Exact linearity of the macroscopic Hall current response in infinitely extended gapped fermion systems

• Published in: arXiv.org 2024-12
• Main Authors: Wesle, Marius, Marcelli, Giovanna, Miyao, Tadahiro, Monaco, Domenico, Teufel, Stefan
• Format: Article
• Language: English
• Subjects: Bulk density; Commutators; Current density; Electric field strength; Electric fields; Fermions; Linearity; Topological insulators; Voltage drop
• ISSN: 2331-8422

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 unique 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 (NEASS) 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 \(\sigma_\mathrm{H}\) 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 \(\sigma_H\) 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.