Thermodynamics of quantum crystalline membranes

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 from those in the classical regime. The role of retardation, namely of frequency dependence, in the effective anharmonic interactions turns out to be crucial in the quantum regime. We identify a crossover temperature, T*, between the classical and quantum regimes, which is similar to 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.