We review recent results on the low-temperature behaviors of the Casimir-Polder and Casimir free energy an entropy for a polarizable atom interacting with a graphene sheet and for two graphene sheets, respectively. These results are discussed in the wide context of problems arising in the Lifshitz theory of van der Waals and Casimir forces when it is applied to metallic and dielectric bodies. After a brief treatment of different approaches to theoretical description of the electromagnetic response of graphene, we concentrate on the derivation of response function in the framework of thermal quantum field theory in the Matsubara formulation using the polarization tensor in (2 + 1)-dimensional space—time. The asymptotic expressions for the Casimir-Polder and Casimir free energy and entropy at low temperature, obtained with the polarization tensor, are presented for a pristine graphene as well as for graphene sheets possessing some nonzero energy gap $\Delta$ and chemical potential $\mu$ under different relationships between the values of $\Delta$ and $\mu$. Along with reviewing the results obtained in the literature, we present some new findings concerning the case $\mu \ne 0$, $\Delta =0$. The conclusion is made that the Lifshitz theory of the Casimir and Casimir-Polder forces in graphene systems using the quantum field theoretical description of a pristine graphene, as well as real graphene sheets with $\Delta >2\mu$ or $\Delta <2\mu$, is consistent with the requirements of thermodynamics. The case of graphene with $\Delta =2\mu \ne 0$ leads to an entropic anomaly, but is argued to be physically unrealistic. The way to a resolution of thermodynamic problems in the Lifshitz theory based on the results obtained for graphene is discussed.

中文翻译：

---

石墨烯系统中的卡西米尔和卡西米尔-波尔德力：量子场理论描述和热力学

我们回顾了有关卡西米尔-帕尔德和卡西米尔自由能的低温行为的最新结果，这是一个可极化原子与石墨烯片相互作用和两个石墨烯片相互作用的熵。在将范德华力和卡西米尔力的李夫希茨理论应用于金属和介电体时，在广泛的问题中讨论了这些结果。在简要介绍了石墨烯电磁响应的理论描述的不同方法之后，我们集中讨论了使用（2 +1）维空间中的极化张量在松原公式中的热量子场论框架下的响应函数的推导。 -时间。利用极化张量获得的低温下卡西米尔-珀尔和卡西米尔自由能和熵的渐近表达式，$\Delta$ 和化学势 $\mu$ 值之间的不同关系 $\Delta$ 和 $\mu$。除了回顾文献中获得的结果之外，我们还提供了有关此案的一些新发现。$\mu \ne 0$， $\Delta =0$。结论是，使用原始石墨烯的量子场理论描述以及石墨烯的真实石墨烯片，对石墨烯系统中卡西米尔和卡西米尔-波尔德力的李夫希兹理论进行了总结。$\Delta >2\mu$ 要么 $\Delta <2\mu$，与热力学要求一致。石墨烯的情况$\Delta =2\mu \ne 0$导致熵异常，但据认为在物理上是不现实的。讨论了基于石墨烯获得的结果解决Lifshitz理论中热力学问题的方法。