The analytic asymptotic expressions for the Casimir free energy and entropy for two parallel graphene sheets possessing nonzero energy gap $\mathrm{\ensuremath{\Delta}}$ and chemical potential $\ensuremath{\mu}$ 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 $\mathrm{\ensuremath{\Delta}}g2\ensuremath{\mu}$, $\mathrm{\ensuremath{\Delta}}=2\ensuremath{\mu}$, and $\mathrm{\ensuremath{\Delta}}l2\ensuremath{\mu}$ 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 $\mathrm{\ensuremath{\Delta}}g2\ensuremath{\mu}$ and $\mathrm{\ensuremath{\Delta}}l2\ensuremath{\mu}$ the Casimir entropy satisfies the third law of thermodynamics (the Nernst heat theorem), whereas for $\mathrm{\ensuremath{\Delta}}=2\ensuremath{\mu}$ 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.

中文翻译：

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两个真实石墨烯片和热力学之间卡西米尔效应的量子场理论描述

在任意低温下推导出具有非零能隙 $\mathrm{\ensuremath{\Delta}}$ 和化学势 $\ensuremath{\mu}$ 的两个平行石墨烯片的 Casimir 自由能和熵的解析渐近表达式。石墨烯在 Matsubara 公式中的热量子场论框架中通过 ($2+1$) 维时空的极化张量来描述。在 $\mathrm{\ensuremath{\Delta}}g2\ensuremath{\mu}$、$\mathrm{\ensuremath{\Delta}}=2\ensuremath{\mu}$ 和$\mathrm{\ensuremath{\Delta}}l2\ensuremath{\mu}$ 考虑到由于 Matsubara 频率求和引起的隐式温度依赖性和由极化张量对温度的依赖性引起的显式温度依赖性，如一个参数。结果表明，对于 $\mathrm{\ensuremath{\Delta}}g2\ensuremath{\mu}$ 和 $\mathrm{\ensuremath{\Delta}}l2\ensuremath{\mu}$，Casimir 熵都满足热力学第三定律（能斯特热定理），而 $\mathrm{\ensuremath{\Delta}}=2\ensuremath{\mu}$ 违反了这个基本要求。所发现异常的物理意义是在金属和介电体之间卡西米尔效应的热力学性质的背景下考虑的。