The analytic asymptotic expressions for the Casimir free energy and entropy for two parallel graphene sheets possessing nonzero energy gap $\mathrm{\Delta }$ 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 $\mathrm{\Delta }>2\mu$, $\mathrm{\Delta }=2\mu$, and $\mathrm{\Delta }<2\mu$ 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 $\mathrm{\Delta }>2\mu$ and $\mathrm{\Delta }<2\mu$ the Casimir entropy satisfies the third law of thermodynamics (the Nernst heat theorem), whereas for $\mathrm{\Delta }=2\mu$ 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.

• Received 8 June 2020
• Accepted 27 June 2020

DOI:https://doi.org/10.1103/PhysRevD.102.016006