𝑞 + 3 ) for arbitrary 𝑞. 36 Bosonic model with Dirac cones Using the bosonization map (3.2) and (3.3), we can construct an equivalent bosonic model for any arbitrary fermionic model. For instance, Ref. [54] constructs a fermionic model on a cubic lattice with Dirac cones: Õ 𝐻 = −𝑡 (𝑠𝑥 (® 𝑟 )𝑐𝑟†®+𝑥ˆ 𝑐𝑟® + 𝑠 𝑦 (® 𝑟 )𝑐𝑟†®+𝑦ˆ 𝑐𝑟® + 𝑠 𝑧 (® 𝑟 )𝑐𝑟†®+𝑧ˆ 𝑐𝑟® + h.c.) (3.38) 𝑟® with 𝑠𝑥 (® 𝑟 ), 𝑠 𝑦 (® 𝑟 ), and 𝑠 𝑧 (® 𝑟 ) defined as 𝑠𝑥 (𝑖, 𝑗, 𝑘) = 1 𝑠 𝑦 (𝑖, 𝑗, 𝑘) = (−1) 𝑖 (3.39) 𝑠 𝑧 (𝑖, 𝑗, 𝑘) = (−1) 𝑖+ 𝑗 . It is a model with nearest neighbor hopping. The spectrum is q 𝐸 = ±2𝑡 cos2 𝑘 𝑥 + cos2 𝑘 𝑦 + cos2 𝑘 𝑧 (3.40) with two Dirac cones at 𝑘® = (𝜋/2, 𝜋/2, 𝜋/2) and 𝑘® = (3𝜋/2, 𝜋/2, 𝜋/2). Applying the bosonization map, the corresponding bosonic Hamiltonian is 𝑡 Õ 𝐻 =− 𝑠𝑥 (𝐿 ( 𝑓𝑥 ))𝑈 𝑓 𝑥 (1 − 𝑊 𝐿( 𝑓 𝑥 ) 𝑊 𝑅( 𝑓 𝑥 ) ) 2 𝑓 𝑥 𝑡 Õ − 𝑠 𝑦 (𝐿 ( 𝑓 𝑦 ))𝑈 𝑓 𝑦 (1 − 𝑊 𝐿( 𝑓 𝑦 ) 𝑊 𝑅( 𝑓 𝑦 ) ) (3.41) 2 𝑓𝑦 𝑡 Õ − 𝑠 𝑧 (𝐿 ( 𝑓𝑧 ))𝑈 𝑓𝑧 (1 − 𝑊 𝐿( 𝑓𝑧 ) 𝑊 𝑅( 𝑓𝑧 ) ), 2 𝑓 𝑧 where 𝑓𝑥 , 𝑓 𝑦 , 𝑓𝑧 refer to faces normal to 𝑥, 𝑦, 𝑧-directions, with gauge constraints (3.6). On the bosonic side, it is very nontrivial to see that the model describes Dirac cones. 3.4 Euclidean 3+1D gauge theories with fermionic duals In the previous section we constructed a 3d bosonization map which works on the kinematic level (that is, is independent of the Hamiltonian). In this section we apply it to some specific models of free fermions and describe the corresponding dual gauge theories. We then construct Euclidean formulations of these gauge theories. We will make use of cup products, and thus will assume that the 3d space is triangulated, as in section 3.2. Accordingly, the (3+1)D lattice will be the product of the 3d triangulation and discrete time. As explained in the Appendix, (higher) cup 37 products can also be defined on the 3d cubic lattice, thus similar considerations can be used to find the Euclidean formulation of gauge theories constructed in Section 3.1. Consider the simplest gauge-invariant Hamiltonian compatible with the modified Gauss law: Õ Õ 𝐻 = −𝐴 𝑈𝑓 − 𝐵 𝑊𝑡 . (3.42) 𝑡 𝑓 The gauge constraint is ∫ 𝛿𝒆∪1 𝒇 0 ©Ö ª Ö = 1. 𝑍𝑓0 𝐺𝑒 ≡ ­ 𝑋𝑓 ® 0 𝑓 ⊃𝑒 𝑓 « ¬ (3.43) The partition function can be calculated by transfer matrix method: Z = Tr 𝑒 −𝛽𝐻 = Tr 𝑇 𝑀 , (3.44) where 𝑇 is the transfer matrix defined as ! 𝑇= Ö 𝛿𝐺 𝑒 ,1 𝑒 −𝛿𝜏𝐻 , (3.45) 𝑒 where 𝛿𝜏 ≡ 𝛽/𝑀, and 𝑀  1 is a large positive integer. The first factor arises from the gauge constraints on the Hilbert space. For calculation purposes, we can rewrite it as 1 𝛿𝐺 𝑒 ,1 = (1 + 𝐺 𝑒 ) 2 1−𝑍 𝑓 0 1−𝑋 𝑓 1−𝜆𝑒 Í 1−𝜆𝑒 Í 1 Õ = (−1) 2 𝑓 0 ∈NE(𝑒) 2 · (−1) 2 𝑓 ⊃𝑒 2 2 𝜆 =±1 (3.46) 𝑒 ∫ with NE(𝑒) ≡ { 𝑓 | 𝛿𝑒 ∪1 𝑓 = 1}. Here 𝜆 𝑒 is the Lagrange multiplier on each edge 𝑒 of the spatial manifold 𝑀 and will be consider as Z2 fields living on “temporal” faces later. To calculate the partition function (3.44), the completed bases are inserted between 𝑇. Define |𝑚(𝜏)i = |{𝑆 𝑓 }i as the configuration of spins (in 𝑍 𝑓 basis). Then the matrix elements of 𝑇 are h𝑚0 (𝜏 + 𝛿𝜏)|𝑇 |𝑚(𝜏)i ! = h𝑚0 (𝜏 + 𝛿𝜏)| Ö 𝑣 𝛿𝐺 𝑒 ,1 𝑒 −𝛿𝜏𝐻 |𝑚(𝜏)i. (3.47) 38 Next we need to use an identity h𝑆 𝑧0 | 𝑓 (𝜎 𝑥 , 𝜎 𝑧 )|𝑆 𝑧 i 1−𝑆 𝑥 1−𝑆 𝑧 0 1−𝑆 𝑧 1 Õ = 𝑓 (𝑆 𝑥 , 𝑆 𝑧 )(−1) 2 ( 2 + 2 ) , 2 𝑆 𝑥 =±1 (3.48) where we assume that 𝜎 𝑥 is to the right of 𝜎 𝑧 in the function 𝑓 (𝜎 𝑥 , 𝜎 𝑧 ). Plugging this into (3.47), we get ! Ö 0 h𝑚 (𝜏 + 𝛿𝜏)| 𝛿𝐺 𝑒 ,1 𝑒 −𝛿𝜏𝐻 |𝑚(𝜏)i 𝑒 ! = h𝑚0 (𝜏 + 𝛿𝜏)| Ö 𝑒 ©Ö Õ 𝑥 ª 𝛿𝐺 𝑒 ,1 ­ |𝑆 𝑓 ih𝑆 𝑥𝑓 | ® 𝑒 −𝛿𝜏𝐻 |𝑚(𝜏)i 𝑥 « 𝑓 𝑆 𝑓 =±1 ¬ 1−𝑆 𝑧 0   1−𝑆 𝑥 Í 𝑓3 𝑓2 (3.49) 1−𝜆𝑒 Í  Õ ∫ + Í 𝑓2 ⊃𝑒 𝑓3 ∈NE(𝑒) 2 2 2 0  𝑒∪𝛿𝑒  𝜆𝑒 ,𝜆𝑒 0 =−1 (−1) ∼ (−1)     𝜆𝑒 =±1   !   ! 1−𝑆 𝑥 1−𝑆 𝑧 0 1−𝑆 𝑧 𝑓 𝑓 𝑓  Ö Ö Õ Î Î 2 2 + 2 𝐵𝛿𝜏 𝑓4 ⊂𝑡 𝑆 𝑧𝑓 𝐴𝛿𝜏𝑆 𝑒𝑥 𝑓1 ∈Δ ( 𝑓 ) 𝑆 𝑧𝑓   4 , 1 𝑒 𝑒 (−1)   𝑡  𝑥   𝑓 𝑆 𝑓 =±1   ∫ Í ∫ 0 where Δ ( 𝑓 ) ≡ { 𝑓 0 | 𝒇 0 ∪1 𝒇 = 1} and the term (−1) 𝜆𝑒 ,𝜆𝑒 0 =−1 𝒆∪𝛿𝒆 comes from pushing all 𝑋 𝑓 operators to the right, which is the same as the last factor of Eq. (3.25). This term can be expressed as Õ∫ 𝑖𝜋 𝑎𝑖 ∪ 𝛿𝑎𝑖 (3.50) ! 𝑖 if we define 𝑎𝑖 ∈ 𝐶 1 (𝑀, Z2 ) as 1-cochain on the spacetial manifold 𝑀 (𝑖 𝑡ℎ layer) with 𝑎𝑖 (𝑒) = 1 for 𝜆 𝑒 = −1 and 𝑎𝑖 (𝑒) = 0 for 𝜆 𝑒 = 1. We can interpret 𝑎𝑖 at the 𝑖 𝑡ℎ layer as a 2-cochain which lives on the “temporal” faces between the 𝑖 𝑡ℎ and (𝑖 + 1) 𝑡ℎ layers. 39 After extracting this factor, the remaining terms are ! ! 1−𝑆 𝑧 0 Õ Ö Ö 𝐵𝛿𝜏 Î 𝑆 𝑧 𝑓3 1−𝜆𝑒 Í 𝑓 ⊂𝑡 𝑓4 4 (−1) 2 𝑓3 ⊂NE(𝑒) 2 𝑒 𝜆 𝑒 =±1 𝑒 𝑡 1−𝑆 𝑧 0 1−𝑆 𝑧 Í 𝑓 𝑓 1−𝜆𝑒 2 + 2 + 𝑒⊂ 𝑓 2 1−𝑆 𝑥 𝑓 2 !    Ö Õ Î 𝐴𝛿𝜏𝑆 𝑒𝑥 𝑓1 ∈Δ ( 𝑓 ) 𝑆 𝑧𝑓   1 ·𝑒 (−1)       𝑓 𝑆 𝑥𝑓 =±1   ! ! 𝑧 0 1−𝑆 Î Õ Ö Ö 𝑓3 1−𝜆𝑒 Í 𝐵𝛿𝜏 𝑓4 ⊂𝑡 𝑆 𝑧𝑓 4 = (−1) 2 𝑓3 ⊂NE(𝑒) 2 𝑒 𝜆 𝑒 =±1 𝑒 𝑡 Ö Î Î Ö  𝑧 𝑧  𝐴𝛿𝜏 𝑆 −𝐴𝛿𝜏 𝑆 0 𝑧 𝑧 𝑓1 ∈Δ ( 𝑓 ) 𝑓 𝑓1 ∈Δ ( 𝑓 ) 𝑓  (𝑒 1 + 𝑒 1𝑆 𝑆 𝜆 𝑒 )   𝑓 𝑓  𝑒  𝑒⊂ 𝑓  !  ! 0 𝑧 1−𝑆 Ö 𝐵𝛿𝜏 Î 𝑆 𝑧 Õ Ö 𝑓3 1−𝜆𝑒 Í 𝑓4 ⊂𝑡 𝑓 𝑓3 ⊂NE(𝑒) 2 2 4 𝑒 (−1) ∼ 𝜆 𝑒 =±1 𝑡 𝑒  1−𝑆 𝑧 Í 𝑓1 Ö 𝑧 0 𝑧 Î 𝑓1 ∈Δ ( 𝑓 ) 2  𝐽𝑆 𝑓 𝑆 𝑓 𝑒 ⊂ 𝑓 𝜆 𝑒 (−1) 𝑒   𝑓   ! Í = Õ (−1) (3.51) 1−𝜆𝑒 Í 𝑒 𝑓3 ⊂NE(𝑒) 2 1−𝑆 𝑧 0 𝑓3 2 1−𝑆 𝑧 0 1−𝑆 𝑧 Í 𝑓 𝑓 1−𝜆𝑒 2 + 2 + 𝑒⊂ 𝑓 2 𝑧 1−𝑆 Í Í 𝑓1 + 𝑓 𝑓1 ∈Δ ( 𝑓 ) 2 𝜆 𝑒 =±1 Í Î Î Í 𝐽 𝑓 𝑆 𝑧𝑓 0 𝑆 𝑧𝑓 𝑒1 ⊂ 𝑓 𝜆 𝑒1 +𝐵𝛿𝜏 𝑡 𝑓4 ⊂𝑡 𝑆 𝑧𝑓 4 𝑒 ! ! !        1−𝑆 𝑧 0 1−𝑆 𝑧 Í 1−𝜆𝑒1 𝑓 𝑓 2 + 2 + 𝑒1 ⊂ 𝑓 2 ! , where tanh 𝐽 = 𝑒 −2𝐴𝛿𝜏 . The last line is the usual action for a 4D Z2 gauge theory except for some sign factors. We regard these factors as coming from a topological action 𝑆top . From the penultimate line in (3.51), we see that 𝑆top contains Õ 𝑧 0  1 − 𝜆 𝑒 Õ 1 − 𝑆 𝑓3  𝑖𝜋  + 2 2  𝑒 𝑓3 ⊂NE(𝑒)  Õ © Õ 1 − 𝑆 𝑧𝑓 ª © 1 − 𝑆 𝑧𝑓 0 1 − 𝑆 𝑧𝑓 Õ 1 − 𝜆 𝑒 ª 1 1  + + ­ ® ·­ ® . 2 2 2 2  𝑓 « 𝑓1 ∈Δ ( 𝑓 ) 𝑒 ⊂ 𝑓 1 ¬ « ¬ (3.52) The first term is Õ 1 − 𝜆 𝑒 Õ Õ 1 − 𝑆 𝑧𝑓 0 3 2 𝑒 = 𝑓 ⊃𝑒 𝑓 ∪1 𝑓3 =1 2 Õ  Õ 1 − 𝜆 𝑒   Õ 1 − 𝑆 𝑧𝑓 0  3 𝑓 𝑒⊂ 𝑓 2 𝑓 ∪1 𝑓3 =1 2 (3.53) 40 ∫ which is equal to 𝛿𝑎𝑖 ∪1 𝑏𝑖+1 if we define 𝑏𝑖 as a 2-cochain on the 𝑖th layer with 1−𝑆 𝑏𝑖 ( 𝑓 ) = 2 𝑓 . The second term is Õ 𝑓 which is we get ∫ 1 − 𝑆 𝑧𝑓1   1 − 𝑆 𝑧𝑓 0 1 − 𝑆 𝑧𝑓 Õ 1 − 𝜆 𝑒  1 + + · 2 2 2 2 𝑒 ⊂𝑓 Õ 𝑓1 | ∫ 𝑓1 ∪1 𝑓 =1 (3.54) 1 𝑏𝑖 ∪1 (𝑏𝑖 + 𝑏𝑖+1 + 𝛿𝑎𝑖 ). Collecting all terms in (3.50), (3.53), and (3.54), 𝑆top ({𝑎𝑖 }, {𝑏𝑖 }) =𝑖𝜋 Õ∫ 𝑎𝑖 ∪ 𝛿𝑎𝑖 + 𝛿𝑎𝑖 ∪1 𝑏𝑖+1 (3.55) 𝑖 + 𝑏𝑖 ∪1 (𝑏𝑖 + 𝑏𝑖+1 + 𝛿𝑎𝑖 ). 𝐽 Í 𝑆𝑧 0𝑆𝑧 Î 𝜆 +𝐵𝛿𝜏 The usual term 𝑒 𝑓 𝑓 𝑓 𝑒1 ⊂ 𝑓 𝑒1 of (up to an unimportant constant) Í Î 𝑇 𝑧 𝑓4 ⊂𝑇 𝑆 𝑓 4 can be written as the exponential 𝑆4D gauge ({𝑎𝑖 }, {𝑏𝑖 }) Õ Õ = − 2𝐽 |𝑏𝑖 ( 𝑓 ) + 𝑏𝑖+1 ( 𝑓 ) + 𝛿𝑎𝑖 ( 𝑓 )| 𝑖 𝑓 − 2𝐵𝛿𝜏 Õ (3.56)  |𝛿𝑏𝑖 (𝑡)| , 𝑡 where | · · · | gives the argument’s parity 0 or 1. Combining (3.55) and (3.56), the Euclidean action becomes (up to an additive constant) 𝑆({𝑎𝑖 }, {𝑏𝑖 }) = 𝑆top ({𝑎𝑖 }, {𝑏𝑖 }) + 𝑆4D gauge ({𝑎𝑖 }, {𝑏𝑖 }), which is analogous to generalized Steenrod square action: ∫ 𝑖𝜋 (𝐵 ∪ 𝐵 + 𝐵 ∪1 𝛿𝐵). (3.57) (3.58) 𝑌 We will verify that these two actions produce the same boundary term under gauge transformation. This action is gauge-invariant (up to boundary terms) under gauge transformations 𝑏𝑖 → 𝑏𝑖 + 𝛿𝜆𝑖 , 𝑎𝑖 → 𝑎𝑖 + 𝛿𝜇𝑖 + 𝜆𝑖 + 𝜆𝑖+1 , (3.59) 41 where 𝜆𝑖 are arbitrary 1-cochains and 𝜇𝑖 are arbitrary 0-cochains. Indeed, the change in the action is ∫ 0 Δ𝑆top Õ >+ 𝜆 + 𝜆 ) ∪ (𝛿𝜆 + 𝛿𝜆 ) = (𝑎𝑖 +  𝛿𝜇 𝑖 𝑖 𝑖+1 𝑖 𝑖+1 (𝑖𝜋) 𝑖 0 >+ 𝜆 + 𝜆 ) ∪ 𝛿𝑎 + (𝛿𝜆 + 𝛿𝜆 ) ∪ 𝑏 + ( 𝛿𝜇 𝑖 𝑖 𝑖+1 𝑖 𝑖 𝑖+1 1 𝑖+1 + 𝛿𝑎 𝑖 ∪1 𝛿𝜆𝑖+1 :0   + (𝛿𝜆𝑖 +  𝛿𝜆 𝑖+1 ) ∪1 𝛿𝜆𝑖+1 + 𝛿𝜆𝑖 ∪1 (𝑏 𝑖 + 𝑏 𝑖+1 + 𝛿𝑎 𝑖 ) Õ∫ = 𝑎𝑖 ∪ (𝛿𝜆𝑖 + 𝛿𝜆𝑖+1 ) + (𝜆𝑖 + 𝜆𝑖+1 ) ∪ (𝛿𝜆𝑖 + 𝛿𝜆𝑖+1 ) 𝑖 + (𝜆𝑖 + 𝜆𝑖+1 ) ∪ 𝛿𝑎𝑖 + 𝑎𝑖 ∪ 𝛿𝜆𝑖+1 + 𝛿𝜆𝑖+1 ∪ 𝑎𝑖 + 𝜆𝑖 ∪ 𝛿𝜆𝑖+1 + 𝛿𝜆𝑖+1 ∪ 𝜆𝑖 + 𝑎𝑖 ∪ 𝛿𝜆𝑖 + 𝛿𝜆𝑖 ∪ 𝑎𝑖 Õ∫ = 𝜆𝑖 ∪ 𝛿𝜆𝑖 + 𝜆𝑖+1 ∪ 𝛿𝜆𝑖+1 = 0, 𝑖 (3.60) where the terms with the same colors cancel out. In the above computation we assumed periodic time, so that there are no boundary terms. If we do not identify time periodically, the variation is a boundary term ∫ (𝜆 0 ∪ 𝜆 0 + 𝛿𝜆0 ∪1 𝑏 0 ) + (𝜆 𝑁 ∪ 𝜆 𝑁 + 𝛿𝜆 𝑁 ∪1 𝑏 𝑁 ), (3.61) which is the same as the boundary term (3.29) in the previous section. We can also check that the action is invariant under a 2-form global symmetry 𝐵 → 𝐵 + 𝛽, (3.62) where a closed 2-cochain 𝛽 can be represented by 2-cochains 𝛽𝑖 (one for each time slice) and 1-cochains 𝛼𝑖 satisfying 𝛽𝑖 + 𝛽𝑖+1 + 𝛿𝛼𝑖 = 0. Using a gauge transformation (4.30) with 𝑖−1 Õ 𝜆𝑖 = 𝛼 𝑗 , 𝜇𝑖 = 0 (3.63) 𝑗=0 for 𝑖 = 0, 1, ..., 𝑁 − 1, we can see that 𝛽𝑖0 = 𝛽0 , which is independent of 𝑖, and Í 0 0 0 0 𝛼0𝑁−1 = 𝑁−1 𝑗=0 𝛼 𝑗 with other 𝛼𝑖 = 0. Notice that 𝛼 𝑁−1 is closed since 𝛽𝑖 = 𝛽𝑖+1 . 42 Under this 2-form symmetry transformation 𝛽0, the action changes by ∫ : 0Õ  Δ𝑆top    𝛼0𝑁−1 ∪ 𝛿𝑎 𝛿𝑎𝑖 ∪1 𝛽0 =  𝑁−1 +  (𝑖𝜋) 𝑖 *0 Õ    + 𝛽 0 ∪1 ( 𝑏𝑖  + 𝑏𝑖+1 ) + 𝛽0 ∪1 𝛿𝑎𝑖   𝑖 𝑖  Õ = Õ∫ (3.64) 𝑎𝑖 ∪ 𝛽0 + 𝛽0 ∪ 𝑎𝑖 + 𝛽0 ∪ 𝑎𝑖 + 𝑎𝑖 ∪ 𝛽0 = 0. 𝑖 Thus the action is invariant under a global 2-form symmetry, as expected. 