Hall conductance and the statistics of flux insertions in gapped interacting lattice systems

Abstract

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 which are in the same gapped phase. We provide a rigorous versions of Laughlin's flux-insertion argument which shows that for short-range entangled systems the Hall conductance is an integer multiple of e^2/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 e^2/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.

arXiv comment35 pages. v2: some arguments in Section 4.3 have been corrected and moved to an appendix. Version accepted for publication in Journal of Mathematical Physics. v3: a comment added at the end of Section 4.3, typos in Appendix C fixed

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• Stability of invertible, frustration-free ground states against large perturbations
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• Locally equivalent quasifree states and index theory
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• Entanglement area law and Lieb-Schultz-Mattis theorem in long-range interacting systems, and symmetry-enforced long-range entanglement
• Universal Decay of Mutual Information and Conditional Mutual Information in Gapped Pure- and Mixed-State Quantum Matter
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