Rigorous semiclassical results for the magnetic response of an electron gas

TL;DR

This paper provides a rigorous semiclassical analysis of the magnetic response of a free electron gas under a magnetic field, revealing temperature-dependent behaviors and oscillations linked to classical orbits.

Contribution

It introduces a rigorous semiclassical framework for analyzing the magnetic susceptibility of an electron gas across various temperature regimes.

Findings

Derivation of asymptotic expansions for magnetization and susceptibility

Identification of de Haas-van Alphen oscillations at specific temperature scales

Connection between quantum oscillations and classical periodic orbits

Abstract

Consider a free electron gas in a confining potential and a magnetic field in arbitrary dimensions. If this gas is in thermal equilibrium with a reservoir at temperature $T >0$, one can study its orbital magnetic response (omitting the spin). One defines a conveniently ``smeared out'' magnetization $M$, and the corresponding magnetic susceptibility $\chi$, which will be analyzed from a semiclassical point of view, namely when $\hbar$ (the Planck constant) is small compared to classical actions characterizing the system. Then various regimes of temperature $T$ are studied where $M$ and $\chi$ can be obtained in the form of suitable asymptotic $\hbar$-expansions. In particular when $T$ is of the order of $\hbar$, oscillations ``\`a la de Haas-van Alphen'' appear, that can be linked to the classical periodic orbits of the electronic motion.