3.5 Gauging fermion parity We have shown that a lattice fermionic system in 3d is dual to a bosonic spin system with the Gauss law constraints. In this section we show how to get rid of the constraints at the expense of coupling fermions to a Z2 gauge field. Our bosonization map is −𝑖𝛾𝑡 𝛾𝑡0 ←→ 𝑊𝑡 ≡ Ö 𝑍𝑓 𝑓 ⊂𝑡 (−1) ∫ 𝑓 𝐸 (𝑖𝛾 𝐿( 𝑓 ) 𝛾 0𝑅( 𝑓 ) ) ←→ 𝑈 𝑓 ≡ 𝑋 𝑓 Ö ∫ 𝑍𝑓0 𝒇 0 ∪1 𝒇 (3.65) 𝑓0 with gauge constraints ∫ 𝛿𝒆∪1 𝒇 0 ©Ö ª Ö = 1. (3.66) 𝑍𝑓0 𝑋𝑓 ® ­ 0 𝑓 𝑓 ⊃𝑒 « ¬ ˜ 𝑌˜ , and 𝑍, ˜ which live on Now, we introduce new Z2 fields (spins), with operators 𝑋, faces and couple to fermions via a Gauss law constraint Ö 𝑍˜ 𝑓 . (3.67) (−1) 𝐹𝑡 = 𝑓 ⊂𝑡 The fermionic hopping operator must be modified to ∫ 𝐸 𝑆 𝑓 = (−1) 𝐸 𝑓 (𝑖𝛾 𝐿( 𝑓 ) 𝛾 0𝑅( 𝑓 ) ) 𝑋˜ 𝑓 (3.68) in order to commute with the Gauss law constraint (3.67). The bosonization map becomes Ö Ö −𝑖𝛾𝑡 𝛾𝑡0 = 𝑍˜ 𝑓 ←→ 𝑊𝑡 ≡ 𝑍𝑓 𝑓 ⊂𝑡 (−1) ∫ 𝐸 𝑓 𝑓 ⊂𝑡 (𝑖𝛾 𝐿( 𝑓 ) 𝛾 0𝑅( 𝑓 ) ) 𝑋˜ 𝑓 ←→ 𝑈 𝑓 ≡ 𝑋 𝑓 Ö 𝑓0 ∫ 𝑍𝑓0 𝒇 0 ∪1 𝒇 (3.69) 43 and, similarly, the gauge constraints become ∫ 𝛿𝒆∪1 𝒇 0 ©Ö ª Ö . (3.70) 𝑍𝑓0 𝑋˜ 𝑓 ←→ ­ 𝑋𝑓 ® 0 𝑓 ⊃𝑒 « 𝑓 ⊃𝑒 ¬ 𝑓 The equations (3.69) and (3.70) define a bosonization map for fermions coupled to a dynamical Z2 gauge field. In this case, their is no constraint on the bosonic variables. Ö We can apply this modified boson/fermion map to a Z2 version of the Levin-Wen rotor model [53] on general triangulation: 𝐻=− Õ 𝑡 𝑄𝑡 − Õ 𝐵𝑒 (3.71) 𝑒 with 𝑄𝑡 = Ö 𝑍𝑓 𝑓 ⊂𝑡 Ö © Ö ∫ 𝒇 ∪ 𝒇 0ª 1 𝑍𝑓0 𝐵𝑒 = ­𝑋 𝑓 ® (3.72) 0 𝑓 𝑓 ⊃𝑒 « ¬ ∫ 𝛿𝒆∪1 𝒇 0 ©Ö ª Ö =­ 𝑋𝑓 ® 𝑍𝑓0 . 0 𝑓 ⊃𝑒 𝑓 « ¬ ∫ Î Î 𝛿𝑒∪1 𝑓 0 , the above bosonic model Since 𝑄 𝑡 and 𝐵𝑒 are just 𝑊𝑡 and ( 𝑓 ⊃𝑒 𝑋 𝑓 ) 𝑓 0 𝑍 𝑓 0 is equivalent to a model of a Z2 gauge field coupled to fermions and a Hamiltonian ÕÖ ÕÖ 𝐻=− 𝑍˜ 𝑓 − 𝑋˜ 𝑓 . (3.73) 𝑡 𝑓 ⊂𝑡 𝑒 𝑓 ⊃𝑒 The fermions are static, since the above Hamiltonian does not include fermionic hopping terms. The only interaction between the fermions and the gauge field is via the Gauss law constraint Ö 𝑍˜ 𝑓 = (−1) 𝐹𝑡 . (3.74) 𝑓 ⊂𝑡 Thus a complicated model bosonic model is mapped to a simple Z2 lattice gauge theory coupled to static fermions. As another application of the modified bosonization map, consider again the bosonic gauge theory on a cubic lattice with the Hamiltonian (3.41) 𝑡 Õ Õ 𝑠𝑖 (𝐿( 𝑓𝑖 ))𝑈 𝑓𝑖 (1 − 𝑊 𝐿 ( 𝑓𝑖 ) 𝑊 𝑅( 𝑓𝑖 ) ) (3.75) 𝐻=− 2 𝑖=𝑥,𝑦,𝑧 𝑓 𝑖 44 and a gauge constraint (3.6). This constrained model is dual to a model of free fermions with Dirac cones. After coupling the fermions to a Z2 gauge field and applying the modified map, we find that the bosonic model (3.75) without any gauge constraints is equivalent to a fermionic model with the Hamiltonian Õ 𝐻 = −𝑡 𝑠𝑥 (® 𝑟 ) 𝑋˜ 𝑥 (® 𝑟 )𝑐𝑟†®+𝑥ˆ 𝑐𝑟® + 𝑠 𝑦 (® 𝑟 ) 𝑋˜ 𝑦 (® 𝑟 )𝑐𝑟†®+𝑦ˆ 𝑐𝑟® 𝑟® (3.76)  † ˜ + 𝑠 𝑧 (® 𝑟 ) 𝑋𝑧 (® 𝑟 )𝑐𝑟®+𝑧ˆ 𝑐𝑟® + h.c. † Î Î with (−1) 𝑐𝑡 𝑐𝑡 = 𝑓 ⊂𝑡 𝑍˜ 𝑓 . The operators 𝑊˜ 𝑒 ≡ 𝑓 ⊃𝑒 𝑋˜ 𝑓 commute with the Hamiltonian, so we can project the Hilbert space into sectors with fixed 𝑊˜ 𝑒 (𝑊˜ 𝑒 is arbitrary Î ±1 as long as it satisfies 𝑒⊃𝑣 𝑊˜ 𝑒 = 1). In the sector 𝑊˜ 𝑒 = 1 for all 𝑒, the Hamiltonian (3.76) returns to (3.38). The model of unconstrained spins with the Hamiltonian (3.75) thus can be regarded as a 3d analog of Kitaev’s honeycomb model. 45 Chapter 4 EXACT BOSONIZATION IN 𝑛 DIMENSIONS 4.1 Triangulation From the 2d and 3d formulae (2.28) and (3.24), it is very natural to conjecture the 𝑛-dimensional boson-fermion duality. The fermions live at the center 𝑛-simplices, i.e. 𝛾Δ 𝑛 , 𝛾Δ0 𝑛 for each Δ 𝑛 . The Pauli matrices live on (𝑛 − 1)-simplices, i.e. 𝑋Δ 𝑛−1 and 𝑍Δ 𝑛−1 for each Δ 𝑛−1 . The 𝑛-dimensional boson-fermion duality should be Ö 𝑊Δ 𝑛 ≡ 𝑍Δ 𝑛−1 ←→ 𝑃𝑡 = −𝑖𝛾Δ 𝑛 𝛾Δ0 𝑛 , Δ 𝑛−1 ⊂Δ 𝑛 ∫ 𝚫𝒏−1 0 ∪𝑛−2 𝚫𝒏−1 ª ©Ö 𝑈Δ 𝑛−1 ≡ 𝑋Δ 𝑛−1 ­ 𝑍Δ 0 ® 𝑛−1 0 «Δ 𝑛−1 ¬ ∫ ∫ 𝚫𝒏−1 𝚫𝒏−1 𝐸 𝐸 ←→ (−1) 𝑆Δ 𝑛−1 = (−1) 𝑖𝛾 𝐿(Δ 𝑛−1 ) 𝛾 0𝑅(Δ 𝑛−1 ) , (4.1) ∫ 𝛿𝚫𝒏−2 ∪𝑛−2 𝚫𝒏−1 0 ª ©Ö 𝐺 Δ 𝑛−2 ≡ 𝑋Δ 𝑛−1 ­ 𝑍Δ 0 ® 𝑛−1 0 Δ 𝑛−1 ⊃Δ 𝑛−2 Δ « 𝑛−1 ∫ ¬ ∫ Ö 𝚫 ∪ 𝚫 +𝚫 ∪ 𝚫 𝚫 𝒏 𝒏 𝑛−2 𝑛−2 𝒏−2 𝒏−2 𝒏−2 ←→ (−1) 𝑤2 = 1, 𝑆 𝛿𝚫𝑛−2 𝑃Δ 𝑛 Ö Δ𝑛 Ö 𝑊Δ 𝑛 = 1 ←→ Δ𝑛 Ö 𝑃Δ 𝑛 , Δ𝑛 where 𝑤 2 ∈ 𝐶𝑛−2 (𝑀𝑛 , Z2 ) is the chain representative of the second Stiefel–Whitney class, 𝐸 ∈ 𝐶𝑛−1 (𝑀𝑛 , Z2 ) denotes a choice of spin structure (𝜕𝐸 = 𝑤 2 ), and for general (𝑛 − 1)-cochain 𝝀𝑛−1 and 𝝀0𝑛−1 , the product of 𝑆 operators is defined as 0 𝑆𝜆 𝑛−1 +𝜆 𝑛−1 ≡ (−1) ∫ 0 𝝀 𝑛−1 ∪𝑛−2 𝝀 𝑛−1 0 𝑆𝜆 𝑛−1 𝑆𝜆 𝑛−1 . (4.2) This 𝑛-dimensional boson-fermion duality (4.1) is the most crucial result of this paper, which will be proved by the end of this section. Commutation relations Consider an 𝑛-simplex Δ 𝑛 = h012 . . . 𝑛i. Its boundary contains all (𝑛 − 1)-simplex (𝜕Δ 𝑛 ) 𝑖 = h0 . . . 𝑖ˆ . . . 𝑛i where 𝑖ˆ means the vertex 𝑖 is omitted. We define the orientation of (𝜕Δ 𝑛 ) 𝑖 as 𝑂 ((𝜕Δ 𝑛 ) 𝑖 ) = (−1) 𝑖 . For “+”-oriented Δ 𝑛 , if 𝑂 ((𝜕Δ 𝑛 ) 𝑖 ) = 1, 46 the boundary (𝜕Δ 𝑛 ) 𝑖 is outward, and if 𝑂 ((𝜕Δ 𝑛 ) 𝑖 ) = −1, the boundary (𝜕Δ 𝑛 ) 𝑖 is inward. For “−”-oriented Δ 𝑛 , the inward and outward boundaries are opposite. 0 𝑆Δ 𝑛−1 and 𝑆Δ 𝑛−1 anti-commute only when Δ 𝑛−1 and Δ 0𝑛−1 are both inward or both outward boundaries of some 𝑛-simplex, i.e. Δ 𝑛−1 , Δ 0𝑛−1 ∈ 𝜕Δ 𝑛 . We are going to prove that this is equivalent to 0 𝑆Δ 𝑛−1 𝑆Δ 𝑛−1 = (−1) ∫ 0 +𝚫 0 ∪ 𝚫𝒏−1 ∪𝑛−2 𝚫𝒏−1 𝒏−1 𝑛−2 𝚫𝒏−1 0 𝑆Δ 𝑛−1 𝑆Δ 𝑛−1 . (4.3) From the definition of the higher cup product (A.2), we have [𝚫 𝒏−1 ∪𝑛−2 𝚫0𝒏−1 ] (0, 1, · · · , 𝑛) Õ = 𝚫 𝒏−1 (0 ∼ 𝑖0 , 𝑖1 ∼ 𝑖 2 , 𝑖3 ∼ 𝑖4 , · · · )𝚫0𝒏−1 (𝑖0 ∼ 𝑖1 , 𝑖2 ∼ 𝑖3 , · · · ) 0≤𝑖0 <𝑖1 <···<𝑖 𝑛−2 ≤𝑛 Õ = 𝚫 𝒏−1 (h0 . . . 𝑗ˆ2 . . . 𝑛i)𝚫0𝒏−1 (h0 . . . 𝑗ˆ1 . . . 𝑛i) (4.4) 0≤ 𝑗 1 < 𝑗2 ≤𝑛| 𝑗 1 , 𝑗 2 ∈even + Õ 𝚫 𝒏−1 (h0 . . . 𝑘ˆ1 . . . 𝑛i)𝚫0𝒏−1 (h0 . . . 𝑘ˆ2 . . . 𝑛i). 0≤𝑘 1 <𝑘 2 ≤𝑛|𝑘 1 ,𝑘 2 ∈odd The ∪𝑛−2 only contains the product of boundaries Δ 𝑖𝑛−1 with the same orientation 0 (inward or outward) and each pair of Δ 𝑖𝑛−1 , Δ 𝑖𝑛−1 with the same orientation appears exactly once. Therefore, the ∪𝑛−2 expression in (4.3) captures the commutation relations of fermionic hopping operators 𝑆Δ 𝑛−1 . It is easy to check that bosonic operators 𝑈Δ 𝑛−1 satisfy the same commutation relations: 0 𝑈Δ 𝑛−1 𝑈Δ 𝑛−1 = (−1) ∫ 0 +𝚫 0 ∪ 𝚫𝒏−1 ∪𝑛−2 𝚫𝒏−1 𝒏−1 𝑛−2 𝚫𝒏−1 0 𝑈Δ 𝑈Δ 𝑛−1 𝑛−1 . (4.5) Therefore, {𝑆Δ 𝑛−1 , 𝑃Δ 𝑛 } and {𝑈Δ 𝑛−1 , 𝑊Δ 𝑛 } in (4.1) have the same commutation relations. Gauge constraints In Appendix C, the identity for fermionic operators is derived: ∫ Ö ∫ 𝚫 ∪ 𝚫 +𝚫 ∪ 𝚫 𝚫 𝑛 𝑛−2 𝑛−2 𝑛−2 𝑛−2 𝑛 𝑤2 𝑛−2 (−1) 𝑆 𝛿Δ 𝑛−2 𝑃Δ 𝑛 = 1. (4.6) Δ𝑛 We can modify the sign of 𝑆Δ 𝑛−1 as ∫ 𝑆Δ𝐸 𝑛−1 ≡ (−1) 𝐸 𝚫𝑛−1 𝑆Δ 𝑛−1 , (4.7) where 𝐸 ∈ 𝐶𝑛−1 (𝑀𝑛 , Z2 ) is a choice of spin structure satisfying 𝜕𝐸 = 𝑤 2 . In these modified operators, the constraint on the fermionic operator becomes Ö ∫ 𝚫 ∪ 𝚫 +𝚫 ∪ 𝚫 𝑛 𝑛−2 𝑛−2 𝑛−2 𝑛−2 𝑛 𝐸 𝑆 𝛿Δ 𝑛−2 𝑃Δ 𝑛 = 1, (4.8) Δ𝑛 47 which is mapped to 𝐺 Δ 𝑛−2 =𝑈𝛿Δ 𝑛−2 Ö ∫ 𝚫𝑛−2 ∪𝑛−2 𝚫𝑛 +𝚫𝑛 ∪𝑛−2 𝚫𝑛−2 𝑊Δ 𝑛 Δ𝑛 = Ö 𝑋Δ 𝑛−1 ( Δ 𝑛−1 ⊃Δ 𝑛−2 Ö Δ 𝑛−1 ∫ 𝑍Δ 𝛿𝚫𝒏−2 ∪𝑛−2 𝚫𝒏−1 0 0 (4.9) ). 𝑛−1 0 We need to impose this gauge constraint 𝐺 Δ 𝑛−2 = 1 on bosonic operators for every (𝑛 − 2)-simplex Δ 𝑛−2 . We also need to impose the even total parity constraint for fermions Ö 𝑃Δ 𝑛 = 1 (4.10) Δ𝑛 Î since it is mapped to the bosonic operator Δ 𝑛 𝑊Δ 𝑛 = 1. After imposing the gauge constraints, the 𝑛-dimensional boson-fermion duality (4.1) is completed. 4.2 Modified Gauss’s law and Euclidean action Gauss’s law as boundary anomaly First, we consider the standard Z2 lattice gauge theory on the 𝑛-dimensional manifold 𝑀𝑛 : Õ Õ 𝐻 0 = −𝐴 𝑋Δ 𝑛−1 − 𝐵 𝑊Δ 𝑛 (4.11) Δ 𝑛−1 Δ𝑛 with the gauge constraint (Gauss’s law) Ö 𝐺 0Δ 𝑛−2 = 𝑋Δ 𝑛−2 = 1. (4.12) Δ 𝑛−1 ⊃Δ 𝑛−2 It is well-kwown that its Euclidean theory is (𝑛 + 1)-dimensional Ising model (with some choice of 𝐴 and 𝐵) [55]: Õ 𝑆Ising ( 𝐴𝑛−1 ) = −𝐽 |𝛿 𝐴𝑛−1 (Δ 𝑛 )|, (4.13) Δ 𝑛 ⊂𝑌 where 𝐴 ∈ 𝐶 𝑛−1 (𝑌 , Z2 ) is a (𝑛 − 1)-cochain on the spacetime manifold 𝑌 . In this case, 𝑆Ising is invariant under the gauge transformation 𝐴𝑛−1 → 𝐴𝑛−1 + 𝛿Λ𝑛−2 for arbitrary (𝑛 − 2)-cochain Λ𝑛−2 ∈ 𝐶 𝑛−2 (𝑌 , Z2 ). Therefore, 𝑆Ising has no boundary anomaly under the standard Gauss’s law. Now, we propose a new class of Z2 lattice gauge theory: Õ Õ 𝐻 = −𝐴 𝑈Δ 𝑛−1 − 𝐵 𝑊Δ 𝑛 Δ 𝑛−1 Δ𝑛 (4.14) 48 with the modified Gauss’s law (gauge constraints) at (𝑛 − 2)-simplices Ö Ö ∫ 𝛿𝚫 ∪ 𝚫 0 𝒏−2 𝑛−2 𝒏−1 𝑍Δ 0 ) = 1. 