Vanishing Hall conductance for commuting Hamiltonians

TL;DR

This paper proves that for certain two-dimensional quantum systems described by commuting Hamiltonians, the process of flux insertion cannot induce charge transport, implying the Hall conductance is zero.

Contribution

It establishes a rigorous proof that commuting Hamiltonians in 2D systems cannot produce a non-zero Hall conductance through flux insertion.

Findings

Flux insertion does not pump charge in these systems

Hall conductance must vanish for the considered Hamiltonians

Ground states exhibit no charge transport under flux insertion

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.