A note on Hall conductance and Hall conductivity in interacting Fermion systems

TL;DR

This paper derives a formula for Hall conductance in interacting Fermion systems using the NEASS approach, connecting microscopic current responses to adiabatic chemical potential changes and the bulk-boundary correspondence.

Contribution

It introduces a new derivation of the Hall conductance formula via linear response to chemical potential changes, linking microscopic and macroscopic perspectives.

Findings

Derived a formula for Hall conductance using NEASS approach.

Connected chemical potential increase in cone-like regions to boundary currents.

Related the microscopic response to the double commutator formula for Hall conductivity.

Abstract

In this note we consider lattice fermions on $\mathbb{Z}^2$ with a gapped ground state and show how to apply the NEASS approach to linear response to derive a formula for the Hall conductance in terms of the ground state expectation of a commutator of modified step functions. This formula is usually derived by a charge pumping argument going back to Laughlin. Here we show that it can also be obtained as the linear response coefficient of the microscopic current response to an adiabatic increase of the chemical potential on a half plane (or more generally on any cone-like region). Indeed, in a manner reminiscent of the bulk-boundary correspondence, we show that raising the chemical potential in any cone-like region gives rise to a current that flows along its boundary and is nearly linear in the increase in chemical potential. We also discuss the connection with the double commutator formula with modified position operators for the Hall conductivity derived in arXiv:2411.06967 as the linear response coefficient of the macroscopic current response to the adiabatic application of a constant electric field.