Rippling transition from electron-induced condensation of curvature   field in graphene
  A quantum field theory approach is applied to investigate the dynamics of flexural phonons in a metallic membrane like graphene, looking for the effects deriving from the strong interaction between the electronic excitations and elastic deformations. Relying on a self-consistent screening approximation to the phonon self-energy, we show that the theory has a critical point characterized by the vanishing of the effective bending rigidity of the membrane at low momentum. We also check that the instability in the sector of flexural phonons takes place without the development of an in-plane static distortion, which is avoided due to the significant reduction of the electron-phonon couplings for in-plane phonons at large momenta. Furthermore, we analyze the scaling properties of the many-body theory to identify the order parameter that opens up at the point of the transition. We find that the vanishing of the effective bending rigidity and the onset of a nonzero expectation value of the mean curvature are concurrent manifestations of the critical behavior. The results presented here imply that, even in the absence of tension, the theory has a critical point at which the flat geometry becomes an unstable configuration of the metallic membrane, with a condensation of the mean curvature field that may well reproduce the smooth distribution of ripples observed in free-standing graphene. 
J. Gonzalez
cond-mat.mes-hall