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A quantum field theory approach is applied to investigate the dynamics offlexural phonons in a metallic membrane like graphene, looking for the effectsderiving from the strong interaction between the electronic excitations andelastic deformations. Relying on a self-consistent screening approximation tothe phonon self-energy, we show that the theory has a critical pointcharacterized by the vanishing of the effective bending rigidity of themembrane at low momentum. We also check that the instability in the sector offlexural phonons takes place without the development of an in-plane staticdistortion, which is avoided due to the significant reduction of theelectron-phonon couplings for in-plane phonons at large momenta. Furthermore,we analyze the scaling properties of the many-body theory to identify the orderparameter that opens up at the point of the transition. We find that thevanishing of the effective bending rigidity and the onset of a nonzeroexpectation value of the mean curvature are concurrent manifestations of thecritical behavior. The results presented here imply that, even in the absenceof tension, the theory has a critical point at which the flat geometry becomesan unstable configuration of the metallic membrane, with a condensation of themean curvature field that may well reproduce the smooth distribution of ripplesobserved in free-standing graphene.

Rippling transition from electron-induced condensation of curvature field in graphene