Statistical Mechanics of Freely Fluctuating Two-Dimensional Elastic Crystals

In this paper, some facts related Joel L. Lebowitz are mentioned. In addition, some of the known theory concerning the statistical physics of freely fluctuating two-dimensional crystals subject to a non-linear elastic hamiltonian is described. Particular topics that are discussed include the existence of two-dimensional crystals in relation to the Hohenberg–Mermin–Wagner theorem, the crumpling transition for freely suspended crystalline membranes and the renormalization of elastic moduli. Although much of what is known has been uncovered since the mid-80s, the topic has become of interest once again due to the discovery of graphene and other two-dimensional crystals. The field is vast so it is the aim of this note to describe some of its fundamental properties.