Universality of the Hall conductivity for a weakly interacting magnetic fermionic gas in the Hartree-Fock approximation

TL;DR

This paper demonstrates that in a weakly interacting two-dimensional fermionic gas under a magnetic field, the Hall conductivity remains universal and quantized, independent of interactions, within the Hartree-Fock approximation.

Contribution

It proves the universality of the Hall conductivity in weakly interacting fermionic systems using a self-consistent Hartree-Fock approach in the thermodynamic limit.

Findings

The integrated density of states varies linearly with magnetic field.

The slope of this variation is quantized and interaction-independent.

The results support the universality of the quantum Hall effect in weakly interacting systems.

Abstract

We consider a two-dimensional gas of interacting fermions in presence of an external constant magnetic field: the system is extended and homogeneous, and thus assumed to be invariant under magnetic translations. Working within the Hartree-Fock approximation, we analyze the system directly in the thermodynamic limit by solving a self-consistent fixed-point equation for the one-particle density matrix. We prove that, provided that the interactions among fermions are sufficiently weak, there exists a unique one-particle density matrix that solves the self-consistency condition. By choosing the Fermi-Dirac distribution as the function in the fixed-point equation, this approach can describe both positive and zero-temperature cases. At zero temperature and when the chemical potential of the non-interacting system lies in a spectral gap of the free Landau operator, our self-consistent solution is an orthogonal projection (an "interacting" effective Fermi projection). We prove that its integrated density of states varies linearly with the external magnetic field, provided the interaction is weak enough: the slope of this variation is quantized and independent of the interaction. According to the St\v{r}eda formula, this can be seen as yet another expression of the universality of the quantum Hall effect in weakly interacting systems, at least within the Hartree-Fock approximation.