𝐺 Δ 𝑛−2 = 𝑋Δ 𝑛−1 ( (4.15) 𝑛−1 Δ 𝑛−1 ⊃Δ 𝑛−2 Δ 𝑛−1 0 This model describes a free fermion system, since it is dual to ∫ Õ Õ Δ 𝑛−1 0 𝐸 𝐻 𝑓 = −𝐴 (−1) 𝑖𝛾 𝐿(Δ 𝑛−1 ) 𝛾 𝑅(Δ 𝑛−1 ) − 𝐵 (−𝑖𝛾Δ 𝑛 𝛾Δ0 𝑛 ) Δ 𝑛−1 = −𝐴 Õ Δ𝑛 𝑆Δ𝐸 𝑛−1 − 𝐵 Δ 𝑛−1 Õ (4.16) 𝑃Δ 𝑛 . Δ𝑛 The modified Gauss’s law (4.15) on a (𝑛 − 2)-simplex Δ 𝑛−2 , or equivalently on the dual (𝑛 − 2)-cochain 𝚫𝑛−2 , can be generalized to an arbitrary (𝑛 − 2)-cochain Í 𝜆 𝑛−2 = 𝑖 𝚫𝑖𝑛−2 , the Gauss’s law is Ö 𝐺 𝜆 𝑛−2 = 𝐺Δ𝑖 𝑛−2 𝑖 Ö =( 𝑋Δ 𝑛−1 )( 𝚫𝑛−1 ∈𝛿𝜆 𝑛−2 Ö Δ 𝑛−1 ∫ 𝑍Δ 𝛿𝜆 𝑛−2 ∪𝑛−2 𝚫𝒏−1 0 0 ∫ )(−1) 𝜆 𝑛−2 ∪𝑛−4 𝜆 𝑛−2 +𝜆 𝑛−2 ∪𝑛−3 𝛿𝜆 𝑛−2 (4.17) 𝑛−1 0 =1, where the sign comes from anti-commutation of 𝑋 and 𝑍 on the same simplex. The derivation uses the following property of higher cup products: 𝐴 ∪𝑎 𝐵 + 𝐵 ∪𝑎 𝐴 = 𝐴 ∪𝑎+1 𝛿𝐵 + 𝛿 𝐴 ∪𝑎+1 𝐵 + 𝛿( 𝐴 ∪𝑎+1 𝐵). (4.18) Consider now the following (𝑛 − 1)-form gauge theory defined on a general triangulated (𝑛 + 1)-dimensional manifold 𝑌 : ∫ Õ 𝑆( 𝐴𝑛−1 ) = − |𝛿 𝐴𝑛−1 (Δ 𝑛 )| + 𝑖𝜋 ( 𝐴𝑛−1 ∪𝑛−3 𝐴𝑛−1 + 𝐴𝑛−1 ∪𝑛−2 𝛿 𝐴𝑛−1 ), (4.19) Δ 𝑛 ⊂𝑌 𝑌 where 𝐴𝑛−1 ∈ 𝐶 𝑛−1 (𝑌 , Z2 ), and the gauge symmetry acts by 𝐴𝑛−1 → 𝐴𝑛−1 + 𝛿Λ𝑛−2 for Λ𝑛−2 ∈ 𝐶 𝑛−2 (𝑌 , Z2 ). The second term is the generalized Steenrod square term defined in [24]. The action is gauge-invariant up to a boundary term: 𝑆( 𝐴𝑛−1 + 𝛿Λ𝑛−2 ) − 𝑆( 𝐴𝑛−1 ) ∫ =𝑖𝜋 (Λ𝑛−2 ∪𝑛−4 Λ𝑛−2 + Λ𝑛−2 ∪𝑛−3 𝛿Λ𝑛−2 + 𝛿Λ𝑛−2 ∪𝑛−2 𝐴𝑛−1 ) 𝜕𝑌 ∫ =𝑖𝜋 (Λ ∪𝑛−4 Λ + Λ ∪𝑛−3 𝛿Λ + 𝛿Λ ∪𝑛−2 𝐴), 𝜕𝑌 (4.20) 49 where we have omited the subscript of 𝐴𝑛−1 and Λ𝑛−2 for simplicity. This boundary term determines the Gauss law for the wave-function Ψ( 𝐴) on the spatial slice 𝑀 = 𝜕𝑌 : Ψ( 𝐴 + 𝛿Λ) = (−1) 𝜔(Λ,𝐴) Ψ( 𝐴), (4.21) ∫ where 𝜔(Λ, 𝐴) = 𝑀 (Λ ∪𝑛−4 Λ + Λ ∪𝑛−3 𝛿Λ + 𝛿Λ ∪𝑛−2 𝐴). The Gauss law is the same as the gauge constraint (4.17) if we identify 𝑍Δ 𝑛−1 as (−1) 𝐴𝑛−1 (Δ 𝑛−1 ) and 𝑋Δ 𝑛−1 acts as the transformation 𝐴𝑛−1 → 𝐴𝑛−1 + 𝚫𝑛−1 . The modified Gauss’s law (4.15) represents the boundary anomaly of topological action (4.19) as we claimed. In the following section, we derive the Euclidean action of the modified Z2 lattice gauge theory (4.14) explicitly, which is analogous to (4.19). Euclidean path integral of lattice gauge theories Start with the Hamiltonian of modified Z2 lattice gauge theory: Õ Õ 𝐻 = −𝐴 𝑈Δ 𝑛−1 − 𝐵 𝑊Δ 𝑛 Δ 𝑛−1 = −𝐴 Õ Δ𝑛 𝑋Δ 𝑛−1 ( Ö ∫ 𝑍Δ 𝚫𝒏−1 0 ∪𝑛−2 𝚫𝒏−1 0 )−𝐵 Õ Ö 𝑛−1 Δ 𝑛−1 0 Δ 𝑛−1 (4.22) 𝑍Δ 𝑛−1 Δ 𝑛 Δ 𝑛−1 ⊂Δ 𝑛 with gauge constraints Ö 𝐺 Δ 𝑛−2 = 𝑋Δ 𝑛−1 ( Δ 𝑛−1 ⊃Δ 𝑛−2 Ö Δ 𝑛−1 ∫ 𝑍Δ 𝛿𝚫𝒏−2 ∪𝑛−2 𝚫𝒏−1 0 0 ) = 1. (4.23) 𝑛−1 0 The partition function is: Z = Tr 𝑒 −𝛽𝐻 = Tr 𝑇 𝑀 , (4.24) where we use Trotter-Suzuki decomposition in imaginary time direction and 𝑇 is the transfer matrix defined as ! Ö 𝑇= 𝛿𝐺 Δ 𝑛−2 ,1 𝑒 −𝛿𝜏𝐻 . (4.25) Δ 𝑛−2 The first factor arises from the gauge constraints on the Hilbert space. The spacetime manifold consists of many time slices labelled by layers {𝑖}. In the 𝑖th layer, we insert a complete basis (in Pauli matrix 𝑍Δ 𝑛−1 ): 𝑏𝑖𝑛−1 ∈ 𝐶 𝑛−1 (𝑀𝑛 , Z2 ) (Z2 fields on 𝑖 each Δ 𝑛−1 of the spatial manifold 𝑀𝑛 such that 𝑍Δ 𝑛−1 = (−1) 𝑏 𝑛−1 (Δ 𝑛−1 ) ). The transfer matrix 𝑇 between the 𝑖th layer and the (𝑖 + 1)th layer contains gauge constraints on every spatial (𝑛 − 2)-simplex Δ 𝑛−2 : 𝑖+1/2 1 + 𝐺 Δ 𝑛−2 1 Õ 𝛿𝐺 Δ 𝑛−2 ,1 = = (𝐺 Δ 𝑛−2 ) 𝑎 𝑛−2 , (4.26) 2 2 𝑖+1/2 𝑎 𝑛−2 =0,1 50 𝑛−2 (𝑀 , Z ) (Z fields on where we introduce the Lagrangian multiplier 𝑎𝑖+1/2 𝑛 2 2 𝑛−2 ∈ 𝐶 𝑖+1/2 each Δ 𝑛−2 of the spatial manifold 𝑀𝑛 ). Notice that 𝑎 𝑛−2 defined between two time slices lives on the spatial (𝑛 − 2)-simplex Δ 𝑛−2 , which can be interpreted as the spacetime (𝑛 − 1)-simplex between the two layers. From the same calculation in Section 3.4, we have Õ 𝑖 (4.27) Z= exp([𝑆Ising + 𝑆top ] ({{𝑎𝑖+1/2 𝑛−2 }, {𝑏 𝑛−1 }})), 𝑖+1/2 {{𝑎 𝑛−2 },{𝑏 𝑖𝑛−1 }} where 𝑖 𝑆Ising ({{𝑎𝑖+1/2 𝑛−2 }, {𝑎 𝑛−1 }}) = Õ −𝐽𝑠 Õ i Õ h 𝑖+1/2 |𝛿𝑏𝑖𝑛−1 (Δ 𝑛 )| − 𝐽𝜏 | 𝑏𝑖𝑛−1 + 𝑏𝑖+1 + 𝛿𝑎 (Δ 𝑛−1 )| 𝑛−1 𝑛−2 Δ𝑛 𝑖 ! (4.28) Δ 𝑛−1 and 𝑖 𝑆top ({{𝑎𝑖+1/2 𝑛−2 }, {𝑏 𝑛−1 }}) ∫ Õ 𝑖+1/2 𝑖+1/2 𝑖+1/2 𝑎𝑖+1/2 = 𝑖𝜋 𝑛−2 ∪𝑛−4 𝑎 𝑛−2 + 𝑎 𝑛−2 ∪𝑛−3 𝛿𝑎 𝑛−2 𝑖 (4.29) 𝑀𝑛 𝑖+1/2 𝑖+1 𝑖 𝑖 𝑖+1 + 𝛿𝑎𝑖+1/2 𝑛−2 ∪𝑛−2 𝑏 𝑛−1 + 𝑏 𝑛−1 ∪𝑛−2 (𝑏 𝑛−1 + 𝑏 𝑛−1 + 𝛿𝑎 𝑛−2 ). Here 𝐽𝑠 , 𝐽𝜏 are constants depending on 𝐴, 𝐵, 𝛿𝜏 in the original Hamiltonian and we assume 𝐽𝑠 = 𝐽𝜏 = 𝐽 for simplicity. | · · · | gives the argument’s parity 0 or 1. The gauge transformations act as 𝑎𝑖𝑛−1 → 𝑎𝑖𝑛−1 + 𝛿𝜆𝑖 , 𝑖+1/2 𝑖 𝑖 𝑖+1 𝑎𝑖+1/2 𝑛−2 → 𝑎 𝑛−2 + 𝛿𝜇 + 𝜆 + 𝜆 , (4.30) where 𝜆𝑖 are arbitrary (𝑛 − 2)-cochains and 𝜇𝑖 are arbitrary (𝑛 − 3)-cochains. If we interpret 𝑎𝑖+1/2 𝑛−2 as spacetime (𝑛 − 1)-cochains, we can rewrite 𝑖 𝑛−1 {{𝑎𝑖+1/2 (𝑌 , Z2 ), 𝑛−2 }, {𝑏 𝑛−1 }} → 𝐴𝑛−1 ∈ 𝐶 (4.31) which is Z2 fields on (𝑛 − 1)-simplices in spacetime manifold 𝑌 . It is natural to write 𝑆Ising in (4.28) as Õ 𝑆Ising = − |𝛿 𝐴𝑛−1 (Δ 𝑛 )|. (4.32) Δ 𝑛 ⊂𝑌 The spacetime manifold 𝑌 = 𝑀𝑛 × [−∞, 0] (spatial and temporal parts) is not a triangulation, since we only triangularize the spatial manifold 𝑀𝑛 under the discretized 51 time. The (higher) cup products are not well-defined in 𝑌 . However, we can still write an expression ∫ 𝑆top = 𝑖𝜋 ( 𝐴𝑛−1 ∪𝑛−3 𝐴𝑛−1 + 𝐴𝑛−1 ∪𝑛−2 𝛿 𝐴𝑛−1 ) (4.33) 𝑌0 in (𝑛+1)-dimensional triangulation 𝑌 0 such that 𝑌 0 is a refinement of 𝑌 . We can check that (4.29) and (4.33) produce the same boundary term under gauge transformations. Part II Constructions of SPT phases 52 53 Chapter 5 (2+1)D SPT CONSTRUCTIONS In this chapter, we construct the supercohomology fSPT and its bosonic dual in (2+1)D. The main result is shown in Fig. 5.1. To construct the supercohomology 𝑓 fSPT protected by 𝐺 × Z2 , we first consider another bSPT protected by 𝐺˜ = 𝐺 o Z2 . 𝐺˜ is called Z2 extension of 𝐺 and will be defined explicitly in Section 5.2. We will show that the bSPT becomes a Z2 gauge theory after gauging the Z2 symmetry. This Z2 gauge theory is equivalent to a fermionic theory according to the exact bosonization developed in Part I. Therefore, from the well-known construction of bSPT, we are able to derive the finite-depth local unitary quantum circuit for supercohomology fSPT phases. This section is started with the reviewing of the well-known bosonic SPT constructions. We will show the correspondence between bSPT phases and fSPT phases explicitly. 𝑓 Figure 5.1: To construct a 𝐺 𝑓 = 𝐺 ×Z2 supercohomology SPT model in 2d, we start with a model for a particular bosonic SPT phase determined by the supercohomology data (𝜌, 𝜈). The symmetry of this bosonic SPT is 𝐺˜ = 𝐺 o Z2 (𝐺 extended by 𝑍 𝑍2 according to 2-cocycle 𝑛 ∈ 𝐻 2 (𝐵𝐺, Z2 )). Next, we gauge the Z2 subgroup of 𝐺˜ to build the shadow model, which is a Z2 lattice gauge theory. We then condense the fermion in the shadow model, or apply the fermionization duality, to obtain a model for the supercohomology SPT phase corresponding to (𝜌, 𝜈). 5.1 Review of group cohomology bSPT constructions The bosonic SPT phases are partially classified by group cohomology [32]. The 𝑑-dimensional interacting bosonic SPT phases with symmetry 𝐺 is characterized by 𝜈 ∈ 𝐻 𝑑+1 (𝐵𝐺, R/Z). Given the 𝐺-spins living at vertices, the SPT state can be 54 written as |Ψ𝑆𝑃𝑇 i = Õ Ö exp(2𝜋𝑖𝜈(1, 𝑔𝑣 1 , 𝑔𝑣 2 · · · 𝑔𝑣 𝑑+1 ))|{𝑔𝑣 }i {𝑔 𝑣 } h𝑣 1 𝑣 2 ···𝑣 𝑑+1 i ≡𝑈 Õ (5.1) |{𝑔𝑣 }i, {𝑔 𝑣 } where the product is over all spatial 𝑑-simplexes. The Hamiltonian is simply Õ 𝐻𝑆𝑃𝑇 = 𝑈 (− 𝑃𝑣 )𝑈 † , (5.2) 𝑣 where 𝑃𝑣 is the projector to the state inhomogeneous form of the cocycle: Í 𝑔 |𝑔i at a vertex 𝑣. We can also use the 𝜈(𝑔0 , 𝑔1 , · · · , 𝑔𝑑 , 𝑔𝑑+1 ) ≡ 𝜔(𝑔0−1 𝑔1 , 𝑔1−1 𝑔2 , · · · 𝑔𝑑−1 𝑔𝑑+1 ). (5.3) We denote the configuration of 𝐺-spins {𝑔𝑣 } as Φ and the SPT state (5.1) becomes ∫ Õ |Ψ𝑆𝑃𝑇 i = exp( 𝜔(𝑑Φ))|Φi, (5.4) Φ 𝐶𝑀 where 𝑑Φ is defined in Fig 5.2 and the integral is over spacetime manifold 𝐶 𝑀 (the "cone" of the spatial manifold 𝑀). Figure 5.2: The configuration of 𝐺-spins and the gauging map. 5.2 Correspendence between bSPT and supercohomology fSPT 𝑓 Let us start with the (2+1)D case with trivial product symmetry 𝐺 × 𝑍2 first [42]. fSPTs have supercohomology data (𝜈, 𝑛) with 𝜈 ∈ 𝐶 3 (𝐵𝐺, R/Z), 𝑛 ∈ 𝐻 2 (𝐵𝐺, Z2 ). They satisfy Gu-Wen equations: 55 1 𝛿𝜈 = 𝑛 ∪ 𝑛 2 𝛿𝑛 = 0. (5.5) The 2-cocycle 𝑛 ∈ 𝐻 2 (𝐵𝐺, Z2 ) gives the central extension 0 → Z2 → 𝐺˜ → 𝐺 → 1. (5.6) The group elements in 𝐺˜ are 𝑔 𝑎 with label 𝑎 = 0, 1 and the group law is 𝑔1𝑎1 𝑔2𝑎2 = (𝑔1 𝑔2 ) 𝑎1 +𝑎2 +𝑛(1,𝑔1 ,𝑔1 𝑔2 ) . (5.7) We can define a cocycle in 𝐻 3 (𝐵𝐺 0, R/Z) by 1 𝛼3 = 𝜈 + 𝑛 ∪ 𝜖 1 , 2 (5.8) where 𝜖1 (𝑔1𝑎1 , 𝑔2𝑎2 ) ≡ 𝑎 1 + 𝑎 2 + 𝑛(1, 𝑔1 , 𝑔2 ) is a 1-cochain in 𝐺 0 satisfying 𝛿𝜖 1 = 𝑛. One can check that this definition is homogeneous, i.e. 𝜖1 (ℎ0𝑔𝑖0, ℎ0𝑔0𝑗 ) = 𝜖 (𝑔𝑖0, 𝑔0𝑗 ). In inhomogeneous form of 𝜖1 , the above definition for 𝜖1 is (𝑎 ) 𝜖1 ((𝑔𝑖(𝑎𝑖 ) ) −1 𝑔 𝑗 𝑗 ) = 𝜖1 ((𝑔𝑖−1 𝑔 𝑗 ) (𝑎𝑖 +𝑎 𝑗 +𝑛(1,𝑔𝑖 ,𝑔 𝑗 )) ) (5.9) ≡ 𝑎𝑖 + 𝑎 𝑗 + 𝑛(1, 𝑔𝑖 , 𝑔 𝑗 ), which is equivalent to 𝜖1 (𝑔 (𝑎) ) = 𝑎 (we have assumed 𝑛 is normal, i.e. 