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arXiv:2107.10316 · Physical review. B./Physical review. B

Vanishing Hall conductance for commuting Hamiltonians

Carolyn Zhang, Michael Levin, Sven Bachmann

Published Feb 8, 2022• 11 citations
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Abstract

We consider the process of flux insertion for ground states of almost local commuting projector Hamiltonians in two spatial dimensions. In the case of finite-dimensional local Hilbert spaces, we prove that this process cannot pump any charge, and we conclude that the Hall conductance must vanish.
DOI
10.1103/physrevb.105.l081103

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We study charge transport for zero-temperature infinite-volume gapped lattice systems in two dimensions with short-range interactions. We show that the Hall conductance is locally computable and is the same for all systems that are in the same gapped phase. We provide a rigorous version of Laughlin’s flux-insertion argument, which shows that for short-range entangled systems, the Hall conductance is an integer multiple of e2/h. We show that the Hall conductance determines the statistics of flux insertions. For bosonic short-range entangled systems, this implies that the Hall conductance is an even multiple of e2/h. Finally, we adapt a proof of quantization of the Thouless charge pump to the case of infinite-volume gapped lattice systems in one dimension.
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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.
We derive a Kubo-like formula for the thermal Hall conductance of a 2d lattice systems which is free from ambiguities associated with the definition of energy magnetization. We use it to define a relative topological invariant of gapped 2d lattice systems at zero temperature. Up to a numerical factor, it can be identified with the difference of chiral central charges for the corresponding edge modes. This establishes the bulk-boundary correspondence for the chiral central charge. We also show that for any local commuting projector Hamiltonian, the relative chiral central charge vanishes, while for free fermionic systems, it is related to the zero-temperature electric Hall conductance via the Wiedemann-Franz law.
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