  We analyze the statistical mechanics of a free-standing quantum crystalline
membrane within the framework of a systematic perturbative renormalization
group (RG). A power-counting analysis shows that the leading singularities of
correlation functions can be analyzed within an effective renormalizable model
in which the kinetic energy of in-plane phonons and subleading geometrical
nonlinearities in the expansion of the strain tensor are neglected. For
membranes at zero temperature, governed by zero-point motion, the RG equations
of the effective model provide a systematic derivation of logarithmic
corrections to the bending rigidity and to the elastic Young modulus derived in
earlier investigations. In the limit of a weakly applied external tension, the
stress-strain relation at $T = 0$ is anomalous: the linear Hooke's law is
replaced with a singular law exhibiting logarithmic corrections. For small, but
finite temperatures, we use techniques of finite-size scaling to derive general
relations between the zero-temperature RG flow and scaling laws of
thermodynamic quantities such as the thermal expansion coefficient $\alpha$,
the entropy S, and the specific heat C. A combination of the scaling relations
with an analysis of thermal fluctuations shows that, for small temperatures,
the thermal expansion coefficient $\alpha$ is negative and logarithmically
dependent on $T$, as predicted in an earlier work. Although the requirement
$\lim_{T \to 0} \alpha = 0$, expected from the third law of thermodynamics is
formally satisfied, $\alpha$ is predicted to exhibit such a slow variation to
remain practically constant down to unaccessibly small temperatures.
