On the Landau diamagnetism

The grand-canonical partition function of an assembly of free spinless electrons in a magnetic field enclosed in a box (Dirichlet boundary conditions) is shown to be an entire function of the fugacityz and the magnetic fieldH, as a consequence of the trace-norm convergence of the perturbation series for the statistical semigroup. This allows to derive analyticity properties of the pressure as a function ofz andH, and to express the coefficients of its power series expansion aroundz=H=0 by means of the unperturbed semigroup. Hence, the magnetic susceptibility at zero field and fixed density is expressed in terms of Green functions of the heat equation. Its asymptotic expansion for Λ→∞ (Fisher) along parallelepipedic domains is obtained up to 0 (S(Λ)/V(Λ)). The volume term of this expansion is the Landau diamagnetism.