  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.
