Anomalous elasticity of 2D materials beyond self--consistent approximation

We study elastic properties of two--dimensional crystalline materials, such as graphene. It is known that in 2D membranes strong thermal fluctuations of flexural phonons lead to dramatic change of phonon spectrum and, as a result, in anomalous material--independent elastic properties, such as non--linear Hooke's law under the low stress and auxetic behavior.
We compute elastic moduli η and Poisson ratio ν of 2D membrane in the approximation of high embedded dimensionality d_c=2+d. We go beyond one-loop approximation and find that famous self--consistent screening approximation is only as good as first--order approximation.
Most remarkably, we analyze a case of disordered membrane and find new disorder--dominant phase, where all the critical exponents are different from the clean case. We find that phase transition happens at finite temperature in contrast to the prediction given by self--consistent screening approximation, that transition is only possible at absolute zero.

Presented work is a continuation of the results reported in papers PhysRevB.97.125402, PhysRevB.92.155428.