𝑛(1, 1, 𝑔) = 0). By this 𝛼3 , we can construct an auxiliary bosonic SPT (with 𝐺˜ symmetry) as = |Ψ𝑏 i Õ Ö 𝑎𝑝 𝑎𝑞 𝑎𝑟 𝑒 2𝜋𝑖𝛼3 (1,𝑔 𝑝 ,𝑔𝑞 ,𝑔𝑟 )𝑂 𝑝𝑞𝑟 |{𝑔𝑣 }, {𝑎 𝑣 }i {𝑔 𝑣 },{𝑎 𝑣 } 𝑓 =h𝑝𝑞𝑟i = Õ Ö h 𝑒 2𝜋𝑖𝜈(1,𝑔 𝑝 ,𝑔𝑞 ,𝑔𝑟 )𝑂 𝑝𝑞𝑟 × (5.10) {𝑔 𝑣 },{𝑎 𝑣 } 𝑓 =h𝑝𝑞𝑟i i (−1) 𝑛(1,𝑔 𝑝 ,𝑔𝑞 ) (𝑎 𝑞 +𝑎𝑟 +𝑛(1,𝑔𝑞 ,𝑔𝑟 ) ) |{𝑔𝑣 }, {𝑎 𝑣 }i, where 𝑓 = h𝑝𝑞𝑟i is a 2-simplex (face) and 𝑂 𝑝𝑞𝑟 denotes the orientation of the simplex h𝑝𝑞𝑟i . This state is invariant under multiplication of constant ℎ0 on all vertices, i.e 𝑔𝑣𝑎 𝑣 → (ℎ𝑔𝑣 ) 𝑎 𝑣 +𝑛(1,ℎ,ℎ𝑔 𝑣 ) . In other words, the |Ψ𝑆𝑃𝑇 i is invariant under the symmetry action: |{𝑔𝑣 }, {𝑎 𝑣 }i → |{ℎ𝑔𝑣 }, {𝑎 𝑣 + 𝑛(1, ℎ, ℎ𝑔𝑣 )}i. (5.11) 56 The next step is to gauge the 0-form symmetry on {𝑎 𝑣 }. This is a duality mapping 𝑎 𝑣 degrees of freedom on vertices to those living on edges. The procedure for gauging the 0-form symmetry or higher-form symmetry is reviewed in Appendix D. Since each configuration {𝑎 𝑣 } can be represented by the cochain 𝒂 𝑣 , the gauging map is defined as Γ(|𝒂 𝑣 i) = |𝛿𝒂 𝑣 i, (5.12) where 𝛿𝒂 𝑣 are Z2 fields living on edges, i.e. 𝛿𝒂 𝑣 (𝑒𝑖 𝑗 ) = 𝑎𝑖 + 𝑎 𝑗 , shown in Fig. D.1 in Appendix D. The bosonic shadow wavefunction is = |Ψ𝑠 i = Γ(|Ψ𝑏 i) Õ Ö  𝑒 2𝜋𝑖𝜈(1,𝑔 𝑝 ,𝑔𝑞 ,𝑔𝑟 )𝑂 𝑝𝑞𝑟 × {𝑔 𝑣 },{𝑎 𝑣 } 𝑓 =h𝑝𝑞𝑟i =  (−1) 𝑛(1,𝑔 𝑝 ,𝑔𝑞 ) (𝑎 𝑞 +𝑎𝑟 +𝑛(1,𝑔𝑞 ,𝑔𝑟 ) ) |{𝑔𝑣 }, {𝛿𝑎 𝑣 }i  Õ Ö  2𝜋𝑖𝜈(1,𝑔 𝑝 ,𝑔𝑞 ,𝑔𝑟 )𝑂 𝑝𝑞𝑟 𝑛¯ 𝑝𝑞 (𝑎 𝑞 +𝑎𝑟 +𝑛¯ 𝑞𝑟 ) 𝑒 (−1) (5.13) {𝑔 𝑣 },{𝑎 𝑣 } h𝑝𝑞𝑟i |{𝑔𝑣 }, {𝑎𝑖 + 𝑎 𝑗 + 𝑛(1, 𝑔𝑖 , 𝑔 𝑗 )}i0, where in the last line, we defined a new basis of states |{𝑔𝑣 }, {𝑎𝑖 + 𝑎 𝑗 + 𝑛(1, 𝑔𝑖 , 𝑔 𝑗 )}i0 ≡ |{𝑔𝑣 }, {𝑎𝑖 + 𝑎 𝑗 }i and 𝑛¯ ∈ 𝐶 1 (𝑀, Z2 ) is a 1-cochain: 𝑛(𝑒 ¯ 𝑖 𝑗 ) = 𝑛¯ 𝑖 𝑗 = 𝑛(1, 𝑔𝑖 , 𝑔 𝑗 ). (5.14) We can introduce the variables 𝑏 𝑒 ≡ 𝑎𝑖 + 𝑎 𝑗 + 𝑛(1, 𝑔𝑖 , 𝑔 𝑗 ) (Z2 fields living on edges) and Pauli operators 𝑍 𝑒 ≡ (−1) 𝑏 𝑒 which measures the 𝑏 𝑒 variables. Doing so, the symmetry action (5.11) becomes 𝑉 (ℎ) : |{𝑔𝑣 }, {𝑏 𝑒 }i0 → |{ℎ𝑔𝑣 }, {𝑏 𝑒 }i0 (5.15) which acts only on the vertex variables 𝑔𝑣 . The action on 𝑏 𝑒 cancels because of the cocycle condition of 𝑛. The bosonic shadow state can be written as Õ Ö h |Ψ𝑠 i = 𝑒 2𝜋𝑖𝜈(1,𝑔 𝑝 ,𝑔𝑞 ,𝑔𝑟 )𝑂 𝑝𝑞𝑟 × {𝑔 𝑣 },{𝑏 𝑒 } h𝑝𝑞𝑟i 𝛿𝑏 𝑒 (h𝑖 𝑗 𝑘i)=𝑛(𝑔𝑖 ,𝑔 𝑗 ,𝑔 𝑘 ) (5.16) (−1) 𝑛¯ 𝑝𝑞 𝑏 𝑞𝑟 |{𝑔𝑣 }, {𝑏 𝑒 }i0,  where we have assumed that the state lives on a simply connected manifold 𝑀 for simplicity. If 𝑀 has noncontractible loops, there will be additional conditions on 57 the holonomy of 𝑏 𝑒 fields. In this gauged state, we notice that there is an additional symmetry: |{𝑔𝑣 }, {𝑏 𝑒 }i0 → (−1) ∫ 𝑛∪𝜆 ¯ 𝑒 |{𝑔𝑣 }, {𝑏 𝑒 } + 𝜆i0 (5.17) for any closed 1-cochain 𝜆 ∈ 𝐶 1 (𝑀, Z2 ) where we identify Z2 fields {𝑏 𝑒 } as an 1-cochain in 𝐶 1 (𝑀, Z2 ). If we choose 𝜆 𝑒 = 𝛿𝒗 for a vertex 𝑣. This symmetry is ∫ |{𝑔𝑣 }, {𝑏 𝑒 }i0 →(−1) 𝑛∪𝒗 |{𝑔𝑣 }, {𝑏 𝑒 } + 𝛿𝒗i0 Ö ∫ 𝛿𝒆 0∪𝒗 Ö ) =( 𝑍𝑒 0 𝑋𝑒 |{𝑔𝑣 }, {𝑏 𝑒 }i0 𝑒0 (5.18) 𝑒⊃𝑣 0 = 𝐺˜ 𝑣 |{𝑔𝑣 }, {𝑏 𝑒 }i , where we have used 𝛿𝑏 𝑒 = 𝑛 and the Pauli matrices 𝑋𝑒 and 𝑍 𝑒 act on the second entry {𝑏 𝑒 }, i.e. 𝑋𝑒 |{𝑔𝑣 }, {𝑏 𝑒 }i0 = |{𝑔𝑣 }, {𝑏 𝑒 } + 𝒆i0 𝑍 𝑒 |{𝑔𝑣 }, {𝑏 𝑒 }i0 = (−1) 𝑏 𝑒 |{𝑔𝑣 }, {𝑏 𝑒 }i0 ∫ (5.19) 0 Î 𝒆 ∪𝛿𝒗 Î ) 𝑒⊃𝑣 𝑋𝑒 . In other words, the bosonic wavefunction is with 𝐺˜ 𝑣 = ( 𝑒 0 𝑍 𝑒 0 gauge-invariant under 𝐺˜ 𝑣 |Ψ𝑠 i = |Ψ𝑠 i (5.20) and we can apply our fermion-boson duality (2.28) (with a slightly different convention) to get the dual fermionic wavefunction. More explicitly, we further simplify the state |Ψ𝑠 i by expressing the state as Pauli matrices acting on the groundstate of toric code: = |Ψ𝑠 i Õ Ö  𝑒 2𝜋𝑖𝜈(1,𝑔 𝑝 ,𝑔𝑞 ,𝑔𝑟 )𝑂 𝑝𝑞𝑟 𝑛¯ 𝑍 𝑞𝑟𝑝𝑞  {𝑔 𝑣 },{𝑎 𝑣 } h𝑝𝑞𝑟i = Õ |{𝑔𝑣 }, {𝛿𝑎 𝑣 } + 𝑛i ¯ 0  Ö  𝑒 2𝜋𝑖𝜈(1,𝑔 𝑝 ,𝑔𝑞 ,𝑔𝑟 )𝑂 𝑝𝑞𝑟 (5.21) {𝑔 𝑣 },{𝑎 𝑣 } h𝑝𝑞𝑟i Ö Ö  ∫ 𝑛∪𝒆 ¯ 0 ¯ 𝑍𝑒 0 𝑋𝑒𝑛(𝑒) |{𝑔𝑣 }, {𝛿𝑎 𝑣 }i0 𝑒0 = Õ Ö  𝑒  𝑒 2𝜋𝑖𝜈(1,𝑔 𝑝 ,𝑔𝑞 ,𝑔𝑟 )𝑂 𝑝𝑞𝑟 𝑈˜ 𝑛¯ |{𝑔𝑣 }, {𝛿𝑎 𝑣 }i0, {𝑔 𝑣 },{𝑎 𝑣 } h𝑝𝑞𝑟i where 𝑛¯ is the 1-cochain defined in (5.14), and |{𝑔𝑣 }, {𝛿𝑎 𝑣 } + 𝑛i ¯ 0 is the shorthand notation of |{𝑔𝑣 }, {𝑎𝑖 + 𝑎 𝑗 + 𝑛(1, 𝑔𝑖 , 𝑔 𝑗 )}i0. The bosonic hopping operator 𝑈˜ 𝑒 is 58 defined as 𝑈˜ 𝑒 = ( Ö ∫ 𝒆∪𝒆 0 𝑍𝑒 0 ) 𝑋𝑒 (5.22) 𝑒0 or more generally for any 1-cochain 𝜆 ∈ 𝐶 1 (𝑀, Z2 ): Ö ∫ 𝜆∪𝒆 0 Ö ˜ 𝑈𝜆 = ( ) 𝑋𝑒 . 𝑍𝑒 0 𝑒0 (5.23) 𝑒|𝜆(𝑒)=1 (5.23) can be derived from (5.22) by ∫ 0 𝑈˜ 𝜆+𝜆 0 ≡ (−1) 𝜆∪𝜆 𝑈˜ 𝜆 0 𝑈˜ 𝜆 . (5.24) |Ψ𝑠 i in (5.21) is still invariant under the symmetry action (5.15) since we just reorganize the state. The final step is to apply the fermionization map (2.28): |Ψ𝑠 i ←→ |Ψ 𝑓 i  Õ Ö  2𝜋𝑖𝜈(1,𝑔 𝑝 ,𝑔𝑞 ,𝑔𝑟 )𝑂 𝑝𝑞𝑟 𝑆𝑛𝐸¯ |{𝑔𝑣 }, {𝑣𝑎𝑐}i 𝑓 , = 𝑒 (5.25) {𝑔 𝑣 } h𝑝𝑞𝑟i ∫ where 𝑆 𝑒𝐸 = (−1) 𝐸 𝒆 𝑖𝛾 𝐿(𝑒) 𝛾 0𝑅(𝑒) and its definition on general 1-cochain is: ∫ 0 𝐸 𝜆∪𝜆 𝐸 𝐸 𝑆𝜆+𝜆 𝑆𝜆 0 𝑆𝜆 , 0 ≡ (−1) (5.26) where 𝜆, 𝜆0 ∈ 𝐶 1 (𝑀, Z2 ). The toric code groundstate (𝑊 𝑓 = 1 ∀ 𝑓 ) is mapped to the vacuum state (𝑃 𝑓 = 1 ∀ 𝑓 ). The symmetry of this fermionic SPT states |Ψ 𝑓 i is |{𝑔𝑣 }, {𝑃 𝑓 }i 𝑓 → |{ℎ𝑔𝑣 }, {𝑃 𝑓 }i 𝑓 (5.27) since the duality map is defined as |{𝑔𝑣 }, {𝑏 𝑒 }i0 → |{𝑔𝑣 }, {𝑃 𝑓 = 𝛿𝑏 𝑒 }i 𝑓 (up to a sign) and the symmetry action on the bosonic state is simply |{𝑔𝑣 }, {𝑏 𝑒 }i0 → |{ℎ𝑔𝑣 }, {𝑏 𝑒 }i0 by (5.15). 59 Chapter 6 (3+1)D CONSTRUCTIONS In (2+1)D, the supercohomology fSPT with symmetry group 𝐺 corresponds to a ˜ where 𝐺˜ is a Z2 extension of 𝐺 by 2-cocycle bSPT with with symmetry group 𝐺, 𝑛 ∈ 𝐻 2 (𝐵𝐺, Z2 ). For (3+1)D, the structure becomes more complicated. We will show that every supercohomology fSPT with symmetry group 𝐺 corresponds to a “2-group” bSPT. The “2-group” symmetry contains both 0-form 𝐺 symmetry and 1-form Z2 symmetry, with nontrivial coupling between them. The 2-group is formed by the group 𝐺 extend by Z2 1-form symmetry, according the 3-cocycle 𝐻 3 (𝐵𝐺, Z2 ). The main result for the duality of 3d bSPT and fSPT phases is shown in Fig. 6.1. In this section, we will first introduce the concepts of 2-group, 2-group extension, and 2-gauge theory. We will use these concepts to construct 2-group bSPT in 3d. After gauging the 1-form Z2 symmetry of this 2-group, the bSPT becomes a Z2 gauge theory. By the bosonization in Part I, this Z2 gauge theory is equivalent a fermionic theory, which is the desired supercohomology fSPT in 3d. 𝑓 Figure 6.1: To construct a 𝐺 𝑓 = 𝐺 ×Z2 supercohomology SPT model in 3d, we start with a model for a particular 2-group SPT phase determined by the supercohomology data (𝜌, 𝜈). Next, we gauge the Z2 1-form symmetry of the 2-group to build the shadow model, which is a Z2 lattice gauge theory. We then condense the fermion in the shadow model, or apply the fermionization duality, to obtain a model for the supercohomology SPT phase corresponding to (𝜌, 𝜈). 6.1 2-group, 2-group extension, and 2-gauge theory 2-group and 2-group extension A 2-group G = (𝐺, 𝐴, 𝑡, 𝛼) is defined as below. 𝐺 and 𝐴 are groups, 𝑡 : 𝐴 → 𝐺 is a group homomorphism, and 𝛼 : 𝐺 → Aut( 𝐴) is an action of 𝐺 on 𝐴 such that it satisfies: 𝑡 (𝛼(𝑔)(𝑎)) = 𝑔𝑡 (𝑎)𝑔 −1 , 𝛼(𝑡 (𝑎))(𝑎0) = 𝑎𝑎0 𝑎 −1 . (6.1) 60 Given a 𝐺-module 𝐴 (group 𝐺 with action 𝛼(𝑔) on a group 𝐴), we consider a double extension: 𝑖 𝑡0 𝜋 1 → 𝐴→ − 𝐴0 − → 𝐺0 → − 𝐺 → 1, (6.2) where (𝐺 0, 𝐴0, 𝑡 0, 𝛼0) is a 2-group and 𝛼0 induces the action 𝛼 from 𝐺 to 𝐴. By the property of exact sequence, we have 𝐺 = 𝑡 0 and 𝐴 = ker 𝑡 0. It’s known that the equivalence classes of the double extension are labeled by elements in 𝐻 3 (𝐵𝐺, 𝐴). We can illustrate how they are related. First, consider a section 𝑠 : 𝐺 → 𝐺 0 and it satisfies the group law projectively: 𝑠(𝑔)𝑠(ℎ) = 𝑓 (𝑔, ℎ)𝑠(𝑔ℎ) (6.3) with 𝑓 : 𝐺 × 𝐺 → ker 𝜋. By associative property, it must satisfy [𝑠(𝑔) 𝑓 (ℎ, 𝑘)𝑠(𝑔) −1 ] 𝑓 (𝑔, ℎ𝑘) = 𝑓 (𝑔, ℎ) 𝑓 (𝑔ℎ, 𝑘). (6.4) Since we have 𝑡 0 = ker 𝜋, we can lift 𝑓 to 𝐹 : 𝐺 × 𝐺 → 𝐴0 and (6.4) is satisfied projectively: [𝛼0 (𝑠(𝑔))(𝐹 (ℎ, 𝑘))]𝐹 (𝑔, ℎ𝑘) = 𝑖(𝛾(𝑔, ℎ, 𝑘))𝐹 (𝑔, ℎ)𝐹 (𝑔ℎ, 𝑘), (6.5) where 𝛾 ∈ 𝑍 3 (𝐵𝐺, 𝐴) is a 3-cocycle. It can be shown that the cohomology class of 𝛾 labels the equivalence class of double extension (6.2). 2-gauge theory on lattice We now define 2-gauge theory, using data (𝐺, 𝐴, 𝛼, 𝛾) decribed in the previous section. For simplicity, we consider 𝐴 = Z2 and therefore only trivial 𝛼 action (𝐺 on Z2 ) exists. The field configuration is an assigenment of an element 𝑔𝑒 ∈ 𝐺 to each edge and of an element 𝑏 𝑓 ∈ Z2 to each triangle. A branching structure on the ¯ with 𝑔¯ ≡ {𝑔𝑒 } and triangulation is chosen. We denote the configuration to be ( 𝑔, ¯ 𝑏) ¯ needs to satisfy some constraints. First, on each triangle Δ 012 , 𝑔¯ 𝑏¯ ≡ {𝑏 𝑓 }. ( 𝑔, ¯ 𝑏) satisfies: 𝑔01 𝑔12 = 𝑔02 . (6.6) ¯ has the following relation: Second, on each tetrahedron, ( 𝑔, ¯ 𝐵) 𝛿𝑏 = 𝑏 012 + 𝑏 013 + 𝑏 023 + 𝑏 123 = 𝛾(𝑔01 , 𝑔12 , 𝑔23 ). (6.7) For the convenience, we will use homogeneous notation 𝜌 for cocycle later, i.e. 𝜌(1, 𝑔, 𝑔ℎ, 𝑔ℎ𝑘) = 𝛾(𝑔, ℎ, 𝑘). 61 The 0-form gauge transformation by ℎ¯ = {ℎ𝑣 } (ℎ𝑣 ∈ 𝐺 at each vertex 𝑣) is 𝑔¯ → 𝑔¯ ℎ : 𝑔𝑖ℎ𝑗 = ℎ𝑖−1 𝑔𝑖 𝑗 ℎ 𝑗 𝑏¯ → 𝑏¯ ℎ : 𝑏𝑖ℎ𝑗 𝑘 = 𝑏𝑖 𝑗 𝑘 + 𝜁 (𝑔𝑖 𝑗 , 𝑔 𝑗 𝑘 , ℎ𝑖 , ℎ 𝑗 , ℎ 𝑘 ) (6.8) with 𝜁 is Z2 -valued function satisfying ¯ = 𝜌( 𝑔¯ ℎ ) − 𝜌( 𝑔). 