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 \(\varepsilon \) 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.