  We consider relativistic (2+1)-QFTs on a product of time with a two-space and
study the vacuum free energy as a functional of the temperature and spatial
geometry. We focus on free scalar and Dirac fields on arbitrary perturbations
of flat space, finding that the free energy difference from flat space is
finite and always \textit{negative} to leading order in the perturbation. Thus
free (2+1)-QFTs appear to always energetically favor a crumpled space on all
scales; at zero temperature this is a purely quantum effect. Importantly, we
show that this quantum effect is non-negligible for the relativistic Dirac
degrees of freedom on monolayer graphene even at room temperature, so we argue
that this vacuum energy effect should be included for a proper analysis of the
equilibrium configuration of graphene or similar materials.
