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arXiv:2102.13057

A Commuting Projector Model with a Non-zero Quantized Hall conductance

DeMarco, Michael, Xiao-Gang Wen

Published Feb 25, 2021• 1 citation
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Abstract

By ungauging a recently discovered lattice rotor model for Chern-Simons theory, we create an exactly soluble path integral on spacetime lattice for $U^κ(1)$ Symmetry Protected Topological (SPT) phases in $2+1$ dimensions with a non-zero Hall conductance. We then convert the path integral on a $2+1$d spacetime lattice into a $2$d Hamiltonian lattice model, and show that the Hamiltonian consists of mutually commuting local projectors. We confirm the non-zero Hall conductance by calculating the Chern number of the exact ground state. It has recently been suggested that no commuting projector model can host a nonzero Hall conductance. We evade this no-go theorem by considering a rotor model, with a countably infinite number of states per site.

DOI
10.48550/arxiv.2102.13057

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By ungauging a recently discovered lattice rotor model for Chern-Simons theory, we create an exactly soluble path integral on spacetime lattice for Uκ(1)U^κ(1)Uκ(1) Symmetry Protected Topological (SPT) phases in 2+12+12+1 dimensions with a non-zero Hall conductance. We then convert the path integral on a 2+12+12+1d spacetime lattice into a 222d Hamiltonian lattice model, and show that the Hamiltonian consists of mutually commuting local projectors. We confirm the non-zero Hall conductance by calculating the Chern number of the exact ground state. It has recently been suggested that no commuting projector model can host a nonzero Hall conductance. We evade this no-go theorem by considering a rotor model, with a countably infinite number of states per site.
Hall conductivities are important characterizations of phases of matter. It is known that nonzero Hall conductivities are difficult to realize in local commuting projector lattice models due to no-go theorems in ( 2 + 1 ) D . In this work we construct local commuting projector models in ( 2 + 1 ) D and ( 3 + 1 ) D with nonzero generalized Hall conductivities for ordinary and higher-form continuous symmetries on tensor product Hilbert space of finite local dimension. The model is given by a standard Z N toric code, but the symmetries do not admit expression in terms of on-site charge operators. The symmetry do not have local charges or currents on the lattice in the absence of boundaries, but there is still a notion of Hall conductivities that coincide with the continuum field theories. We construct protected gapless boundaries of the lattice models using modified Villain formalism. The generalized Hall conductivities are computed by surface currents as well as bulk flux insertion and many-body Chern number.
A bstract The Kapustin-Fidkowski no-go theorem forbids U(1) symmetric topological orders with non-trivial Hall conductivity in (2+1)d from admitting commuting projector Hamiltonians, where the latter is the paradigmatic method to construct exactly solvable lattice models for topological orders. Even if a topological order would intrinsically have admitted commuting projector Hamiltonians, the theorem forbids so once its interplay with U(1) global symmetry which generates Hall conductivity is taken into consideration. Nonetheless, in this work, we show that for all (2+1)d U(1) symmetric abelian topological orders of such kind, we can construct a lattice Hamiltonian that is controllably solvable at low energies, even though not “exactly” solvable; hence, this no-go theorem does not lead to significant difficulty in the lattice study of these topological orders. Moreover, for the fermionic topological orders in our construction, we introduce the lattice notion of spin-c structure — a concept important in the continuum that has previously not been adequately introduced in the lattice context.
We construct exactly solved commuting projector Hamiltonian lattice models for all known (2+1)-dimensional (2+1D) fermionic symmetry protected topological phases (SPTs) with on-site unitary symmetry group ${G}_{f}=G\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{f}$, where GGG is finite and Z2f{\mathbb{Z}}_{2}^{f}Z2f​ is the fermion parity symmetry. In particular, our models transcend the class of group supercohomology models, which realize some, but not all, fermionic SPTs in 2+1D. A natural ingredient in our construction is a discrete form of the spin structure of the 2D spatial surface MMM on which our model is defined, namely a ``Kasteleyn'' orientation of a certain graph associated with the lattice. As a special case, our construction yields commuting projector models for all eight members of the Z8{\mathbb{Z}}_{8}Z8​ classification of 2D fermionic SPTs with G=Z2G={\mathbb{Z}}_{2}G=Z2​.
We introduce a family of commuting-projector Hamiltonians whose degrees of freedom involve Z3{\mathbb{Z}}_{3}Z3​ parafermion zero modes residing in a parent fractional-quantum-Hall fluid. These commuting-projector models inherit nontrivial Hall conductance from the parent quantum-Hall states in which they are defined, and thus can describe chiral topological phases. The two simplest models in this family emerge from dressing Ising-paramagnet and toric-code spin models with parafermions; we study their edge properties, anyonic excitations, and ground-state degeneracy. We show that the first model realizes a symmetry-enriched topological phase (SET) for which Z2{\mathbb{Z}}_{2}Z2​ spin-flip symmetry from the Ising paramagnet permutes the anyons. Interestingly, the interface between this SET and the parent quantum-Hall phase realizes symmetry-enforced Z3{\mathbb{Z}}_{3}Z3​ parafermion criticality with no fine-tuning required. The second model exhibits a non-Abelian phase that is consistent with SU(2)4\text{SU}{(2)}_{4}SU(2)4​ topological order, and can be accessed by gauging the Z2{\mathbb{Z}}_{2}Z2​ symmetry in the SET. Employing Levin-Wen string-net models with Z2{\mathbb{Z}}_{2}Z2​-graded structure, we generalize this picture to construct a large class of commuting-projector models for Z2{\mathbb{Z}}_{2}Z2​ SETs and non-Abelian topological orders exhibiting the same relation. Our construction provides the first commuting-projector-Hamiltonian realization of chiral bosonic non-Abelian topological order.
We propose a bosonic U^{κ}(1) rotor model on a three dimensional spacetime lattice. With the inclusion of a Maxwell term, we show that the low-energy properties of our model can be obtained reliably via a semiclassical approach. Those properties are the same as that of the Chern-Simons field theory, S=∫d^{3}x(K_{IJ}/4π)A_{I}dA_{J}. We require the lattice variables on each link to be compact (i.e., take values on circles), which enforces the quantization of the K matrix as a symmetric integer matrix with even diagonals. Our lattice model also has exact 1-symmetries, which gives rise to the 1-form symmetry in the Chern-Simons field theory. In particular, some of those 1-symmetries are anomalous (i.e., non-on-site) in the expected way. The anomaly can be probed via the breaking of those lattice 1-symmetries by the boundaries.
We construct an exactly sovable commuting projector Hamiltonian for (2+1)D bosonic topological insulator which is one of symmetry-protected topological (SPT) phases protected by U(1) and time-reversal Z2T\mathbb{Z}_2^TZ2T​ symmetry, where the symmetry group is U(1)⋊Z2T\rtimes\mathbb{Z}_2^T⋊Z2T​. The model construction is based on the decorated domain-wall interpretation of the E∞E_{\infty}E∞​-page of a spectral sequence of a cobordism group that classifies the SPT phases in question. We demonstrate nontriviality of the model by showing an emergence of a Kramers doublet when the system is put on a semi-infinite cylinder (−∞,0]×S1(-\infty,0]\times S^1(−∞,0]×S1 with an inserted πππ-flux. The surface anomaly manifests itself as a non-onsite representation of the U(1)⋊Z2T\rtimes\mathbb{Z}_2^T⋊Z2T​ symmetry. Anomaly matching on a boundary is discussed within a simple boundary theory.