𝛿𝜁 ( 𝑔, ¯ ℎ) ¯ (6.9) The solution for 𝜁 always exists since we can consider 𝜌 as the label for DijkgraafWitten theory and gauge transformation of 𝐺 fields doesn’t change the cohomology class of 𝜌. Our choice is 𝜁 is (derived in appendix E): 𝜁 (𝑔12 , 𝑔23 , ℎ1 , ℎ2 , ℎ3 ) =𝜌(1, 𝑔12 , 𝑔12 𝑔23 , 𝑔12 𝑔23 ℎ3 ) + 𝜌(1, 𝑔12 , 𝑔12 ℎ2 , 𝑔12 𝑔23 ℎ3 ) (6.10) + 𝜌(1, ℎ1 , 𝑔12 ℎ2 , 𝑔12 𝑔23 ℎ3 ). The 1-form symmetry depends on 1-form Z2 -valued cochain 𝜆¯ = {𝜆 𝑒 }. The transformation is as usual 𝑔𝑒 → 𝑔𝑒 𝑏 𝑓 → 𝑏 𝑓 + 𝛿𝜆 𝑒 . (6.11) The theory defined above is called 2-gauge theory. The classifying space of a 2-group G, denoted as 𝐵G, is a 2-gauge theory. It can be described by a Δ-complex structure. 𝐵G contains one vertex and edges labelled by 𝑔 ∈ 𝐺. Its 2-simplices h012i are labeled by (𝑔01 , 𝑔12 , 𝑔02 , 𝑏 012 ) such that 𝑔01 𝑔12 = 𝑔02 and 𝑏 012 = 0, 1. Its 3-simplices h0123i contains boundary 2-simplices h012i, h013i, h023i, and h123i such that 𝛾(𝑔01 , 𝑔12 , 𝑔23 ) =𝜌(1, 𝑔01 , 𝑔01 𝑔12 , 𝑔01 𝑔12 𝑔23 ) =𝑏 012 + 𝑏 013 + 𝑏 023 + 𝑏 123 (mod 2). (6.12) For 𝑛 ≥ 4, we glue the 𝑛-simplex to any (𝑛 − 1)-cycle. More explicitly, we glue the boundary of a 𝑛-simplex h01 . . . 𝑛i with (𝑛 − 1)-simplices h0 . . . 𝑖ˆ . . . 𝑛i for 𝑖 = 0, 1, . . . , 𝑛, where 𝑖ˆ means that 𝑖 is omitted. According to Dijkgraaf and Witten, topological gauge theories with gauge group 𝐺 in (𝑑 + 1) spacetime dimensions are classified by 𝐻 𝑑+1 (𝐵𝐺, 𝑈 (1)). This was generalized to 2-group gauge theories by Kapustin and Thorngren [50]: the 2-gauge theories with 2-group G in (𝑑 + 1) spaectime dimensions is classified by 𝐻 𝑑+1 (𝐵G, 𝑈 (1)). 62 6.2 Auxiliary “2-group” bSPT In this section, we construct the supercohomology fermionic SPT phases in (3+1)dimensions from bosonic 2-group SPT phases. Here we give a quick introduction to 2-group SPTs. A 2-group G = (𝐺, Z2 , 𝛾) is labelled by two usual groups 𝐺, Z2 , and a 3-cocycle 𝛾 ∈ 𝐻 3 (𝐵𝐺, Z2 ). A 2-gauge theory on a lattice has 𝐺 elements 𝑔𝑒 ∈ 𝐺 on edges 𝑒 with flat condition 𝑔01 𝑔12 = 𝑔02 on each face h𝑖 𝑗 𝑘i and Z2 fields 𝑏 𝑓 = {0, 1} on faces 𝑓 such that on each tetrahedron h0123i 𝛾(𝑔01 , 𝑔12 , 𝑔23 ) = 𝛿𝑏 𝑓 (h0123i), (6.13) which is the most crucial condition for 2-gauge theories. We will construct a 2-group SPT wavefunction for a 2-group G and given 𝛼4 ∈ 𝐻 4 (𝐵G, 𝑈 (1)). On a spatial slice 𝑀, the “matter field” Φ contains {𝑔𝑣 } (elements of 𝐺) on vertices and {𝑎 𝑒 } (elements of Z2 ) on edges. We consider Φ = ({𝑔𝑣 }, {𝑎 𝑒 }) as a gauge transformation operator (6.8) and (6.11) with {ℎ𝑖 } = {𝑔𝑣 } and {𝜆 𝑒 } = {𝑎 𝑒 }, and apply this gauge transformation on the trivial configuration in 2-gauge theory on the spacetime manifold 𝐶 𝑀 (the "cone" of the spatial manifold 𝑀), with 𝑔𝑒 = 1 ∀𝑒 and 𝑏 𝑓 = 0 ∀ 𝑓 [50]. We obtain a configuration 𝑑Φ in the 2-gauge theory, as shown as Fig. 6.2. We will evaluate the 2-group cocycle 𝛼4 ∈ 𝐻 4 (𝐵G, 𝑈 (1)) on this 𝑑Φ. Figure 6.2: We have used 𝜁 (1, 1, 1, 𝑔, ℎ) = 0, which is equivalent to normalized 𝜌, i.e. 𝜌(1, 1, 𝑔, ℎ) = 0 [50]. The global symmetry transformation rules (ℎ, 𝜆) ∈ (𝐺, 𝑍 1 (𝑀, Z2 )) for matter fields Φ on spatial manifold 𝑀 can be written as 𝑔𝑣 → ℎ𝑔𝑣 𝑎𝑖 𝑗 → 𝑎𝑖 𝑗 + 𝜆𝑖 𝑗 + 𝜅 ℎ (𝑔𝑖 , 𝑔 𝑗 ), (6.14) 63 where 1-cochain 𝜅 is the second descendant of 𝛾 satisfying 𝛿𝜅 ℎ = 𝜁 (1, 1, ℎ𝑔𝑖 , ℎ𝑔 𝑗 , ℎ𝑔 𝑘 ) − 𝜁 (1, 1, 𝑔𝑖 , 𝑔 𝑗 , 𝑔 𝑘 ). (6.15) It can be checked that 𝑑Φ is invariant under this symmetry transformation. The global symmetry contains constant 0-form transformation by ℎ and closed 1-form (i.e. 𝛿𝜆 = 0) transformation by 𝜆. It is convenient to work with the homogeneous cocycle 𝜌, i.e. 𝜌(1, 𝑔, 𝑔ℎ, 𝑔ℎ𝑘) = 𝛾(𝑔, ℎ, 𝑘) with 𝜌(ℎ𝑔1 , ℎ𝑔2 , ℎ𝑔3 , ℎ𝑔4 ) = 𝜌(𝑔1 , 𝑔2 , 𝑔3 , 𝑔4 ). With the choice of 𝜁 in (6.10), we have 𝜁 (1, 1, 𝑔𝑖 , 𝑔 𝑗 , 𝑔 𝑘 ) = 𝜌(1, 𝑔𝑖 , 𝑔 𝑗 , 𝑔 𝑘 ). (6.16) We can further define 𝜅 ℎ (𝑔1 , 𝑔2 ) ≡ 𝜌(1, ℎ−1 , 𝑔1 , 𝑔2 ). (6.17) This 𝜅 ℎ satisfies (6.15) by the cocycle condition of 𝜌. We can evaluate the value of 𝛼4 on spacetime manifold 𝐶 𝑀 in an identical way as Chen-Liu-Gu-Wen’s bosonic SPT phase [32]. The groundstate wavefunction on 𝑀 can be written as   ∫ Õ |𝛼4 i = exp 2𝜋𝑖 𝛼4 (𝑑Φ) |Φi. (6.18) Φ 𝐶𝑀 From [46], 𝛼4 ∈ 𝐻 4 (𝐵G, 𝑈 (1)) can be constructed from Gu-Wen supercohomology data (𝜈, 𝜌) with 𝜈 ∈ 𝐶 4 (𝐵𝐺, 𝑈 (1)) and 𝜌 ∈ 𝑍 3 (𝐵𝐺, Z2 ) satisfying 𝛿𝜈 = 21 𝜌 ∪1 𝜌: 1 1 𝛼4 = 𝜈 + 𝜌 ∪1 𝜖 2 + 𝜖2 ∪ 𝜖2 , 2 2 (6.19) where 𝜖2 is the 2-cochain in 𝐶 2 (𝑀, Z2 ) satisfying 𝛿𝜖 2 = 𝜌. To simplify future expressions, we introduce cochains 𝜈¯ ∈ 𝐶 4 (𝑀, R/Z) and 𝜌¯ ∈ 𝐶 3 (𝑀, Z/2) 𝜈(1234) ¯ = 𝜈(Δ ¯ 1234 ) ≡ 𝜈(1, 𝑔1 , 𝑔2 , 𝑔3 , 𝑔4 ), (6.20) 𝜌(123) ¯ = 𝜌(Δ ¯ 1234 ) ≡ 𝜌(1, 𝑔1 , 𝑔2 , 𝑔3 ). (6.21) It is important to note that these two cochains cannot be pulled-back from some cochain on 𝐵𝐺. With this, we choose the cochain 𝜖 2 to be 𝜖2 (Δ 012 ) = 𝑎 12 𝜖2 (Δ 123 ) =𝛿𝑎(Δ 123 ) + 𝜌(𝑔 ¯ 1 , 𝑔2 , 𝑔3 ) (6.22) (6.23) 64 for time-like faces Δ 012 and spacelike faces Δ 123 , respectively1. We can then use this 𝛼4 to construct the 2-group SPT wavefunction as the following:   ∫ Ψ𝛼4 (Φ = ({𝑔𝑣 }, {𝑎 𝑒 })) = exp 2𝜋𝑖 𝛼4 (𝑑Φ) 𝐶𝑀 Ö (6.24) = 𝑒 2𝜋𝑖 𝜈(01234)𝑂 𝑡 (−1) 𝜖2 (012)𝜖2 (234) 𝑡=h1234i (−1) 𝜌(0234)𝜖2 (012)+𝜌(0134)𝜖2 (123)+𝜌(0124)𝜖2 (234) . Inserting 𝜖2 from (6.23), the final form of the auxillary bosonic SPT in terms of supercohomology data (𝜈 and 𝜌) is Õ Ö ¯ 𝑡 |Ψ𝑏 i = 𝑒 2𝜋𝑖 𝜈(1234)𝑂 (−1) 𝑎(12)𝛿𝑎(234) {𝑔 𝑣 },{𝑎 𝑒 } 𝑡=h1234i ¯ (𝛿𝑎(123)+ 𝜌(123))+ ¯ 𝜌(124) ¯ (𝛿𝑎(234)+ 𝜌(234)) ¯ × (−1) 𝜌(134) |{𝑔𝑣 }, {𝑎 𝑒 }i. (6.25) Shadow theory The wavefunction we have written down contains the 1-form Z2 symmetry. This guarantees that when we apply the gauging map Γ defined in (D.7), the gauged wavefunction will be the ground state the twisted toric code enriched by 𝐺. = |Ψ𝑠 i = Γ(|Ψ𝑏 i) Õ Ö ¯ 𝑡 𝑒 2𝜋𝑖 𝜈(1234)𝑂 (−1) 𝑎(12)𝛿𝑎(234) {𝑔 𝑣 },{𝑎 𝑒 } 𝑡=h1234i (6.26) ¯ (𝛿𝑎(123)+ 𝜌(123))+ ¯ 𝜌(124) ¯ (𝛿𝑎(234)+ 𝜌(234)) ¯ × (−1) 𝜌(134) |{𝑔𝑣 }, {𝛿𝑎 𝑒 }i. Next, we perform a basis transformation |{𝑔𝑣 }, {𝛿𝑎 𝑒 }i ≡ |{𝑔𝑣 }, {𝛿𝑎 𝑒 } + 𝜌i ¯ 0. Furthermore, we define Pauli matrices which acts on faces 𝑓 (the second entry of states) as 𝑋 𝑓 |{𝑔𝑣 }, {𝑏 𝑓 }i0 = |{𝑔𝑣 }, {𝑏 𝑓 } + 𝒇 i0 𝑍 𝑓 |{𝑔𝑣 }, {𝑏 𝑓 }i0 = (−1) 𝑏 𝑓 |{𝑔𝑣 }, {𝑏 𝑓 }i0 . 1 This choice is consistent with choosing the cocycle 𝜌 such that 𝜌(1, 1, 𝑔 , 𝑔 ) 1 2 (6.27) = 0 (in our notations, the vertex 0 represents the 𝑡 = −∞ point and the group element at this vertex is 1 ∈ 𝐺 65 The bosonic shadow state (6.26) becomes: = |Ψ𝑠 i Õ Ö ¯ 𝑡 𝑒 2𝜋𝑖 𝜈(1234)𝑂 (−1) 𝑎(12)𝛿𝑎(234) {𝑔 𝑣 },{𝑎 𝑒 } 𝑡=h1234i 𝜌(134) ¯ = 𝑍123 Õ 𝜌(124) ¯ ¯ 0 𝑍234 |{𝑔𝑣 }, {𝛿𝑎 𝑒 } + 𝜌i  Ö  ¯ 𝑡 𝑒 2𝜋𝑖 𝜈(1234)𝑂 {𝑔 𝑣 },{𝑎 𝑒 } 𝑡=h1234i Ö  ∫ Ö  ∫ 𝜌∪ ¯ 1𝑓0 𝜌( ¯ 𝑓) 𝑍𝑓0 𝑋𝑓 (−1) 𝑎 𝑒 ∪𝛿𝑎 𝑒 |{𝑔𝑣 }, {𝛿𝑎 𝑒 }i0 𝑓 𝑓0 = Õ Ö  (6.28)  ¯ 𝑡 𝑒 2𝜋𝑖 𝜈(1234)𝑂 {𝑔 𝑣 },{𝑎 𝑒 } 𝑡=h1234i 𝑈˜ 𝜌¯ (−1) ∫ 𝒂 𝑒 ∪𝛿𝒂 𝑒 |{𝑔𝑣 }, {𝛿𝑎 𝑒 }i0, where the bosonic hopping operator 𝑈˜ 𝑓 is: 𝑈˜ 𝑓 = ( Ö ∫ 𝑍𝑓0 𝒇 ∪1 𝒇 0 )𝑋𝑓 (6.29) 𝑓0 or more generally for any 2-cochain 𝜷 ∈ 𝐶 2 (𝑀, Z2 ): Ö ∫ 𝜷∪ 𝒇 0 Ö 1 𝑈˜ 𝛽 = ( 𝑍𝑓0 ) 𝑋𝑓 . 𝑓0 (6.30) 𝑓 | 𝜷( 𝑓 )=1 (6.30) can be derived from (6.29) by 𝑈˜ 𝛽+𝛽 0 ≡ (−1) ∫ 𝜷∪1 𝜷 0 ˜ 𝑈 𝛽 0 𝑈˜ 𝛽 . (6.31) Physically, emergent fermions in the twisted toric code are decorated on to junctions of 𝐺-domain walls according to 𝜌. ¯ Hence, 𝑈˜ 𝜌¯ describes the hopping of such fermions as the domain walls fluctuate. The state (6.28) is invariant under the symmetry action: 𝑉 (ℎ) : |{𝑔𝑣 }, {𝑏 𝑓 }i0 → |{ℎ𝑔𝑣 }, {𝑏 𝑓 }i0 (6.32) which is derived from (6.14) and definition of states |· · ·i0. The next step is to fermionize the bosonic shadow state (6.28). 66 6.3 Fermionic SPT Fermionization of the bosonic shadow wavefunction Consider a 𝐺-spin paramagnet and a decoupled copy of the twisted toric code: Õ Õ Õ 𝐻𝑠0 = − 𝑃𝑣 − 𝐺˜ 𝑒 − 𝑊𝑡 . (6.33) 𝑣 𝑒 𝑡 Here 𝑃𝑣 at each vertex 𝑣 is the projector onto the symmetric state √1 Í |𝐺 | 𝑔 𝑣 |𝑔𝑣 i: 1 Õ 0 |𝑔 ih𝑔𝑣 |. |𝐺 | 𝑔 ,𝑔 0 𝑣 𝑃𝑣 = 𝑣 (6.34) 𝑣 𝐺˜ 𝑒 is defined as 𝐺˜ 𝑒 = ( Ö ∫ 𝑍𝑓0 𝒇 0 ∪1 𝛿𝒆 𝑓0 ) Ö 𝑋𝑓 , (6.35) 𝑓 ⊃𝑒 which commutes with 𝑈˜ 𝑓 and 𝑊𝑡 . 𝐺˜ 𝑒 acts trivially on the state ∫ Õ |Ψ𝑠0 i = (−1) 𝒂 𝑒 ∪𝛿𝒂 𝑒 |{𝑔𝑣 }, {𝛿𝑎 𝑒 }i0, (6.36) {𝑔 𝑣 },{𝑎 𝑒 } i.e. 𝐺˜ 𝑒 |Ψ𝑠0 i = |Ψ𝑠0 i. In other words, |Ψ𝑠0 i is the groundstate of (6.33). Therefore, 𝐺˜ 𝑒 acts trivially on (6.28) (because 𝐺˜ 𝑒 commutes with 𝑈˜ 𝑓 ): 𝐺˜ 𝑒 |Ψ𝑠 i = |Ψ𝑠 i. (6.37) We can now apply the 3d boson-fermion duality (3.24) to |Ψ𝑠 i in (6.28): |Ψ𝑏 i ←→ |Ψ 𝑓 i  Õ Ö  = 𝑒 2𝜋𝑖 𝜈(01234)𝑂 𝑡 𝑆 𝐸𝜌¯ |{𝑔𝑣 }, {𝑣𝑎𝑐}i 𝑓 , (6.38) {𝑔 𝑣 } 𝑡=h1234i ∫ where 𝑆 𝐸𝑓 = (−1) 𝐸 𝒇 𝑖𝛾 𝐿( 𝑓 ) 𝛾 0𝑅( 𝑓 ) and its definition on general 2-cochain is 𝑆 𝐸𝛽+𝛽 0 ≡ (−1) ∫ 𝛽∪𝛽 0 𝐸 𝐸 𝑆 𝛽0 𝑆 𝛽 , (6.39) where 𝛽, 𝛽0 ∈ 𝐶 2 (𝑀, Z2 ). The groundstate of twisted toric code has been mapped to the vacuum state. The symmetry of this fermionic SPT states |Ψ 𝑓 i is |{𝑔𝑣 }, {𝑃𝑡 }i 𝑓 → |{ℎ𝑔𝑣 }, {𝑃𝑡 }i 𝑓 (6.40) since the duality map is defined as |{𝑔𝑣 }, {𝑏 𝑓 }i0 → |{𝑔𝑣 }, {𝑃𝑡 = 𝛿𝑏 𝑓 }i 𝑓 (up to a sign) and the symmetry action on the bosonic state is simply |{𝑔𝑣 }, {𝑏 𝑓 }i0 → |{ℎ𝑔𝑣 }, {𝑏 𝑓 }i0 by (6.32). 