We consider relativistic (2 þ 1)-dimensional quantum field theories (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 negative to leading order in the perturbation. Thus, free (2 þ 1)-dimensional QFTs appear to always energetically favor a crumpled space on all scales; this is true both as a purely quantum effect at zero temperature and as a purely thermal effect at high temperature. 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. DOI: 10.1103/PhysRevLett.120.261601 Introduction.-The presence of matter gives a surface embedded in an ambient space an energy. This matter may be external to the surface-like the pressure of air on a soap bubble-or may comprise the material nature of the surface itself-like a membrane with surface tension and bending energy. These energies determine the equilibrium (i.e., static) configuration of such a surface: for instance, the presence of surface tension tends to make membranes favor (smooth) minimal-area configurations, while finitetemperature thermodynamic effects may render membranes unstable to crumpling or rippling [1][2][3].In this Letter, we initiate a study of the free energy contribution to the equilibrium configuration of a surface due to free relativistic quantized matter fields living on it. In particular, we include zero-temperature (Casimir) effects. Such relativistic quantum fields occur in various physical settings: for example, in graphene and related materials, the electronic structure gives rise to an effective description in terms of relativistic Dirac fermions propagating on the two-dimensional crystal [4][5][6]. In cosmology, domain wall defects may exist [7] and could carry upon them relativistic degrees of freedom(d.o.f.). More exotically, in braneworld models, our Universe is itself a surface on which the standard model fields live [8,9].The setting is then (2 þ 1)-dimensional quantum field theory (QFT) on a product of time with a two-space. By studying both the free nonminimally coupled scalar and the free Dirac fermion, we will see that such a field lowers the

We consider (2+1)-QFT at finite temperature on a product of time with a static spatial geometry. The suitably defined difference of thermal vacuum free energy for the QFT on a deformation of flat space from its value on flat space is a UV finite quantity, and for reasonable fall-off conditions on the deformation is IR finite too. For perturbations of flat space we show this free energy difference goes quadratically with perturbation amplitude and may be computed from the linear response of the stress tensor. As an illustration we compute it for a holographic CFT finding that at any temperature, and for any perturbation, the free energy decreases. Similar behaviour was previously found for free scalars and fermions, and for unitary CFTs at zero temperature, suggesting (2+1)-QFT may generally energetically favour a crumpled spatial geometry. We also treat the deformation in a hydrostatic small curvature expansion relative to the thermal scale. Then the free energy variation is determined by a curvature correction to the stress tensor and for these theories is negative for small curvature deformations of flat space.

We examine the renormalized free energy of the free Dirac fermion and the free scalar on a (2+1)-dimensional geometry ℝ × Σ, with Σ having spherical topology and prescribed area. Using heat kernel methods, we perturbatively compute this energy when Σ is a small deformation of the round sphere, finding that at any temperature the round sphere is a local maximum. At low temperature the free energy difference is due to the Casimir effect. We then numerically compute this free energy for a class of large axisymmetric deformations, providing evidence that the round sphere globally maximizes it, and we show that the free energy difference relative to the round sphere is unbounded below as the geometry on Σ becomes singular. Both our perturbative and numerical results in fact stem from the stronger finding that the difference between the heat kernels of the round sphere and a deformed sphere always appears to have definite sign. We investigate the relevance of our results to physical systems like monolayer graphene consisting of a membrane supporting relativistic QFT degrees of freedom.

We compare the behavior of the vacuum free energy (i.e. the Casimir energy) of various (2 + 1)-dimensional CFTs on an ultrastatic spacetime as a function of the spatial geometry. The CFTs we consider are a free Dirac fermion, the conformally-coupled scalar, and a holographic CFT, and we take the spatial geometry to be an axisymmetric deformation of the round sphere. The free energies of the fermion and of the scalar are computed numerically using heat kernel methods; the free energy of the holographic CFT is computed numerically from a static, asymptotically AdS dual geometry using a novel approach we introduce here. We find that the free energy of the two free theories is qualitatively similar as a function of the sphere deformation, but we also find that the holographic CFT has a remarkable and mysterious quantitative similarity to the free fermion; this agreement is especially surprising given that the holographic CFT is strongly-coupled. Over the wide ranges of deformations for which we are able to perform the computations accurately, the scalar and fermion differ by up to 50% whereas the holographic CFT differs from the fermion by less than one percent.

We compare the behavior of the vacuum free energy (i.e. the Casimir energy) of various (2 + 1)-dimensional CFTs on an ultrastatic spacetime as a function of the spatial geometry. The CFTs we consider are a free Dirac fermion, the conformally-coupled scalar, and a holographic CFT, and we take the spatial geometry to be an axisymmetric deformation of the round sphere. The free energies of the fermion and of the scalar are computed numerically using heat kernel methods; the free energy of the holographic CFT is computed numerically from a static, asymptotically AdS dual geometry using a novel approach we introduce here. We find that the free energy of the two free theories is qualitatively similar as a function of the sphere deformation, but we also find that the holographic CFT has a remarkable and mysterious quantitative similarity to the free fermion; this agreement is especially surprising given that the holographic CFT is strongly-coupled. Over the wide ranges of deformations for which we are able to perform the computations accurately, the scalar and fermion differ by up to 50% whereas the holographic CFT differs from the fermion by less than one percent.