What spatial geometry does the (2+1)-dimensional QFT vacuum prefer?

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 negative to leading order in the perturbation. Thus free (2+1)-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.


I. 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 finite-temperature 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.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+1)-d QFT on a product of time with a two-space.By studying both the free nonminimally coupled scalar 1 and the free Dirac fermion we will see that such a field lowers the free energy of the surface on which it lives when the surface is deformed away from being intrinsically flat 2 .This energy difference is UV finite (and thus well-defined) and present at any temperature including T = 0 (in which case it can be interpreted as a Casimir effect) both for massless and massive fields, and any scalar non-minimal coupling.It is then natural to wonder whether a classical membrane action is able to counteract this quantum tendency to crumple.We will perform a naïve analysis of this question for monolayer graphene, which is indeed seen to ripple on short scales [12,13].We show that at room temperature the quantum vacuum energy of the Dirac fermions give a scale at which one would expect crumpling effects on the order of the lattice spacing.The effective membrane description which would validate our analysis breaks down at this scale, so our results make no definitive statement about the rippling of graphene.However, they do indicate that a careful consideration of these quantum effects is important for a proper treatment of equilibrium configurations of graphene and similar materials even at room temperature.

II. FREE ENERGY DIFFERENCE
We consider a spacetime which is a product of time with a two-space Σ (for now taken to be general).Since we are interested in QFT at finite temperature T we work in Euclidean time, so the metric is 3 with τ periodic with period β = 1/T .We will consider a free scalar φ and Dirac spinor ψ with equations of motion respectively, where R is the Ricci scalar and / D is understood as being defined by the spin connection (our conventions can be found in the Supplemental Material).
in holographic CFTs and perturbatively in general [10,11]. 3Unless otherwise stated we use natural units = c = k B = 1 with c the "effective" speed of light of the relativistic fields (not necessarily equal to the actual speed of light).
The free energy F [Σ] is a functional of the geometry Σ (and temperature T ) and is given in terms of the partition function Z[Σ] as F = −T ln Z.We are specifically interested in the difference ∆F between the free energy on Σ and some reference background space Σ at the same temperature, which satisfies where Φ stands for the matter field (scalar or fermion) being integrated over in the path integral, ∆S = S Σ − S Σ is the difference between the action on S 1 ×Σ and S 1 ×Σ, and the expectation value is defined by the path integral on the background geometry S 1 × Σ.To evaluate ∆F , recall that for free fields the path integrals in (3) yield functional determinants, giving where q = −1/2 (+1) for the scalar (fermion), O is an elliptic self-adjoint scalar operator on Σ given explicitly in (11) below, and the determinant is evaluated over Matsubara frequencies on the thermal circle (with appropriate periodicity or antiperiodicity in the scalar and fermion cases respectively).For the scalar, ( 4) is obtained straightforwardly.The fermion case is more subtle, and we leave full details to the Supplemental Material.In short, a direct path integral yields Z = det(i / D − iM ).However, by exploiting the direct product structure of the metric (1) along with the fact that the two-dimensional rotation group only has a single generator, we may eliminate the spinor structure and reduce the determinant to that of an elliptic operator of the form (4) with the determinant taken over the space of complex functions with antiperiodicity on the thermal circle.
The free energy can then be evaluated via heat kernel methods [14]: defining the heat kernel as K L (t) ≡ Tr(e −tL ) = i e −tλi (with λ i the eigenvalues of L), one obtains where This expression is UV divergent unless ∆K L (t) vanishes at t = 0; this condition can be ensured by an appropriate choice of background Σ.Specifically, the heat kernel expansion gives [14] where V Σ and χ Σ are the volume 4 and Euler number of Σ, respectively, and c 1 , c 2 , and c 3 are dimensionless constants independent of the geometry (though they depend on the choice of matter field).Thus requiring that ∆F be UV-finite imposes that we choose a background geometry Σ with the same volume and topology as Σ.It is worth emphasizing that although the undifferenced functional determinant det L is UV-divergent, we do not need to invoke any regularization to evaluate the differenced free energy 5 .Now we specialize to our case of interest.Ultimately we wish to take Σ to be a deformation of flat space, Σ.Since these are two-dimensional we introduce conformally flat coordinates x A , in terms of which the metrics on Σ and Σ take the form In order to have good control over the spectrum of L (which is essential for computing the heat kernel), we compactify these to tori Σ L , Σ L via the identifications x A ∼ x A + L A with L 1 = L and L 2 = rL.We consider a family of deformations f L (x) so that as L → ∞ (with r fixed) we recover (7) with the x A uncompactified.Moreover, at any finite L, we may choose f L such that V Σ L = V Σ L .By the arguments above, this condition will ensure that for every L, the free energy difference between the deformed and flat torus will be UV finite. 6ur object of interest is the free energy difference with this IR regulator removed: with For notational convenience we will henceforth forego writing the arguments of these functionals explicitly, and we will drop the subscripts L on f L and ∞ on ∆F ∞ .Using (4), we finally obtain where the sum over Matsubara frequencies yields and the operators O are given explicitly in terms of f as with ∇ A and ¯ the covariant derivative and Hodge dual on the flat background f = 0, and the subscripts s and f denoting the scalar and fermion.

