Abstract

We consider (2+1)-d relativistic QFT on a product of time with a static two-space and study the vacuum free energy as a functional of temperature and spatial geometry. Looking at both free scalars and fermions, with and without mass (and in the scalar case including a curvature coupling) we surprisingly find that any perturbation of a flat space is always energetically preferred to flat space. This is a UV finite effect, insensitive to any cut-off. We see the same behaviour for (2+1)-holographic CFTs which we compute via the gravity dual. In all these theories the same is true non-perturbatively for low curvature deformations of flat space. We consider the physical application of this to membranes carrying relativistic degrees of freedom, the vacuum energy of which then induce a tendency for the membrane to crumple. An interesting case is monolayer graphene, which experimentally is observed to ripple, and on large scales can be understood as a membrane carrying massless Dirac degrees of freedom.