67 We are going to derive the parent Hamiltonian corresponding to the fermionic SPT state (6.38). We start on bosonic side. The bosonic shadow state |Ψ𝑠 i in (6.28) can be expressed by acting finite-depth local unitary quantum circuit 𝑈ˆ 𝑠 , which will be defined shortly, on the twisted toric code groundstate |Ψ𝑠0 i in (6.36): |Ψ𝑠 i = 𝑈ˆ 𝑠 |Ψ𝑠0 i. (6.41) Since |Ψ𝑠0 i is the groundstate of 𝐻𝑠0 in (6.33), the parent Hamiltonian of |Ψ𝑠 i is 𝐻𝑠 = 𝑈ˆ 𝑠 𝐻𝑠0𝑈ˆ 𝑠† . (6.42) From (6.28), the quantum circuit 𝑈ˆ 𝑠 can be defined as  Ö  𝜌(124) ¯ 𝜌(134) ¯ ˆ 𝑈𝑠 ≡ exp(2𝜋𝑖 𝜈(01234)𝑂 𝑡 )𝑍123 𝑍234 𝑡=h1234i Ö 𝜌( ¯ 𝑓) 𝑋𝑓 𝑓 (6.43) 𝑡=h1234i Ö =  Ö   𝜌(123)+ ¯ 𝜌(134) ¯ 𝑊𝑡 (exp(2𝜋𝑖 𝜈(01234)𝑂 𝑡 )) 𝑈˜ 𝜌¯  Ö  ∫ 𝜌∪ ¯ 2𝑡 , 𝑊𝑡 𝑡 𝑡=h1234i where 𝜈¯ ∈ 𝐶 3 (𝑀, R/Z) is an operator (R/Z-valued function) acting on tetrahedrons 𝑡 = h1234i of the spatial manifold 𝑀: 𝜈(𝑡) ¯ ≡ 𝜈(1, 𝑔1 , 𝑔2 , 𝑔3 , 𝑔4 ). Notice that we have include the factor of 𝑊𝑡 before 𝑈˜ 𝜌¯ . When acting on the |Ψ𝑏0 i, this factor becomes trivial since the groundstate of twisted toric code is a superposition of states with 𝑊𝑡 = 1 ∀𝑡. However, this 𝑊𝑡 will be important when we show the symmetry property of this circuit. Then, we can use our fermionization procedure to get the fermionic SPT Hamiltonian and wavefunction. Õ Õ 𝐻 0𝑓 = − 𝑃𝑣 − (−𝑖𝛾𝑡 𝛾𝑡0) 𝑣 𝑈ˆ 𝑓 = Ö = Ö 𝑡 𝐸 (exp(2𝜋𝑖 𝜈(𝑡)𝑂 ¯ 𝑡 )) 𝑆 𝜌¯  Ö  ∫ 𝜌∪ ¯ 2𝑡 𝑃𝑡 𝑡 𝑡 ! exp(2𝜋𝑖 𝜈(𝑡)𝑂 ¯ 𝑡) 𝑡 (6.44) Í ∫ Ö 0 ∫  © ª (−1) 𝐸 𝑓 𝑖𝛾 𝐿( 𝑓 ) 𝛾 0𝑅( 𝑓 ) ® ­ (−1) 𝑓 < 𝑓 0 ∈𝜌¯ 𝒇 ∪1 𝒇 𝑓 ∈ 𝜌¯ « ¬ ! ∫ Ö ¯ 2𝑡 (−𝑖𝛾𝑡 𝛾𝑡0) 𝜌∪ 𝑡 𝐻 𝑓 =𝑈ˆ 𝑓 𝐻 0𝑓 𝑈ˆ †𝑓 . 68 As a reminder, the explicitly formula for 𝑆 𝐸𝜌¯ is described in (3.19). 𝑓 ∈ 𝜌¯ means that Î 𝜌( ¯ 𝑓 ) = 1. The product 𝑓 ∈ 𝜌¯ 𝑆 𝑓 itself depends on the order of multiplying 𝑆 𝑓 since the fermionic hopping operators don’t commute with each other. The sign factor in Î front of 𝑓 ∈ 𝜌¯ 𝑆 𝑓 resolves this ambiguity of the multiplying order. 𝑃𝑣 is the projector Í 0 to state √1 𝑔 𝑣 |𝑔𝑣 i. 𝛾𝑡 and 𝛾𝑡 are majorana fermion operators, forming a complex |𝐺 | fermion at each tetrahedron 𝑡. The groundstate |Ψ 𝑓 i of 𝐻 𝑓 is the groundstate |Ψ0𝑓 i of 𝐻 0𝑓 evolved by circuit 𝑈ˆ 𝑓 . When we apply the quantum circuit 𝑈ˆ 𝑓 on |Ψ0𝑓 i, we can ignore the last factor in 𝑈ˆ 𝑓 since it’s 1. This is the (3+1)D fermionic SPT state and Hamiltonian. The circuit is symmetric: 𝑉 (𝑔)𝑈ˆ 𝑓 𝑉 (𝑔) † = 𝑈ˆ 𝑓 (6.45) and the stacking law of two fermionic SPT phases is 1 𝑈ˆ 𝑓 (𝜈1 , 𝜌1 )𝑈ˆ 𝑓 (𝜈2 , 𝜌2 ) = 𝑈ˆ 𝑓 (𝜈1 + 𝜈2 + 𝜌1 ∪2 𝜌2 , 𝜌1 + 𝜌2 ). 2 (6.46) 69 Chapter 7 SUMMARY In Part I, we have described a 𝑛-dimensional (𝑛 ≥ 2) analog of the Jordan-Wigner transformation. We started by 2d square lattice and showed the bosonization formalism explicitly: an arbitrary fermionic system can be mapped to Pauli matrices while preserving the locality of the Hamiltonian. After introducing the (higher) cup product notations, the bosonization can be extended to arbitrary triangulation in all dimensions easily. For simply-connected space, this bosonization gives a duality between any fermionic system in arbitrary 𝑛 spatial dimensions and a new class of (𝑛 − 1)-form Z2 gauge theories in 𝑛 dimensions with a modified Gauss’s law (gauge constraint). Several examples of 2d bosonization, including free fermions on square and honeycomb lattices and the Hubbard model, and 3d bosonization, including a solvable Z2 lattice gauge theory with Dirac cones in the spectrum, have been discussed. The key property of this bosonization formalism is the explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the manifold, which is a feature for defining fermions. To establish this, we have derived a new formula for Stiefel-Whitney homology classes on lattices. We also derive the Euclidean actions for the corresponding lattice gauge theories from the bosonization. The topological actions contain Chern-Simons terms for (2 + 1)D or Steenrod Square terms for general dimensions. In Part II, we apply the bosonization technique to construct various bosonic or fermionic SPT phases. We showed that for any supercohomolgy SPT with symmetry 𝑓 𝐺×Z2 , we are able to find a special bosonic SPT which can be dualized to a fermionic ˜ In 2d, group 𝐺˜ model. The bosonic SPT is protected by a different symmetry 𝐺. is simply a Z2 extension of 𝐺. In 3d, 𝐺˜ becomes a 2-group: 0-form 𝐺 extended by 1-form Z2 . The bosonization developed in Part II create a duality between any supercohomology fermionic SPT phase and a higher-group bosonic SPT phase. 70 Appendix A CHAINS AND COCHAINS NOTATIONS AND (HIGHER) CUP PRODUCTS ON A TRIANGULATION AND A CUBIC LATTICE In this section, we review some concepts in algebraic topology and also introduce notations used in this paper. We will always work with an arbitrary triangulation of a simply-connected 𝑛-dimensional manifold 𝑀𝑛 equipped with a branching structure (orientations on edges without forming a loop in any triangle). The vertices, edges, faces, and tetrahedra are denoted 𝑣, 𝑒, 𝑓 , 𝑡, respectively. A general 𝑑-simplex is denoted as Δ 𝑑 . We can label the vertices of Δ 𝑑 as 0, 1, 2, . . . , 𝑑 such that the orientations of edges are from the smaller number to the larger number. We denote this 𝑑-simplex as Δ 𝑑 = h01 . . . 𝑑i. A finite linear combination of 𝑑-simplices with coefficients in Z2 = {0, 1} is called a 𝑑-chain. 𝑑-chains form an abelian group denoted 𝐶𝑑 (𝑀𝑛 , Z2 ). A Z2 -valued 𝑑-chain can be identified with a finite set of 𝑑-simplices (the set consisting of those simplices whose coefficients are nonzero). A Z2 -valued 𝑑-cochain is a function from the set of 𝑑-simplices to Z2 . The set of all 𝑑-cochains is an abelian group denoted 𝐶 𝑑 (𝑀𝑛 , Z2 ), For example, a 0-cochain assigns 0 or 1 to all vertices, and a 1-cochain assigns 0 or 1 to all edges. A 𝑑-cochain can be evaluated on any 𝑑-chain by evaluating the cochain on each simplex of the 𝑑-chain and adding up the results modulo 2. For every vertex 𝑣 we define its dual 0-cochain 𝒗, which takes value 1 on 𝑣, and 0 otherwise, i.e. 𝒗(𝑣 0) = 𝛿𝑣,𝑣 0 . Similarly, e is an 1-cochain 𝒆(𝑒0) = 𝛿 𝑒,𝑒 0 , and so forth. All dual cochains will be denoted in bold. An evaluation of a cochain 𝒄 on a chain 𝑐0 ∫ will sometimes be denoted 𝑐 0 𝒄. For example, if 𝑐0 is a 2-chain and 𝒄 is a 2-cochain, ∫ Í 𝒄 ≡ 𝒄(𝑐0) = 𝑓 ∈𝑐 0 𝒄( 𝑓 ). When the integration range is not written, 𝒄 is assumed 𝑐0 ∫ ∫ to be the top dimension and 𝒄 ≡ 𝑀 𝒄. A 𝑑-cochain 𝒄 𝑑 ∈ 𝐶 𝑑 (𝑀𝑛 , Z2 ) can be 𝑛 thought of as a collection of Z2 -valued variables, one for each 𝑑-simplex Δ 𝑑 , whose values given by 𝒄 𝑑 (Δ 𝑑 ). We will limit ourselves to the case of Z2 -valued cochains, since this is all we need in this paper. The boundary operator is denoted by 𝜕. For an 𝑛-simplex Δ 𝑛 , 𝜕Δ 𝑛 consists of all boundary (𝑛 − 1)-simplices of Δ 𝑛 . The coboundary operator is denoted by 𝛿 (not to be confused with the Kronecker delta previously). For example, for a 0-cochain 𝒗, 𝛿𝒗 is a function on the set of edges which takes value 1 on 𝑒 if 𝜕𝑒 contains 𝑣, and 71 takes value 0 otherwise: 𝛿𝒗(𝑒𝑖 𝑗 ) = 𝒗(𝜕𝑒𝑖 𝑗 ) = 𝒗(𝑣 𝑖 + 𝑣 𝑗 ) = 𝛿𝑣,𝑣 𝑖 + 𝛿𝑣,𝑣 𝑗 . The notation Δ 1𝑛 ⊃ Δ 2𝑛 0 or Δ 2𝑛 0 ⊂ Δ 1𝑛 means that the simplex Δ 1𝑛 contains Δ 2𝑛 0 as one of its faces. For example, a simplex 𝑓 = h012i contains the edges 𝑒 = h01i, h02i, h12i ⊂ 𝑓 . In the case of a general triangulation, our bosonization procedure is based on the properties of the cup product ∪ and the higher cup product ∪1 . These mathematical operations have been defined by Steenrod [23] (see also Appendix B in [56] for a review) for an arbitrary simplicial complex, but not for a cubic lattice. Below we will define these operations on a triangulation and then describe a version which works for a cubic lattice. On a lattice triangulation , the cup product ∪ of a 𝑝-cochain 𝛼 𝑝 and a 𝑞-cochain 𝛽𝑞 is a ( 𝑝 + 𝑞)-cochain defined as [57]: [𝛼 𝑝 ∪ 𝛽𝑞 ] (h0, 1, . . . , 𝑝 + 𝑞i) = 𝛼 𝑝 (h0, 1, . . . , 𝑝i) 𝛽𝑞 (h𝑝, 𝑝 + 1, . . . , 𝑝 + 𝑞i) (A.1) = 𝛼 𝑝 (0 ∼ 𝑝) 𝛽𝑞 ( 𝑝 ∼ 𝑝 + 𝑞). The definition of the higher cup product [20, 23] is [𝛼 𝑝 ∪𝑎 𝛽𝑞 ] (0, 1, · · · , 𝑝 + 𝑞 − 𝑎) = Õ 𝛼 𝑝 (0 ∼ 𝑖 0 , 𝑖1 ∼ 𝑖2 , 𝑖3 ∼ 𝑖 4 , · · · ) 0≤𝑖 0 <𝑖1 <···<𝑖 𝑎 ≤ 𝑝+𝑞−𝑎 (A.2) × 𝛽𝑞 (𝑖0 ∼ 𝑖1 , 𝑖2 ∼ 𝑖3 , · · · ), where 𝑖 ∼ 𝑗 represents the integers from 𝑖 to 𝑗, i.e. 