Tuesday, September 6, 2011

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

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 of the Epstein zeta function. Various resummations of this expansion are considered. The general procedure developed is then applied to two models: massless particles in a box both at zero and nonzero chemical potential, and electrons in a thin metal film. 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 size of the crystal are described.


Fig. 3. On the top panel: I. The chemical potential of electrons in the thin metal film at the effective electron mass m*=m, the temperature Image

, and the average particle number N=1.6×1016 what corresponds to the undeformed metal film with the area Image

, the width Lx=1 nm, and the chemical potential Image

. The small plots depict the total chemical potential at Image

(II) and Image

(III). The small plot (IV) depicts the quasiclassical part Image

of the chemical potential at Image

. On the bottom panel: (I) The quasiclassical contribution to the average number of conduction electrons at Image

and Image

. The total number of conduction electrons in the thin metal film with the area Image

at the fixed chemical potential Image

, and the temperatures Image

(II), Image

(III), Image

(IV), and Image

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