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

TL;DR

This paper rigorously establishes the quantization and topological invariance of Hall conductance in two-dimensional gapped lattice systems, linking it to flux insertion statistics and extending known results to infinite-volume and bosonic systems.

Contribution

It provides a rigorous proof that Hall conductance is quantized and invariant within gapped phases, and connects it to flux statistics in infinite-volume lattice systems.

Findings

Hall conductance is locally computable and invariant within the same gapped phase.

For short-range entangled systems, Hall conductance is an integer multiple of e^2/h.

In bosonic systems, Hall conductance is an even multiple of e^2/h.

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.