The very existence of isolated graphene is a challenge to the Mermin-Wagner theorem that forbid long-range (lattice) ordering in truly two-dimensional (2D) systems, which should be unstable against lattice fluctuations leading to crumpling, folding etc. In real materials, however, the anharmonic coupling between the in-plane lattice mode and the out-of-plane flexural modes was shown, in the high-temperature classical regime, to stabilize the system [1-4]. The generalization of these results in the low temperature quantum regime leads however to unphysical results signalizing that new physics is involved.
In this contribution we present a microscopic theory to investigate the thermodynamical properties of graphene, modeled as a crystalline membranes, in the zero/low-temperature regime. Using Quantum Field Theory, we generalize the self-consistent screening approximation (SCSA) approach  at the quantum level. A key role is played by the retarded nature of the anharmonic coupling between in-plane and out-of-plane lattice modes that, in the quantum limit, turns to have crucial different consequences than in the classical regime.
We identify a crossover temperature T* between classical and quantum regimes, which is T* ~ 70-79 K for graphene . Below T* the heat capacity and thermal expansion coefficients results to decrease as power laws with decreasing temperature, vanishing at T=0, reconciling the SCSA theory with the third law of thermodynamics .
 A. Fasolino, J. Los, and M.I. Katsnelson, Nat. Mater. 6, 858 (2007).  D. Nelson and L. Peliti, J. Phys. 48, 1085 (1987).  P. Le Doussal and L. Radzihovsky, Phys. Rev. Lett. 69, 1209 (1992).  D. Gazit, Phys. Rev. E 80, 041117 (2009).  K. V. Zakharchenko, R. Roldan, A. Fasolino, and M.I. Katsnelson, Phys. Rev. B82, 125435 (2010).  B. Amorim, R. Roldan, E. Cappelluti, A. Fasolino, F. Guinea, and M.I. Katsnelson, arXiv:1403.2637 (2014).