III. PERTURBATION THEORY
We introduce a perturbation parameter to expand our deformation Σ of the flat Σ as Preservation of the volume requires We denote by λ i and h i (x) the eigenvalues and eigenfunctions of O, so in order to compute ∆K O (t) perturbatively we must compute the perturbative corrections to λ i .We expand O as explicit expressions for O (1) and O (2) can be obtained by expanding the form of O given in (11).Continuing to use bars to denote unperturbed objects, we likewise expand its eigenvalues and eigenfunctions as so the hi are eigenfunctions of the flat space Laplacian with corresponding eigenvalues λi .We choose these eigenfunctions to be normalized as Then defining hj , hj + k; λk = λi P ik P (1) standard perturbation theory yields the eigenvalue shifts ii , λ Note that we have glossed over a subtlety: recall from QM perturbation theory that the presence of degenerate subspaces imposes additional constraints on the unperturbed eigenfunctions hi for the expansion (15b) to be consistent.The first order eigenvalue problem requires we arrange our basis hi such that P (1) ij is diagonal within such subspaces (i.e. if λi = λj but i = j, then P If any degeneracies remain at first order, we must further ensure at second order that P (2) ij be diagonal in the remaining degenerate subspaces.We discuss this issue explicitly in the Supplemental Material.
Finally, we expand the heat kernel as with ii , (20a) IV. RESULTS In order to perform our computations we Fourier decompose the perturbation where the sum runs over all pairs of integers N = {n 1 , n 2 }, and the second line defines k A = lim L→∞ 2πn A /L A .An explicit calculation on the torus for fixed L reveals that (for both the scalar and fermion) while the eigenvalues are indeed shifted at first order in , their contribution to the heat kernel vanishes: K (1) = 0.The leading order perturbation to the heat kernel is then the second order term K (2) .A lengthy but straightforward computation yields the finite-L expressions presented in the Supplemental Material; in the limit L → ∞ they become and 0 dζ e (ζ ) 2 .Thus using (9) we find ζ decay of I(ζ) also implies IR finiteness in the massless case M = 0 for both the fermion and minimally-coupled scalar (ξ = 0) 7 .Finally, a key physical point is that, as can be seen by explicitly plotting 8 , qI(ζ) < 0 for all ζ > 0 (and all ξ for the scalar), implying that for any (nonconstant) f the free energy difference is strictly negative to leading order in : ∆F < 0.
The form of the expression (25), along with the asymptotic behaviors of Θ(ζ) and I(ζ), makes it possible to derive scaling relations.Specifically, defining M = /(cM ) to be the (reduced) Compton wavelength, T = c/(k B T ) to be a thermal wavelength, and to be the characteristic length scale of f , ∆F scales as in Table I.Thus at small temperatures -in which ∆F becomes the energy difference ∆E -the effect is a purely quantum one: ∆E ∼ − 2 c/ for M .On the other hand, at high temperatures the energy decrease is a thermal effect: e.g.∆F f ∼ − 2 k B T for M .As a final note, the small-temperature limit T max[ , M ] is in fact analytically tractable: Poisson resummation gives Θ(T 2 t) = β/ √ 4πt up to terms that are exponentially suppressed in β 2 /t, which allows us to compute a(k) explicitly as 9 We have seen that at any temperature, free relativistic (2+1)-dimensional degrees of freedom energetically prefer deformations of flat space to flat space itself.Let us now consider how this effect competes with a membrane's bending energy, which at zero temperature favors a flat geometry.
Consider three-dimensional flat space with Cartesian coordinates {X A , Z} and parametrize a surface in it by Then for small and suitable v A , the intrinsic metric on the membrane in 7 For the massless scalar with non-minimal coupling ξ = 0, ∆Fs is IR divergent since the flat space zero eigenvalue aquires a negative contribution due to the scalar curvature coupling.This is reflected in the ln( M / ) corrections mentioned in Table I. 8 It is possible to prove that I f < 0 without resorting to plotting it; we have not been able to find as elegant of a proof for Is. 9 In the massless limit these agree precisely with the energy in [11] for the massless scalar CFT (ξ = 1/8) and free fermion CFT (with their appropriate central charges c T = (3/2)/(4π) 2 and 3/(4π) 2 respectively).
the coordinates x A is as in equation (7) with The bending energy due to extrinsic curvature is where κ is the bending rigidity.If the membrane is deformed from flat over a region of characteristic size M , then the (positive) bending energy E B and (negative) vacuum energy E Q (at zero temperature) for N free relativistic quantum fields are parametrically given as The ground state equilibrium configuration of the membrane should minimize E = E B + E Q .One might expect that because E B is lower order in than E Q , a perturbative analysis guarantees that E > 0 for any deformation of flat space.However, note that E B and E Q have different scale dependence, with E Q dominating at sufficiently small scales.Thus if crumple ≡ N c/κ, E Q can be comparable to and even dominate E B while still in the perturbative regime 1.Whether or not E actually decreases for (sufficiently large) deformations of flat space -therefore implying that the membrane's equilibrium configuration is crumpled at a scale crumplethen depends on nonlinear and higher-derivative contributions to its bending action and whether or not these are relevant at the scale crumple at amplitudes O( 2 ).
