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

Abstract

© 2020 American Physical Society. The analytic asymptotic expressions for the Casimir free energy and entropy for two parallel graphene sheets possessing nonzero energy gap Δ and chemical potential μ 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 Δ>2μ, Δ=2μ, and Δ<2μ 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 Δ>2μ and Δ<2μ the Casimir entropy satisfies the third law of thermodynamics (the Nernst heat theorem), whereas for Δ=2μ 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.