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 \(\varepsilon \) is of order \(\mathcal {O}(\varepsilon ^\infty )\) , i.e. the system is an insulator in the usual sense. For the Hall current density we show instead that it is linear in \(\varepsilon \) up to terms of order \(\mathcal {O}(\varepsilon ^\infty )\) . The proportionality factor \(\sigma _\textrm{H}\) 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 \( \sigma _{\textrm{H}}\) in expectation up to \(\mathcal {O}(\varepsilon ^\infty )\) , 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.
