Diamagnetic expansions for perfect quantum gases

Abstract

Abstract:In this work we study the diamagnetic properties of a perfect quantum gas in the presence of a constant magnetic field of intensity B. We investigate the Gibbs semigroup associated to the one particle operator at finite volume, and study its Taylor series with respect to the field parameter ω := eB/c in different topologies. This allows us to prove the existence of the thermodynamic limit for the pressure and for all its derivatives with respect to ω (the so-called generalized susceptibilities).MSC 2000: 82B10, 82B21, 81V99

The Faraday effect revisited: Thermodynamic limit

“…The hard part is to show that the polynomial growth in L of the trace norm does not appear in the actual trace. Similar difficulties involving magnetic semigroups were encountered in [1,6,7,10].…”

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.…”

“…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).…”

A Rigorous Proof of the Landau-Peierls Formula and much more

“…One can see this by performing an integration by parts in (3.2) and using the fact that the kernel of R 2 ∞ (ω, ξ) is jointly continuous. Moreover, one can prove [4,5,6] that the thermodynamic limit of the grand-canonical pressure is jointly smooth on (z, ω) ∈ (−e βE0 , ∞) × R.…”

“…One can prove using the same techniques as in [15] that these operators are locally trace class and have a jointly continuous kernel on R 3 × R 3 . By a closely related method as the one in [4], [5], it is proved in [39] that we can invert the thermodynamic limit with the partial derivatives w.r.t. ω of the grand-canonical pressure.…”