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 non-random 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

“…The regularity properties announced in (ii)-(iii) of Lemma 1.1 are far from being optimum. On the one hand, one can prove that z → P (β, z, b) can be analytically extended to [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.…”

“…Owing to this splitting of Λ N , we expect only the contribution to Tr(R N,b (ξ)) coming from the core regioñ Λ N to give rise to the limit in (2.6). With this aim in view, the geometric perturbation method consists in approximating the resolvent R N,b (ξ) with the operator U N,b (ξ) defined by: 7) where χΛ N and χΛ N are the characteristic functions ofΛ N andΛ N respectively. Afterwards, it remains to control the behavior when N → ∞ of the trace of the r.h.s.…”

“…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).…”

On the atomic orbital magnetism: A rigorous derivation of the Larmor and Van Vleck contributions

“…Under this approximation, we suppose that the distance R > 0 between two consecutive ions is sufficiently large so that the Fermi gas 'feels' mainly the potential energy generated by one single nucleus. To investigate the atomic orbital magnetism, our starting-point is the expression derived in [12,Thm. 1.2] for the bulk zero-field orbital susceptibility of the Fermi gas valid for any 'temperature' β > 0 and R > 0.…”

“…The starting-point is the expression (3.9) for the bulk zero-field orbital susceptibility (under the grand-canonical conditions) which holds for any β > 0 and R > 0. The derivation of such an expression is the main subject of [12]. Its proof relies on the gauge invariant magnetic perturbation theory applied on the magnetic resolvent operator.…”