One-loop omega-potential of quantum fields with ellipsoid constant-energy surface dispersion law

TL;DR

This paper develops rapidly convergent expansions for the one-loop omega-potential of quantum fields with ellipsoid dispersion laws, analyzing contributions and applying to models like electrons in thin films and graphene.

Contribution

It introduces a general method for expanding the omega-potential with ellipsoid dispersion laws, including quasiclassical, branch cut, and oscillating parts, with applications to specific physical models.

Findings

Derived explicit low- and high-temperature expansions.

Established relation between cut contribution and Casimir energy.

Obtained oscillation descriptions of chemical potential in thin films.

Abstract

Rapidly convergent expansions of a one-loop contribution to the partition function of quantum fields with ellipsoid constant-energy surface dispersion law are derived. The omega-potential is naturally decomposed into three parts: the quasiclassical contribution, the contribution from the branch cut of the dispersion law, and the oscillating part. The low- and high-temperature expansions of the quasiclassical part are obtained. An explicit expression and a relation of the contribution from the cut with the Casimir term and vacuum energy are established. The oscillating part is represented in the form of the Chowla-Selberg expansion for the Epstein zeta function. Various resummations of this expansion are considered. The developed general procedure is applied to two models: massless particles in a box both at zero and non-zero chemical potential; electrons in a thin metal film. The rapidly convergent expansions of the partition function and average particle number are obtained for these models. In particular, the oscillations of the chemical potential of conduction electrons in graphene and a thin metal film due to a variation of sizes of the crystal are described.