We investigate the thermodynamic properties and the lattice stability of
two-dimensional crystalline membranes, such as graphene and related compounds,
in the low temperature quantum regime T→0. A key role is played by
the anharmonic coupling between in-plane and out-of plane lattice modes that,
in the quantum limit, has very different consequences than in the classical
regime. The role of retardation, namely of the frequency dependence, in the
effective anharmonic interactions turns out to be crucial in the quantum
regime. We identify a crossover temperature, T∗, between classical and
quantum regimes, which is ∼70−90 K for graphene. Below T∗, the
heat capacity and thermal expansion coefficient decrease as power laws with
decreasing temperature, tending to zero for T→0 as required by the
third law of thermodynamics.Comment: 13 pages, 1 figur