𝑖, 𝑖 + 1, . . . , 𝑗, and {𝑖0 , 𝑖1 , . . . , 𝑖 𝑎 } are chosen such that the arguments of 𝛼 𝑝 and 𝛽𝑞 contain 𝑝 + 1 and 𝑞 + 1 numbers separately. For arbitrary Z2 -cochains 𝐴 and 𝐵, the cup products satisfy this identity: 𝐴 ∪𝑎 𝐵 + 𝐵 ∪𝑎 𝐴 = 𝛿( 𝐴 ∪𝑎+1 𝐵) + 𝛿 𝐴 ∪𝑎+1 𝐵 + 𝐴 ∪𝑎+1 𝛿𝐵 (A.3) with ∪0 ≡ ∪ and ∪−1 ≡ 0. To generalize these formulas to the cubic lattice, we first develop an intuition for the cup product ∪. On a triangle Δ 012 , the usual cup product for two 1-cochains 𝜆 and 𝜆0 is 𝜆 ∪ 𝜆0 (012) = 𝜆(01)𝜆0 (12). (A.4) 72 We can think of it as starting from vertex 0, passing through edges 01 and 12 consecutively, and ending at vertex 2, all the while following the orientation of the edges. Following the same logic, it is intuitive to define the cup product on a square 0134 (the bottom face in Fig. A.1) as 𝜆 ∪ 𝜆0 (0134 ) = 𝜆(01)𝜆0 (14) + 𝜆(03)𝜆0 (34). (A.5) The two terms come from two oriented paths from vertex 0 to vertex 4. If 𝜆 and 𝛽 6 z 7 U y B 5 2 x 3 4 L R F 0 1 D Figure A.1: There are six faces for each cube 𝑐. U,D,F,B,L,R stand for faces on direction "up","down","front","back","left","right". We assign the face U, F, R to be inward and D, B, L to be outward. The ∪1 product on two 2-cochain is defined by 𝛽 ∪1 𝛽0 (𝑐) = 𝛽(𝐿) 𝛽0 (𝐵) + 𝛽(𝐿) 𝛽0 (𝐷) + 𝛽(𝐵) 𝛽0 (𝐷) + 𝛽(𝑈) 𝛽0 (𝐹) + 𝛽(𝑈) 𝛽0 (𝑅) + 𝛽(𝐹) 𝛽0 (𝑅) are a 1-cochain and a 2-cochain, the usual cup product is 𝜆 ∪ 𝛽(0123) = 𝜆(01) 𝛽(123) 𝛽 ∪ 𝜆(0123) = 𝛽(012)𝜆(23). (A.6) On the cubic lattice, the corresponding cup products are defined as follows: 𝜆 ∪ 𝛽(𝑐) = 𝜆(01) 𝛽(1457 ) + 𝜆(02) 𝛽(2567 ) + 𝜆(03) 𝛽(3467 ) 𝛽 ∪ 𝜆(𝑐) (A.7) = 𝛽(0236 )𝜆(67) + 𝛽(0125 )𝜆(57) + 𝛽(0134 )𝜆(47), where 𝑐 is a cube whose vertices are labeled in Fig. A.1. For a cup product involving 0-cochains 𝛼, the definition is straightforward: 𝛼 ∪ 𝛽(0134 ) = 𝛼(0) 𝛽(0134 ) 𝛽 ∪ 𝛼(0134 ) = 𝛽(0134 )𝛼(4) 𝛼 ∪ 𝜆(01) = 𝛼(0)𝜆(01) 𝜆 ∪ 𝛼(01) = 𝜆(01)𝛼(1). (A.8) 73 With the above definitions, it can be checked that the following equalities hold for cubic cochains of degrees 0, 1, and 2: 𝒆 1 ∪ 𝛿𝒆 2 = 𝛿𝒆 1 ∪ 𝒆 2 + 𝛿(𝒆 1 ∪ 𝒆 2 ) 𝒗 ∪ 𝛿 𝒇 = 𝛿𝒗 ∪ 𝒇 + 𝛿(𝒗 ∪ 𝒇 ). (A.9) The next step is to define the ∪1 product on the cubic lattice. It need not satisfy all the properties that ∪1 has on a triangulation. The only properties of ∪1 that we need are the anti-commutativity for faces with the same direction and the identity we used in (3.18), (3.26), and (3.60): ∫ ∫ 𝒆 1 ∪ 𝛿𝒆 2 + 𝛿𝒆 2 ∪ 𝒆 1 = 𝛿𝒆 1 ∪1 𝛿𝒆 2 (mod 2). (A.10) Therefore, we only need to define ∪1 product for two 2-cochains so that it satisfies (A.10). Our convention for ∪1 is shown in Fig. A.1: 𝛽 ∪1 𝛽0 (𝑐) = 𝛽(𝐿) 𝛽0 (𝐵) + 𝛽(𝐿) 𝛽0 (𝐷) + 𝛽(𝐵) 𝛽0 (𝐷) + 𝛽(𝑈) 𝛽0 (𝐹) + 𝛽(𝑈) 𝛽0 (𝑅) + 𝛽(𝐹) 𝛽0 (𝑅). (A.11) Once the ∪ and ∪1 products are defined on the cubic lattice, the bosonization procedure on a general triangulation can be applied to the cubic lattice. The formalism derived in Sec. 3.1 is just a special case of Sec. 3.2. In the derivation, (3.12) and (3.14) are modified as follows: 𝑆 𝛿𝑒 = (−1) Í 𝑓 < 𝑓 0 ∈ 𝛿𝒆 𝒇 ∪1 𝒇 0 Ö 𝑆𝑓 𝑓 ∈𝛿𝒆 Ö = (A.12) 𝑊𝑐 𝑐|𝑒∈{01,14,02,47,67,26} 𝑍 𝑓1 𝑍 𝑓2 Ö ( 𝒇 + 𝒇 2 )∪1 𝒇 0 + 𝒇 0 ∪1 ( 𝒇 1 + 𝒇 2 ) 𝑍𝑓01 𝑓 0 ⊂𝑐 =    𝑊𝑐 ,  if 𝑒 ∈ {01, 14, 02, 47, 67, 26}   0,  otherwise (A.13) for faces 𝑓1 and 𝑓2 join at the edge 𝑒. We implicitly choose 𝑤 2 = 0. We can use the ∪1 product defined above to reproduce the fermionic hopping terms defined by framing in Fig. 3.1. The hopping term defined by Eq. (3.10) is 74 z D y 1 x B 4 3 2 8 R 5 6 L F 7 U Figure A.2: (Color online) We rotate the axis U,D,F,B,R,L in Fig. A.1 to match the result in Fig. 3.1. Notice that the cube above is dual lattice and edges 1, 2, 3... in the dual lattice represent faces in the original lattice. 𝑈𝑓 = 𝑋𝑓 Ö ∫ 𝑍𝑓0 𝒇 0 ∪1 𝒇 . (A.14) 𝑓0 Fig. A.2 is dual to Fig. A.1. Therefore, faces in Fig. A.1 become edges in Fig. A.2. Consider the hopping term along dual edge 3. On the dual vertex to the right, it represents the face R. From terms 𝛽(𝐹) 𝛽0 (𝑅) and 𝛽(𝑈) 𝛽0 (𝑅), the hopping term contains 𝑍5 (from F) and 𝑍6 (from U). On the dual vertex to the left, it represents the face L. Since there is no 𝛽(𝐷) 𝛽0 (𝐿) or 𝛽(𝐵) 𝛽0 (𝐿) term, it contributes nothing. So we have 𝑈3 = 𝑋3 𝑍 5 𝑍 6 . (A.15) Similarly, for edge 2, the hopping term has 𝑍7 (from 𝛽(𝐿) 𝛽0 (𝐵)) and 𝑍8 (from 𝛽(𝑈) 𝛽0 (𝐹)) 𝑈2 = 𝑋2 𝑍 7 𝑍 8 . (A.16) For edge 1, the hopping term has 𝑍3 (from 𝛽(𝐿) 𝛽0 (𝐷)) and 𝑍4 (from 𝛽(𝐵) 𝛽0 (𝐷)) 𝑈1 = 𝑋1 𝑍 3 𝑍 4 . (A.17) We get the exact same hopping terms defined by "framing" in Fig. 3.1. We can interpret the choice of framing as a definition of ∪1 product on the cubic lattice. 75 Appendix B A FORMULA FOR STIEFEL-WHITNEY HOMOLOGY CLASSES In this section, we prove the following Lemma: Lemma 1. In 𝑛-dimension manifold with triangulation and branching structure, the homology class of 𝑤 2 can be represented by a (𝑛 − 2)-chain 𝑤 2 ∈ 𝐶𝑛−2 (𝑀𝑛 , Z2 ): Õ 𝑤2 = 𝑐(Δ 𝑛−2 )Δ 𝑛−2 , (B.1) Δ 𝑛−2 where 𝑐(Δ 𝑛−2 ) =1 + Õ Õ 𝚫𝑛−2 (h0 · · · 𝑗ˆ1 · · · 𝑗ˆ2 · · · 𝑛i) “−”-oriented Δ 𝑛 =h0...𝑛i 𝑗1 < 𝑗 2 | 𝑗1 , 𝑗 2 ∈even Õ + Õ 𝚫𝑛−2 (h0 · · · 𝑘ˆ1 · · · 𝑘ˆ2 · · · 𝑛i). (B.2) “+”-oriented Δ 𝑛 =h0...𝑛i 𝑘 1 <𝑘 2 |𝑘 1 ,𝑘 2 ∈odd First, let us recall the theorem proved in [58]. Let 𝑠 be a 𝑝-simplex, say 𝑠 = h𝑣 0 , 𝑣 1 , . . . , 𝑣 𝑝 i. Let 𝑘 be another simplex which has s as a face; i.e., 𝑠 ⊂ 𝑘 (𝑠 may be equal to 𝑘). Let 𝐵−1 = set of vertices of 𝑘 less than 𝑣 0 , 𝐵0 = set of vertices of 𝑘 between 𝑣 0 and 𝑣 1 , (B.3) 𝐵𝑚 = set of vertices of 𝑘 between 𝑣 𝑚 and 𝑣 𝑚+1 , 𝐵 𝑝 = set of vertices of 𝑘 greater than 𝑣 𝑝 . We say that 𝑠 is regular in 𝑘, if #(𝐵𝑚 ) = 0 for every odd 𝑚. Let 𝜕𝑝 (𝑘) denote the mod 2 chain which consists of all 𝑝-dimensional simplices 𝑠 in 𝑘 so that 𝑠 is regular in 𝑘. For example, h012i and h023i are regular in h0123i and therefore 𝜕2 (h0123i) = h012i + h023i. The theorem is [58]: Theorem 1. Í 𝑘 | dim 𝑘 ≥(𝑛−2) 𝜕𝑛−2 (𝑘) is a (𝑛 − 2)-chain which represents 𝑤 2 . In particular, for any 𝑛0-simplex Δ 𝑛 0 = h0 . . . 𝑛0i, all (𝑛0 − 1)-simplices regular in Δ 𝑛 0 are h0 . . . 𝑖ˆ . . . 𝑛i ∀𝑖 ∈ odd (B.4) 76 and all (𝑛0 − 2)-simplices regular in Δ 𝑛 0 are h0 . . . 𝑖ˆ . . . 𝑗ˆ . . . 𝑛i ∀𝑖 ∈ odd, 𝑗 ∈ even, 𝑖 < 𝑗 . (B.5) We now use this theorem to prove lemma 1. Proof of Lemma 1. For every (𝑛 − 2)-simplex Δ 𝑛−2 , it is regular in itself. This contributes the 1 in the coefficient of 𝑐(Δ 𝑛−2 ) in (B.2). For every (𝑛 − 1)-simplex Δ 𝑛−1 , it is a boundary of two 𝑛-simplices Δ 𝑛𝐿 and Δ 𝑛𝑅 , with Δ 𝑛−1 being an outward boundary of Δ 𝑛𝐿 and an inward boundary of Δ 𝑛𝑅 . We define that Δ 𝑛−1 belongs to Δ 𝑛𝑅 and the summation of dim 𝑘 = 𝑛 − 1, 𝑛 in theorem 1 can be written as: Õ Õ 𝜕𝑛−2 (Δ 𝑛−1 ) + 𝜕𝑛−2 (Δ 𝑛 ) Δ 𝑛−1 Δ𝑛  Õ Õ   𝜕𝑛−2 (Δ 𝑛 ) + . 𝜕 (Δ ) = 𝑛−2 𝑛−1    Δ𝑛  Δ 𝑛−1 ∈Δ 𝑛 |Δ 𝑛−1 is inward   If Δ 𝑛 = h0 . . . 𝑛i is “ + ”-oriented, the terms in the summation is Õ 𝜕𝑛−2 (h0 . . . 𝑛i) + 𝜕𝑛−2 (h0 . . . 𝑖ˆ . . . 𝑛i) (B.6) 0≤𝑖≤𝑛|𝑖∈odd Õ = h0 . . . 𝑖ˆ . . . 𝑗ˆ . . . 𝑛i 𝑖, 𝑗 |𝑖< 𝑗, 𝑖∈odd, 𝑗 ∈even Õ + ( Õ h0 . . . 𝑗ˆ . . . 𝑖ˆ . . . 𝑛i + 0≤𝑖≤𝑛|𝑖∈odd 𝑗 <𝑖| 𝑗 ∈odd Õ = Õ h0 . . . 𝑖ˆ . . . 𝑗ˆ . . . 𝑛i) (B.7) 𝑗 >𝑖| 𝑗 ∈even h0 . . . 𝑖ˆ . . . 𝑗ˆ . . . 𝑛i, 𝑖, 𝑗 |𝑖< 𝑗, 𝑖∈odd, 𝑗 ∈odd where we have used the definition of regular simplex defined above. Similarly, we can derive that if Δ 𝑛 = h0 . . . 𝑛i is “ − ”-oriented, the term is Õ h0 . . . 𝑖ˆ . . . 𝑗ˆ . . . 𝑛i. (B.8) 𝑖, 𝑗 |𝑖< 𝑗, 𝑖∈even, 𝑗 ∈even Combining (B.7) and (B.8) with the 1 from dim 𝑘 = 𝑛 − 2 in theorem 1, we have Õ 𝑤2 = 𝑐(Δ 𝑛−2 )Δ 𝑛−2 , (B.9) Δ 𝑛−2 where 𝑐(Δ 𝑛−2 ) = 1+ Õ Õ 𝚫𝑛−2 (h0 · · · 𝑗ˆ1 · · · 𝑗ˆ2 · · · 𝑛i) “−”-oriented Δ 𝑛 =h0...𝑛i 𝑗1 < 𝑗2 | 𝑗 1 , 𝑗 2 ∈even + Õ Õ “+”-oriented Δ 𝑛 =h0...𝑛i 𝑘 1 <𝑘 2 |𝑘 1 ,𝑘 2 ∈odd 𝚫𝑛−2 (h0 · · · 𝑘ˆ1 · · · 𝑘ˆ2 · · · 𝑛i). (B.10) 77  78 Appendix C IDENTITY FOR FERMIONIC ALGEBRA In this section, we will derive the constraints on fermionic operators: ∫ Ö ∫ 𝚫 ∪ 𝚫 +𝚫 ∪ 𝚫 𝚫 𝑛 𝑛−2 𝑛−2 𝑛−2 𝑛−2 𝑛 (−1) 𝑤2 𝑛−2 𝑆 𝛿Δ 𝑛−2 𝑃Δ 𝑛 = 1. (C.1) Δ𝑛 This follows directly from the following two lemmas. Lemma 2. ∫ The Majorana operators in 𝑆 𝛿Δ 𝑛−2 cancel out with Majorana operators Î 𝚫𝑛−2 ∪𝑛−2 𝚫𝑛 +𝚫𝑛 ∪𝑛−2 𝚫𝑛−2 in Δ 𝑛 𝑃Δ 𝑛 . Í𝑑 ∫ 𝑖−1 𝑖 Lemma 3. The sign difference of 𝑆 𝛿Δ 𝑛−2 and the product of 𝑃Δ 𝑛 is −(−1) 𝑖=1 𝚫𝑛−1 ∪𝑛−2 𝚫𝑛−1 where we order (𝑛 − 1)-simplices {Δ 𝑛−1 |Δ 𝑛−1 ⊃ Δ 𝑛−2 } counterclockwise as 𝑑−1 𝑑 𝚫1𝑛−1 , 𝚫2𝑛−1 , . . . , 𝚫𝑛−1 , 𝚫𝑛−1 ≡ 𝚫0𝑛−1 , as shown in Fig. C.2. This sign is a chain representative of 2nd Stiefel-Whitney class: ∫ Í𝑑 ∫ 𝑖−1 Δ 𝚫𝑛−1 ∪𝑛−2 𝚫𝑖𝑛−1 𝑖=1 − (−1) = (−1) 𝑤2 𝑛−2 . (C.2) Proof of Lemma 2. Let us denote Δ 𝑛 = h01 . . . 𝑛i formed by Δ 𝑛−2 and two (𝑛 − 1)𝐿 and Δ 𝑅 , shown in Fig. C.1(a). We know that 𝑆 0 simplex Δ 𝑛−1 𝛿Δ 𝑛−2 contains 𝛾Δ 𝑛 𝛾Δ 𝑛 𝑛−1 𝐿 , Δ𝑅 if and only if Δ 𝑛−1 𝑛−1 are one inward boundary and one outward boundary of 𝑛-simplex Δ 𝑛 , as indicated in Fig. C.1(b) and (c). For the product of 𝑃Δ 𝑛 , we simplify the integral as ∫ ∫ 𝚫𝑛−2 ∪𝑛−2 𝚫𝑛 + 𝚫𝑛 ∪𝑛−2 𝚫𝑛−2 = 𝛿𝚫𝑛−2 ∪𝑛−1 𝚫𝑛 . (C.3) The contribution of Δ 𝑛 = h01 . . . 