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

TL;DR

This work establishes a rigorous Hartree–Fock framework for a 2D weakly interacting fermionic gas in a constant magnetic field, proving the existence and uniqueness of a translation‑invariant self‑consistent density matrix in the thermodynamic limit for small coupling. By embedding the analysis in magnetic pseudodifferential calculus and exploiting magnetic translation symmetry, the authors show that zero‑temperature self‑consistent Fermi projections carry a linear, interaction‑independent integrated density of states, leading to a quantized Hall conductance via the Středa formula. The results thus demonstrate universality of the Hall conductance in weakly interacting fermionic systems within the Hartree–Fock approximation, aligning with non‑interacting and lattice‑based quantum Hall theories. The methodology combines fixed‑point analysis, Helffer–Sjöstrand functional calculus, and a detailed magnetic PDO framework to connect many‑body HF theory with topological transport properties. These findings provide a rigorous bridge between mean‑field electronic structure in magnetic fields and topological aspects of quantum Hall physics.

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 Středa 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.

Theorems & Definitions (34)

• Proposition 1.1
• Corollary 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Lemma 2.1: Gabor frame
• Remark 2.2: Gabor frame in $L^2(\mathbb{R}^2)$
• proof : Proof of Lemma \ref{['pseudo:frame']}
• Lemma 2.3
• proof