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We analyze the statistical mechanics of a free-standing quantum crystallinemembrane within the framework of a systematic perturbative renormalizationgroup (RG). A power-counting analysis shows that the leading singularities ofcorrelation functions can be analyzed within an effective renormalizable modelin which the kinetic energy of in-plane phonons and subleading geometricalnonlinearities in the expansion of the strain tensor are neglected. Formembranes at zero temperature, governed by zero-point motion, the RG equationsof the effective model provide a systematic derivation of logarithmiccorrections to the bending rigidity and to the elastic Young modulus derived inearlier investigations. In the limit of a weakly applied external tension, thestress-strain relation at $T = 0$ is anomalous: the linear Hooke's law isreplaced with a singular law exhibiting logarithmic corrections. For small, butfinite temperatures, we use techniques of finite-size scaling to derive generalrelations between the zero-temperature RG flow and scaling laws ofthermodynamic quantities such as the thermal expansion coefficient $\alpha$,the entropy S, and the specific heat C. A combination of the scaling relationswith an analysis of thermal fluctuations shows that, for small temperatures,the thermal expansion coefficient $\alpha$ is negative and logarithmicallydependent on $T$, as predicted in an earlier work. Although the requirement$\lim_{T \to 0} \alpha = 0$, expected from the third law of thermodynamics isformally satisfied, $\alpha$ is predicted to exhibit such a slow variation toremain practically constant down to unaccessibly small temperatures.

Perturbative renormalization and thermodynamics of quantum crystalline membranes