𝑛i to (C.3) is 𝐿 [(𝚫𝑛−1 + 𝚫 𝑅 ) ∪𝑛−1 𝚫 𝒏 ] (h01 . . . 𝑛i) Õ 𝑛−1 𝐿 𝑅 = (𝚫𝑛−1 + 𝚫𝑛−1 )(0 ∼ 𝑖0 , 𝑖1 ∼ 𝑖2 , 𝑖3 ∼ 𝑖4 , · · · )𝚫𝑛 (𝑖0 ∼ 𝑖1 , 𝑖2 ∼ 𝑖 3 , · · · ) 0≤𝑖0 <𝑖 1 <···<𝑖 𝑛−1 ≤𝑛 = Õ 𝐿 𝑅 (𝚫𝑛−1 + 𝚫𝑛−1 )(h0 . . . 𝑗ˆ . . . 𝑛i)𝚫𝑛 (h01 . . . 𝑛i) 0≤ 𝑗 ≤𝑛| 𝑗 ∈odd = Õ 𝐿 𝑅 (𝚫𝑛−1 + 𝚫𝑛−1 )(h0 . . . 𝑗ˆ . . . 𝑛i) 0≤ 𝑗 ≤𝑛| 𝑗 ∈odd (C.4) 79 𝐿 , Δ 𝑅 are one inward boundary and one outward boundary which is 1 if and only Δ 𝑛−1 𝑛−1 of the 𝑛-simplex Δ 𝑛 . This shows that product of 𝑃Δ 𝑛 contain 𝑃Δ 𝑛 ∼ 𝛾Δ 𝑛 𝛾Δ0 𝑛 if and 𝐿 , Δ𝑅 only if Δ 𝑛−1 𝑛−1 are one inward boundary and one outward boundary of the 𝑛-simplex Δ 𝑛 . This cancels out with 𝑆 𝛿Δ 𝑛−2 exactly.  𝐿 and Figure C.1: (a) The 𝑛-simplex Δ 𝑛 is formed by Δ 𝑛−2 and two (𝑛−1)-simplex Δ 𝑛−1 𝑅 0 0 0 0 Δ 𝑛−1 . (b) The product of 𝑆Δ 𝑛−2 is (𝑖𝛾𝑏 𝛾𝑎 )(𝑖𝛾𝑐 𝛾𝑏 ) = (𝑖𝛾𝑐 𝛾𝑎 )(−𝑖𝛾𝑏 𝛾𝑏 ) = (𝑖𝛾𝑐 𝛾𝑎0 )𝑃𝑏 . (c) The product of 𝑆Δ 𝑛−2 is (𝑖𝛾𝑎 𝛾𝑏0 )(𝑖𝛾𝑏 𝛾𝑐0 ) = (𝑖𝛾𝑎 𝛾𝑐0 )(−𝑖𝛾𝑏 𝛾𝑏0 ) = (𝑖𝛾𝑎 𝛾𝑐0 )𝑃𝑏 . (d) The product of 𝑆Δ 𝑛−2 is (𝑖𝛾𝑎 𝛾𝑏0 )(𝑖𝛾𝑐 𝛾𝑏0 )(𝑖𝛾𝑐 𝛾𝑑0 ) = 𝑖𝛾𝑎 𝛾𝑑0 . (e) The product of 𝑆Δ 𝑛−2 is (𝑖𝛾𝑏 𝛾𝑎0 )(𝑖𝛾𝑏 𝛾𝑐0 )(𝑖𝛾𝑑 𝛾𝑐0 ) = 𝑖𝛾𝑑 𝛾𝑎0 . Figure C.2: By the operations defined in Fig. C.1, we can simplify the product ∫ Î 𝚫𝑛−2 ∪𝑛−2 𝚫𝑛 +𝚫𝑛 ∪𝑛−2 𝚫𝑛−2 𝑆Δ 𝑑 · · · 𝑆Δ 2 𝑆Δ 1 = 𝑆Δ 𝑑 𝑆Δ 1 . Δ 𝑛 ≠𝑎,𝑏 𝑃Δ 𝑛 𝑛−1 𝑛−1 𝑛−1 𝑛−1 𝑛−1 Proof of Lemma 3. We compare the sign between Í 0 0 𝚫 ∪ 𝚫0 Ö 𝑆 𝛿Δ 𝑛−2 = (−1) 𝚫𝑛−1 <𝚫𝑛−1 |Δ 𝑛−1 ,Δ 𝑛−1 ⊃Δ 𝑛−2 𝑛−1 𝑛−2 𝑛−1 𝑆Δ 𝑛−1 (C.5) Δ 𝑛−1 ⊃Δ 𝑛−2 and Ö Δ𝑛 ∫ 𝚫𝑛−2 ∪𝑛−2 𝚫𝑛 +𝚫𝑛 ∪𝑛−2 𝚫𝑛−2 𝑃Δ 𝑛 , (C.6) 80 where we have used the definition of 𝑆𝜆 𝑛−1 in (4.2). As shown in Fig. C.2, 𝑆Δ 𝑑 · · · 𝑆Δ 2 𝑆Δ 1 𝑛−1 𝑛−1 𝑛−1 Ö = 𝑆Δ 𝑑 𝑆Δ 1 𝑛−1 𝑛−1 ∫ 𝚫𝑛−2 ∪𝑛−2 𝚫𝑛 +𝚫𝑛 ∪𝑛−2 𝚫𝑛−2 . 𝑃Δ 𝑛 Δ 𝑛 ≠𝑎,𝑏 We can check that = −(−1) 𝑆Δ 𝑑 𝑆Δ 1 ∫ 𝚫1𝑛 ∪𝑛−2 𝚫𝑛𝑑 +𝚫𝑛𝑑 ∪𝑛−2 𝚫1𝑛 ∫ Ö 𝑛−1 𝑛−1 𝚫𝒏−2 ∪𝑛−2 𝚫𝑛 +𝚫𝑛 ∪𝑛−2 𝚫𝑛−2 𝑃Δ 𝑛 , Δ 𝑛 =𝑎,𝑏 and therefore 𝑆Δ 𝑑 · · · 𝑆Δ 2 𝑆Δ 1 𝑛−1 𝑛−1 = −(−1) ∫ 𝚫1𝑛 ∪𝑛−2 𝚫𝑛𝑑 +𝚫𝑛𝑑 ∪𝑛−2 𝚫1𝑛 Ö 𝑛−1 ∫ 𝚫𝑛−2 ∪𝑛−2 𝚫𝒏 +𝚫𝑛 ∪𝑛−2 𝚫𝑛−2 𝑃Δ 𝑛 . Δ𝑛 Together with (C.5), we have 𝑆 𝛿Δ 𝑛−2 Ö ∫ 𝚫𝑛−2 ∪𝑛−2 𝚫𝑛 +𝚫𝑛 ∪𝑛−2 𝚫𝑛−2 𝑃Δ 𝑛 Δ𝑛 =(−1) ∫ 𝑑 𝚫1𝑛−1 ∪𝑛−2 𝚫𝑛−1 + Í𝑑 ∫ = − (−1) 𝑖=1 Í𝑑 ∫ 𝑖=2 𝑖 𝚫𝑖−1 𝑛−1 ∪𝑛−2 𝚫𝑛−1 𝑖 𝚫𝑖−1 𝑛−1 ∪𝑛−2 𝚫𝑛−1  −(−1) ∫ 𝚫1𝑛 ∪𝑛−2 𝚫𝑛𝑑 +𝚫𝑛𝑑 ∪𝑛−2 𝚫1𝑛  (C.7) . From the definition of ∪𝑛−2 product (4.4), 𝑑 ∫ Õ = 𝑖=1 𝑑 Õ Õ 𝑖 𝚫𝑖−1 𝑛−1 ∪𝑛−2 𝚫𝑛−1 𝑖 𝚫𝑖−1 𝑛−1 ∪𝑛−2 𝚫𝑛−1 (Δ 𝑛 ) (C.8) 𝑖=1 Δ 𝑛 Õ = Õ 𝚫𝑛−2 (h0 · · · 𝑗ˆ1 · · · 𝑗ˆ2 · · · 𝑛i) “−”-oriented Δ 𝑛 =h0...𝑛i 𝑗 1 < 𝑗2 | 𝑗1 , 𝑗 2 ∈even + Õ Õ 𝚫𝑛−2 (h0 · · · 𝑘ˆ1 · · · 𝑘ˆ2 · · · 𝑛i). “+”-oriented Δ 𝑛 =h0...𝑛i 𝑘 1 <𝑘 2 |𝑘 1 ,𝑘 2 ∈odd The distinct orientations of “−”-oriented Δ 𝑛 and “+”-oriented Δ 𝑛 in the summation come from the fact that 𝑗1 , 𝑗2 and 𝑘 1 , 𝑘 2 in (4.4) have opposite orders. Eq. (C.8) is related to 𝑤 2 by the following lemma 1 in appendix B. Therefore, we derive − (−1) Í𝑑 ∫ 𝑖=1 𝑖 𝚫𝑖−1 𝑛−1 ∪𝑛−2 𝚫𝑛−1 ∫ 𝚫 = (−1) 𝑤2 𝑛−2 . (C.9)  81 Appendix D GAUGING THE SYMMETRIES Let’s review the gauging procedure described in [59]. In 2d, the gauging map is to map a state |𝑎, 𝑏i on adjacent sites to a states |𝑎 + 𝑏i on the edge between this two sites (shown as D.1). The 𝑋 operator at the vertex 𝑣 will be mapped to the product Figure D.1: The gauging map is a function from a state with Z2 fields (qubits) living on vextices to a state with Z2 fields (qubits) living on edges. We define the Hilbert spaces as H0 and H1 [59]. of 𝑋 operators on the edges connected to the vertex: Ö 𝑋𝑣 → 𝐴𝑣0 ≡ 𝑋𝑒 . (D.1) 𝑒⊃𝑣 By noticing the summation of edges around a face is always 0, we can write down constraints in the Hilbert space H1 (mapped states) Ö 𝑊𝑓 ≡ 𝑍𝑒 = 1 (D.2) 𝑒⊂ 𝑓 for all face 𝑓 . As discussed in [59], the gauge map is a duality map between the following two subspaces: 𝑠𝑦𝑚 H0 = {|𝜓i ∈ H0 : 𝑆|𝜓i = |𝜓i}, (D.3) 82 where 𝑆 = Î 𝑣 𝑋𝑣 and 𝑠𝑦𝑚 H1 = {|𝜓i ∈ H1 : 𝐵(𝛾)|𝜓i = |𝜓i}, (D.4) where 𝛾 represents an arbitrary colosed loop on a square lattice annd 𝐵(𝛾) = Î 𝑒∈𝛾 𝑍 𝑒 . Consider the trivial system with Z2 symmetry: Õ 𝐻=− 𝑋𝑣 . (D.5) 𝑣 The ground state of this Hamiltonian will be mapped to the groundstate of Õ Õ 𝐻=− 𝐴𝑣0 − 𝑊𝑓 𝑣 (D.6) 𝑓 which is exactly 2d toric code. Figure D.2: Gauging 1-form symmetry [59]. For 3d lattice, [59] defined the gauging procedure in the similar way (shown in Fig. D.2). For any 1-cochain 𝑎 ∈ 𝐶 1 (𝑀, Z2 ) (𝑀 is a manifold with triangulation), the gauging map Γ0 : H1 → H2 is defined as Γ0 (|𝑎i) = |𝛿𝑎i. (D.7) Similarly, the Hilbert space H2 has a constraint on each cube (tetrahedron) 𝑡 Ö 𝑊𝑡 ≡ 𝑍 𝑓 = 1. (D.8) 𝑓 ⊂𝑡 83 This map Γ0 induces a duality between the following two subspaces. In H1 , the symmetric subspace is defined by 𝑠𝑦𝑚 H1 = {|𝜓i ∈ H1 : 𝑆(M)|𝜓i = |𝜓i ∀M}. (D.9) Here the 1-form symmetry operator is written as 𝑆(M) for an arbitrary closed 2manifold M with Pauli 𝑋𝑒 acting on edges intersected by M. In H2 , the gauge symmetric subspace is 𝑠𝑦𝑚 H2 = {|𝜙i ∈ H2 : 𝑇 (N )|𝜙i = |𝜙i∀ N }, (D.10) where N is a closed 2-manifold consisting of faces of the lattice and N acts Pauli 𝑍 𝑓 on these faces. Note that closed 2-manifold M, N live on dual lattice and direct lattice respectively. The 𝑋 operator on a edge 𝑒 will be mapped to the product of 𝑋 operators on faces around the edge 𝑒: Ö 𝑋𝑒 → 𝐴0𝑒 ≡ 𝑋𝑓 , (D.11) 𝑓 ⊃𝑒 where this mapping represents Γ0 (𝑋𝑒 |𝑎i) = 𝐴0𝑒 |𝛿𝑎i by definition. Therefore, the trivial Hamiltonian 𝐻=− Õ (D.12) 𝑋𝑒 𝑒 is mapped to 𝐻=− Õ 𝑒 𝐴0𝑒 − Õ 𝑊𝑡 . 𝑡 This is (2,1)-toric code and doesn’t have any fermionic excitation. (D.13) 84 Appendix E DERIVATION OF 𝜁 (FIRST DESCENDANT OF 𝜌) In this section, we derive the choice of 𝜁 in (6.10) 𝜁 (𝑔12 , 𝑔23 , ℎ1 , ℎ2 , ℎ3 ) =𝜌(1, 𝑔12 , 𝑔12 𝑔23 , 𝑔12 𝑔23 ℎ3 ) + 𝜌(1, 𝑔12 , 𝑔12 ℎ2 , 𝑔12 𝑔23 ℎ3 ) (E.1) + 𝜌(1, ℎ1 , 𝑔12 ℎ2 , 𝑔12 𝑔23 ℎ3 ). Consider a tetrahedron h0123i with group elements 𝑔, ℎ, 𝑘 on edges 01, 12, 23. The value of cocycle 𝜈 on this tetrahedron is expressed as 𝜈(1, 𝑔, 𝑔ℎ, 𝑔ℎ𝑘). First, if we now perform a gauge transformation on vertex 0 by group element 𝑐 0 (and identity element on all other vertices). 𝑔, ℎ, 𝑘 becomes 𝑐−1 0 𝑔, ℎ, 𝑘 and the value of cocycle is −1 −1 𝜌(1, 𝑐−1 0 𝑔, 𝑐 0 𝑔ℎ, 𝑐 0 𝑔ℎ𝑘) = 𝜌(𝑐 0 , 𝑔, 𝑔ℎ, 𝑔ℎ𝑘) = 𝜌(1, 𝑐 0 , 𝑔, 𝑔ℎ𝑘) − 𝜌(1, 𝑐 0 , 𝑔, 𝑔ℎ) (E.2) − 𝜌(1, 𝑐 0 , 𝑔𝑘, 𝑔ℎ𝑘) + 𝜌(1, 𝑔, 𝑔ℎ, 𝑔ℎ𝑘). From the last line, to satisfy (6.9), we can define the gauge transformation of 𝐵𝑖 𝑗 𝑘 by 𝑐 0 at the vertex 𝑖 of a face h𝑖 𝑗 𝑘i as 𝜌(1, 𝑐 0 , 𝑔𝑖 𝑗 , 𝑔 𝑗 𝑘 ) (or explicit 𝜁 (𝑔𝑖 𝑗 , 𝑔 𝑗 𝑘 , 𝑐 0 , 1, 1) = 𝜌(1, 𝑐 0 , 𝑔𝑖 𝑗 , 𝑔𝑖 𝑗 𝑔 𝑗 𝑘 ) ). Secondly, we perform a gauge transformation on vertex 1 by group element 𝑐 1 . 𝑔, ℎ, 𝑘 becomes 𝑔𝑐 1 , 𝑐−1 1 ℎ, 𝑘 and the value of the cocycle is 𝜌(1, 𝑔𝑐 1 , 𝑔ℎ, 𝑔ℎ𝑘) =𝜌(1, 𝑔, 𝑔𝑐 1 , 𝑔ℎ) − 𝜌(1, 𝑔, 𝑔𝑐 1 , 𝑔ℎ𝑘) + 𝜌(𝑔, 𝑔𝑐 1 , 𝑔ℎ, 𝑔ℎ𝑘) + 𝜌(1, 𝑔, 𝑔ℎ, 𝑔ℎ𝑘) (E.3) =𝜌(1, 𝑔, 𝑔𝑐 1 , 𝑔ℎ) − 𝜌(1, 𝑔, 𝑔𝑐 1 , 𝑔ℎ𝑘) + 𝜌(1, 𝑐 1 , ℎ, ℎ𝑘) + 𝜌(1, 𝑔, 𝑔ℎ, 𝑔ℎ𝑘). The first two terms in the last line indicate that the gauge transformation on the vertex 𝑗 of a face h𝑖 𝑗 𝑘i is 𝜌(1, 𝑔𝑖 𝑗 , 𝑔𝑖 𝑗 𝑐 1 , 𝑔 𝑗 𝑘 ) (or 𝜁 (𝑔𝑖 𝑗 , 𝑔 𝑗 𝑘 , 1, 𝑐 1 , 1) = 𝜌(1, 𝑔𝑖 𝑗 , 𝑔𝑖 𝑗 𝑐 1 , 𝑔 𝑗 𝑘 )) and the third term in the last line is consistent with the previous case. 85 Similarly for the gauge transformation on the vertex 2 by group element 𝑐 2 , we have 𝜌(1, 𝑔, 𝑔ℎ𝑐 2 , 𝑔ℎ𝑘) =𝜌(𝑔, 𝑔ℎ, 𝑔ℎ𝑐 2 , 𝑔ℎ𝑘) − 𝜌(1, 𝑔, 𝑔ℎ, 𝑔ℎ𝑐 2 ) + 𝜌(1, 𝑔, 𝑔ℎ𝑐 2 , 𝑔ℎ𝑘) + 𝜌(1, 𝑔, 𝑔ℎ, 𝑔ℎ𝑘) (E.4) =𝜌(1, ℎ, ℎ𝑐 2 , ℎ𝑘) − 𝜌(1, 𝑔, 𝑔ℎ, 𝑔ℎ𝑐 2 ) + 𝜌(1, 𝑔, 𝑔ℎ𝑐 2 , 𝑔ℎ𝑘) + 𝜌(1, 𝑔, 𝑔ℎ, 𝑔ℎ𝑘). The second term in the last line implies 𝜁 (𝑔𝑖 𝑗 , 𝑔 𝑗 𝑘 , 1, 1, 𝑐 2 ) = 𝜌(1, 𝑔𝑖 𝑗 , 𝑔𝑖 𝑗 𝑔 𝑗 𝑘 , 𝑔𝑖 𝑗 𝑔 𝑗 𝑘 𝑐 2 ) and the other terms are consistent with previous cases. We can check the gauge transformation on vertex 3 of the tetrahedron. However, it just gives us a consistence check. The previous three cases are complete gauge transformation. Combining previous 3 cases, we apply gauge transformation ℎ3 , ℎ2 , and ℎ1 on vertices 3, 2, 1 of a face h123i in sequence. First, we add 𝜌(1, 𝑔12 , 𝑔12 𝑔23 , 𝑔12 𝑔23 ℎ3 ) to 𝐵123 and 𝑔12 , 𝑔23 → 𝑔12 , 𝑔23 ℎ3 . Second, we gain 𝜌(1, 𝑔12 , 𝑔12 ℎ2 , 𝑔12 𝑔23 ℎ3 ) and 𝑔12 , 𝑔23 ℎ3 → 𝑔12 ℎ2 , ℎ−1 2 𝑔23 ℎ3 . Finally, we get 𝜌(1, ℎ1 , 𝑔12 ℎ2 , 𝑔12 𝑔23 ℎ3 ). Totally, the gauge transformation on 𝐵123 is 𝜁 (𝑔12 , 𝑔23 , ℎ1 , ℎ2 , ℎ3 ) = 𝜌(1, 𝑔12 , 𝑔12 𝑔23 , 𝑔12 𝑔23 ℎ3 ) + 𝜌(1, 𝑔12 , 𝑔12 ℎ2 , 𝑔12 𝑔23 ℎ3 ) + 𝜌(1, ℎ1 , 𝑔12 ℎ2 , 𝑔12 𝑔23 ℎ3 ). (E.5) 86 BIBLIOGRAPHY [1] Han-Dong Chen, Chen Fang, Jiangping Hu, and Hong Yao. Quantum phase transition in the quantum compass model. Phys. Rev. B, 75:144401, Apr 2007. [2] Alexei Kitaev. Anyons in an exactly solved model and beyond. Annals of Physics (New York), 321(1), 1 2006. [3] Xiao-Yong Feng, Guang-Ming Zhang, and Tao Xiang. Topological characterization of quantum phase transitions in a spin-1/2 model. Phys. Rev. Lett., 98:087204, Feb 2007. 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