Quantum field theoretical description of the Casimir effect between two real graphene sheets and thermodynamics

TL;DR

This paper derives low-temperature asymptotic expressions for the Casimir free energy and entropy between two graphene sheets with energy gap and chemical potential, revealing thermodynamic consistency conditions and anomalies.

Contribution

It provides the first detailed quantum field theoretical analysis of the Casimir effect between real graphene sheets, including the effects of energy gap and chemical potential.

Findings

Casimir entropy satisfies the third law for certain parameter regimes

Anomaly occurs at the condition =2

Different asymptotic expressions are derived for >2, =2, and <2

Abstract

The analytic asymptotic expressions for the Casimir free energy and entropy for two parallel graphene sheets possessing nonzero energy gap $\Delta$ and chemical potential $\mu$ are derived at arbitrarily low temperature. Graphene is described in the framework of thermal quantum field theory in the Matsubara formulation by means of the polarization tensor in (2+1)-dimensional space-time. Different asymptotic expressions are found under the conditions $\Delta>2\mu$, $\Delta=2\mu$, and $\Delta<2\mu$ taking into account both the implicit temperature dependence due to a summation over the Matsubara frequencies and the explicit one caused by a dependence of the polarization tensor on temperature as a parameter. It is shown that for both $\Delta>2\mu$ and $\Delta<2\mu$ the Casimir entropy satisfies the third law of thermodynamics (the Nernst heat theorem), whereas for $\Delta=2\mu$ this fundamental requirement is violated. The physical meaning of the discovered anomaly is considered in the context of thermodynamic properties of the Casimir effect between metallic and dielectric bodies.