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.

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 fr

relativistic qu
ntized 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 etting 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, whi h 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 g neral).Since we are interested in QFT at finite temperature T we work in Euclidean time, so the metric is 3
ds 2 = dτ 2 + ds 2 Σ (1)
with τ periodic with period β = 1/T .We will consider a free scalar φ and Dirac spinor ψ with equations of motion
−∇ 2 + ξR + M 2 φ = 0 , / D + M ψ = 0 (2)
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
e −β∆F = Z[Σ] Z[Σ] = DΦ e −SΣ[Φ] DΦ e −S Σ [Φ] = e −∆S Σ ,(3)
where Φ stands for the matter field (

alar or fermion) being inte
rated 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 at for free fields the path integrals in (3) yield functional determinants, giving
Z = (det L) q with L = −∂ 2 τ + calar (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 th 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 Supplementa 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 n 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
β∆F = q ∞ 0 dt t ∆K L (t),(5)
where
∆K L (t) ≡ K L (t) − K L (t).
This expression is UV divergent unless ∆K L (t) vanishe an appropriate choice of background Σ.Specifically, the heat kernel expansion gives [14]
K L (t) = β c 1 V Σ t 3/2 + c 2 χ Σ + c 3 V Σ M 2 t 1/2 + O(t 1/2 ) ,(6)
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
4 Suitably IR regulated if Σ is non-compact.
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 fl t coordinates x A , in terms of which the metrics on Σ and Σ take the form
ds 2 Σ = e 2f (x) δ AB dx A dx B , ds 2 Σ = δ AB dx A dx B . (7)
In order to have good control ove essent rnel), 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 rec 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 ree energy difference with this IR regulator removed:
β∆F ∞ [f ] ≡ q lim L→∞ ∞ 0 dt t ∆K L [f L ; L](t),(8)
with
∆K L [f L ; L](t) ≡ K L [f L ; L](t) − K L [0; L](t).
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
β∆F [f ] = q lim L→∞ ∞ 0 dt t e −M 2 t Θ(T 2 t)∆K O (t),(9)
where the sum over Matsubara frequencies yields
Θ(ζ) = ∞ n=−∞ e −(2π) 2 (n−q+1/2) 2 ζ (10)
and the operators O are given explicitly in terms of f as
O s = −e −2f ∇ 2 + 2ξ 2 4 ,(11b)
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
f = f (1) + 2 f (2) + O( 3 ). (12)
Preservation of the volume requires
d 2 x f (1) = 0, d 2 x f (2) + f (1) 2 = 0. (13)
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
O = −∇ 2 + O (1) + 2 O (2) + O( 3 );(14)
explicit expressions for O O gi ed objects, we likewise expand its eigenvalues and eigenfunctions as
λ i = λi + λ (1) i + 2 λ (2) i + O( 3 ),(15a)h i = hi + j b (1) ij hj + 2 j b (2) ij hj + O( 3 ),(15b)
so the hi are eigenfun alues λi .We choose these eigenfunctions to be n ij .(16)
Then defining
P (1) ij = hi |O (1)
hj ,
P (2) ij lds the eigenvalue shifts
λ (1) i = P (1)
ii , λ
= P (2) ii (no sum). ((2) i)18
Note that we have glossed over a subtlety: recall from QM pertur
is t

all ζ > 0 (a
d all ξ for the scalar), implying that for any (nonconstant) f the free ene e 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 fina s in fact analy llows us to compute a(k) explicitly as 9
a (T =0) s (k) = πk 3 12 128ξ 2 − 8(1 − 16ξ) M 2 k 2 + 48M 4 k 4 arccot 2M k ,(26a)a (T =0) f (k) = πk 3 64 2M k + 24M 3 k 3 + 1 − 8M 2 k 2 − 48M 4 k 4 arccot 2M k . (26b) V. MEMBRANE CRUMPLING
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
X A = x A + v A (x B ), Z = √ h(x A ).
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
−∇ 2 f = det(∂ A ∂ B h).
The bending energy due to extrinsic curvature is
H = κ d 2 x(∇ 2 h) 2 , (27)
where κ is the bending rigidity.If the m ∼ κ, E Q ∼ − 2 N c .(28)
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 ha e different scale dependence, with E Q dominating at sufficiently small scales.Thus if crumple ≡ N c/κ, E Q ca 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 eff ctive membrane description is valid for cell , while from Table I the scaling properties (28) are valid at room temperatu .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, nge 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 requ res 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
{γ µ , γ ν } = 2δ µν ,(29)
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
S E [ ψ, ψ] = d 3 x √ g ψ(i / D − iM )ψ,(30)
where / D = γ µ (e µ ) a D a with {(e µ ) a } for µ = 1, 2, 3 a vielbein,
D a = ∇ a + 1 2 ω aµν S µν ,(31)
∇ 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 c nditions; see e.g.[16] for a discussion of such subtleties associated with spinors in Euclidean space.

Performing the path integral yields
Z = D ψ Dψ e −S E [ ψ,ψ] = det(i / D − iM ).(32)
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 eigen hus the spectrum of i / D on the background ( 1) is symmetric about zero 10 , so we have
Z 2 = det(i / D − iM ) det(−i / D − iM ) = det( / D 2 − M 2 ).
(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
/ D 2 − M 2 = − )/2 are left and right Weyl projectors on Σ and L is as given in (4).Deco 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 las equality we noted that because L is m), 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 .
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(k) ≡ −qT k 4∞dt e −M 2 t Θ(T 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-

Note is not necessarily the same as a difference of separately renormalized free energies on Σ and Σ, which could be scheme-dependent and therefore unphysical.
For finite L one may choose between periodic and antiperiodic boundary conditions for the fermion on the torus cycles; since our torus is only an IR regulator, and this distinction vanishes in the limit L → ∞, we take the periodic case for simplicity.
Finite-L Heat KernelsHere at finite L. First, in order to deal with the issue of eigenfunction degeneracy, it is convenient to take r 2 irrational (so that no eigenvalue λi has degeneracy greater than four) and choose the label i to consist of { N + , S}, whereThe values of S index the degenerate subspaces; for a given N + , there are) possible such values.The eigenvalues of −∇ 2 are then given by N + as λand have degeneracy d N + , while we write the eigenfu

tions aswhere the sum
uns over all d N + possible choices of S and for fixed N + , cis an arbitrary d N + ×d N + unitary matrix.In other words, for given N + the h N + , S form an arbitrary orthonormal basis of the degeneracy subspace of −∇ 2 with eigenvalue λ N + ; the freedom to choose this basis is what allows us to satisfy the perturbation theory constraints on h N + , S .Using this formalism, we may compute the secondorder correction to the heat kernel at finite L using (20b).After some rearrangement, we find10 Note that the direct product structure of (1) was crucial; in a general odd-dimensional geometry the spectrum of i / D need not be symmetric[17].where we defined S N + ≡ {sN − N and 0 otherwise.Note that the precise form of the matrices cdoes not matter sin bspaces in the first terms in the above expressions is an artifact of needing to treat the degenerate subspaces properly.We may now take the limit L → ∞.The first term in each expression above vanishes in this limit (essentially because L −4 N → L −2 d 2 k → 0); for the same reason, the terms containing δλ N , λ N − N also vanish.We then obtain equation (22) withwhere q = | q|, k = | k|, k × q = k 1 q 2 − k 2 q 1 , and P denotes a Cauchy principal value (which comes about since terms in which the denominator vanishes are excluded in the discrete sums).After integration, we obtain (23a) and (23b).
. D Nelson, L Peliti, J. de Physique. 4810851987

D Nelson, T Piran, S We