Near Linearity of the Macroscopic Hall Current Response in Infinitely Extended Gapped Fermion Systems

We consider an infinitely extended system of fermions on a d-dimensional lattice with (magnetic) translation-invariant short-range interactions. We further assume that the system has a gapped ground state. Physically, this is a model for the bulk of a generic topological insulator at zero temperature, and we are interested in the current response of such a system to a constant external electric field. Using the non-equilibrium almost-stationary states approach, we prove that the longitudinal current density induced by a constant electric field of strength $\epsilon$ is of order $O\left({\epsilon }^{\infty }\right)$ , i.e. the system is an insulator in the usual sense. For the Hall current density we show instead that it is linear in $\epsilon$ up to terms of order $O\left({\epsilon }^{\infty }\right)$ . The proportionality factor ${\sigma }_{\text{H}}$ is by definition the Hall conductivity, and we show that it is given by a generalization of the well known double commutator formula to interacting systems. As a by-product of our results, we find that the Hall conductivity is constant within gapped phases, and that for $d=2$ the relevant observable that "measures" the Hall conductivity in experiments, the Hall conductance, not only agrees with ${\sigma }_{\text{H}}$ in expectation up to $O\left({\epsilon }^{\infty }\right)$ , but also has vanishing variance. A notable difference to several existing results on the current response in interacting fermion systems is that we consider a macroscopic system exposed to a small constant electric field, rather than to a small voltage drop.