A rigorous approach to the magnetic response in disordered systems

Abstract

This paper is a part of an ongoing study on the diamagnetic behavior of a 3-dimensional quantum gas of non-interacting charged particles subjected to an external uniform magnetic field together with a random electric potential. We prove the existence of an almost-sure nonrandom thermodynamic limit for the grand-canonical pressure, magnetization and zero-field orbital magnetic susceptibility. We also give an explicit formulation of these thermodynamic limits. Our results cover a wide class of physically relevant random potentials which model not only crystalline disordered solids, but also amorphous solids.

On the zero-field orbital magnetic susceptibility of Bloch electrons in graphene-like solids: Some rigorous results

“…, see e.g. [3,4,7]. On the other hand, the use of the gauge invariant magnetic perturbation theory to prove (iii) allows us actually to get that b → P (β, z, b) is a C ∞ -function.…”

“…This is made possible by the use of the so-called gauge invariant magnetic perturbation theory (see e.g. [10,31] and [3,12,13,7,8] for further applications), followed by the Bloch-Floquet decomposition. After carrying out some convenient transformations needed to perform the zero-temperature limit, then in the semiconducting situation, we get from (1.11) a complete formula for the zero-field orbital susceptibility at zero temperature and fixed density which holds for an arbitrary number of bands with possible degeneracies, see (1.15).…”