Quantum elasticity of graphene: Thermal expansion coefficient and specific heat

We explore thermodynamics of a quantum membrane, with a particular application to suspended graphene membrane and with a particular focus on the thermal expansion coefficient. We show that an interplay between quantum and classical anharmonicity-controlled fluctuations leads to unusual elastic properties of the membrane. The effect of quantum fluctuations is governed by the dimensionless coupling constant, $g_0 \ll 1$, which vanishes in the classical limit ($\hbar \to 0$) and is equal to $\simeq 0.05$ for graphene. We demonstrate that the thermal expansion coefficient $\alpha_T$ of the membrane is negative and remains nearly constant down to extremely low temperatures, $T_0\propto \exp (-2/g_0)$. We also find that $\alpha_T$ diverges in the classical limit: $\alpha_T \propto - \ln(1/g_0)$ for $g_0 \to 0$. For graphene parameters, we estimate the value of the thermal expansion coefficient as $\alpha_T \simeq - 0.23\:{\rm eV}^{-1}$, which applies below the temperature $T_{\rm uv} \sim g_0 \varkappa_0 \sim 500$\:K (where $\varkappa_0 \sim 1$\:eV is the bending rigidity) down to $T_0 \sim 10^{-14}$\:K. For $T<T_0$, the thermal expansion coefficient slowly (logarithmically) approaches zero with decreasing temperature. This behavior is surprising since typically the thermal expansion coefficient goes to zero as a power-law function. We discuss possible experimental consequences of this anomaly. We also evaluate classical and quantum contributions to the specific heat of the membrane and investigate the behavior of the Gr\"uneisen parameter.