It is instructive to consider the case of a graphene monolayer, for which the bending rigidity is κ ∼ 1 eV, the unit cell has size cell ∼ 1 Å, and the relativistic fields are two Dirac fermions with effective speed c ∼ c light /100, with c light the actual speed of light [5,13].Our effective membrane description is valid for cell , while from Table I the scaling properties (28) are valid at room temperature for T =300 K ∼ 10 3 cell .Computing the crumpling scale, we find crumple ∼ 10 cell , which is sufficiently close to cell to make our effective membrane description suspect.Thus while this naïve analysis is insufficient to imply the existence of a crumpled equi-librium configuration for graphene, it does indicate that long range quantum properties of the conduction electrons (which give rise to the effective Dirac fermions) are important for understanding the energetics of equilibrium monolayer graphene even at room temperature; such effects are presumably highly challenging to correctly incorporate into Monte-Carlo simulations.Indeed, it is intriguing to note that for freely suspended graphene at room temperature, one does see low amplitude ripples on short scales ∼ 50 Å, close our crumple [12].
We emphasize that in future two-dimensional crystal materials whose electronic structures similarly give rise to Dirac fermions (or perhaps scalars or vectors), the situation may be different.In particular, if one wishes to have such a monolayer material that is flat on scales above the unit cell scale cell , one presumably requires N c/κ cell , which may be regarded as a bound on the speed or number of relativistic species, given the bending mechanics of the crystal.
We will follow the Clifford algebra conventions of [15]: in Euclidean signature, the Clifford algebra is which allows us to take the γ µ to be Hermitian.With such conventions, a natural choice of representation of the gamma matrices in three dimensions is γ µ = σ µ with σ µ the Pauli matrices, though we note that none of our statements will depend on such a choice.Scalars are formed from spinors χ, ψ as the bilinears χψ with χ = χ † , and the massive Euclidean Dirac action on a curved space with metric g ab is where / D = γ µ (e µ ) a D a with {(e µ ) a } for µ = 1, 2, 3 a vielbein, ∇ a the usual Riemann connection compatible with g ab , S µν = [γ µ , γ ν ]/4 the generators of the Lorentz group, and ω aµν = (e µ ) b ∇ a (e ν ) b the spin connection.Note that the operator i / D is self-adjoint, but the i in the mass term renders the Euclidean action non-Hermitian.This factor of i is necessary to ensure that the action obeys the Osterwalder-Schrader positivity conditions; see e.g.[16] for a discussion of such subtleties associated with spinors in Euclidean space.
Performing the path integral yields Because i / D is self-adjoint, its eigenvalues are real.Moreover, in the direct product geometry (1), eigenspinors of i / D can be decomposed into Fourier modes ψ = e −iΩnτ ψ Σ , with ψ Σ a spinor on Σ and Ω n = (2n + 1)π/β a Matsubara frequency (with n ∈ Z).It is then straightforward to show that if e −iΩnτ ψ Σ is an eigenspinor of i / D with eigenvalue λ, then e iΩnτ γ τ ψ Σ is an eigenspinor with eigenvalue −λ.Thus the spectrum of i / D on the background ( 1) is symmetric about zero 10 , so we have (33) (See e.g.[18] for more on this trick in d = 4.) Now, writing the metric on Σ in the conformally flat form (7), one can evaluate / D 2 − M 2 .Noting that there is only one generator S 12 = (i/2)γ τ of rotations in two dimensions, we obtain where P L,R = (1 ± γ τ )/2 are left and right Weyl projectors on Σ and L is as given in (4).Decomposing ψ = e −iΩnτ ψ Σ , we see that L and L * act only on left-and right-helicity Weyl spinors P L ψ Σ , P R ψ Σ , respectively.Since these spinors only have one component each, we may just interpret L and L * as acting only on complex functions (albeit with antiperiodic boundary conditions on the thermal circle).We therefore have det(−LP L − L * P R ) = det(−L) det(−L * ) = (det L) 2 , (35) where in the second expression we take the determinants only over the space of functions on which L and L * act, and in the last equality we noted that because L is selfadjoint (with respect to the usual L 2 norm), L and L * have the same spectrum (and thus determinant).Thus the partition function for the fermion can be evaluated by just taking the functional determinant of a scalar differential operator acting on complex functions.

TABLE I .
The scaling of ∆F for the minimally coupled free scalar field and Dirac fermion for different relative magnitudes of , M , and T .Note that for the non-minimally coupled scalar (i.e.ξ = 0), factors of ln( M / ) appear in the last two rows.
A few comments are in order.Firstly we see the leading variation in ∆F is quadratic in .Next, we have thatI(ζ) is finite and Θ(ζ) is O(ζ −1/2 ) at small ζ,and thus ∆F is UV-finite.Likewise, since I(ζ) and Θ(ζ) are O(1) at large ζ, ∆F is also IR-finite for M > 0; in fact, the large-