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The Nernst heat theorem for an atom interacting with graphene: Dirac model with nonzero energy gap and chemical potential

We derive the low-temperature behavior of the Casimir-Polder free energy for a polarizable atom interacting with graphene sheet which possesses the nonzero energy gap $Δ$ and chemical potential $μ$. The response of graphene to the electromagnetic field is described by means of the polarization tensor in the framework of Dirac model on the basis of first principles of thermal quantum field theory in the Matsubara formulation. It is shown that the thermal correction to the Casimir-Polder energy consists of three contributions. The first of them is determined by the Matsubara summation using the polarization tensor defined at zero temperature, whereas the second and third contributions are caused by an explicit temperature dependence of the polarization tensor and originate from the zero-frequency Matsubara term and the sum of all Matsubara terms with nonzero frequencies, respectively. The asymptotic behavior for each of the three contributions at low temperature is found analytically for any value of the energy gap and chemical potential. According to our results, the Nernst heat theorem for the Casimir-Polder free energy and entropy is satisfied for both $Δ> 2μ$ and $Δ< 2μ$. We also reveal an entropic anomaly arising in the case $Δ= 2μ$. The obtained results are discussed in connection with the long-standing fundamental problem in Casimir physics regarding the proper description of the dielectric response of matter to the electromagnetic field.