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

In this note we consider lattice fermions on Z2 with a locally unique gapped ground state and show how to apply the non-equilibrium almost stationary state (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 [Wesle et al., Commun. Math. Phys. 406, 199 (2025)] as the linear response coefficient of the macroscopic current response to the adiabatic application of a constant electric field.