Contents

October 2, 2019

Deformations of Lifshitz Holography in -dimensions

Miok Park and Robert B. Mann

Department of Physics,

University of Waterloo,

Waterloo, Ontario N2L 3G1,

Perimeter Institute for Theoretical Physics,

31 Caroline Street North,

Waterloo, Ontario N2L 2Y5,

Abstract

We investigate deformations of Lifshitz holography in dimensional spacetime. After discussing the situation for general Lifshitz scaling symmetry parameter , we consider and the associated marginally relevant operators. These operators are dynamically generated by a momentum scale and correspond to slightly deformed Lifshitz spacetimes via a holographic picture. We obtain renormalization group flow at finite temperature from UV Lifshitz to IR AdS, and evaluate how physical quantities such as the free energy density and the energy density depend on in the quantum critical regime as .

## 1 Introduction

One of the most innovative ideas in theoretical physics in recent years is the anti-de Sitter spaceetime/Conformal Field Theory (AdS/CFT) correspondence[1], which posits an isomorphism between symmetries of a spacetime and those of matter fields – for instance, the conformal group of a -dimensional CFT aries as the group of isometries of . This idea has been extended to a duality between gauge theories and gravitational theories in one larger dimension, yielding a new way to understand physics. The most exciting aspect of this picture is to provide a technique for obtaining a weakly coupled and calculable dual description of strongly coupled matter fields in terms of gravity.

Several years prior to the conception of the AdS/CFT correspondence in high energy physics, investigations on phase transitions of modern materials indicated that a precarious point exists between two stable phases of matter such as superconductors and ferroelectrics or ferromagnets in which the temperature of a system has been driven to absolute zero by the application of some external parameter such as pressure or an applied magnetic field. Unlike classical critical points, where critical fluctuations are limited to a narrow region around the phase transition, at such ’quantum critical points’, the critical fluctuations are quantum mechanical in nature and exhibit a generalized scale invariance in both time and space. Understanding this puzzling behavior has become a major research effort in condensed matter physics. Such systems exhibit universally distinct characteristics upon fanning out to finite temperatures, and so the effect of quantum criticality is felt without ever reaching absolute zero. A recent attempt to understand quantum critical theory involves extending the AdS/CFT correspondence [2], [3] to condensed matter systems, yielding a considerably broader range of scope for gauge/gravity duality.

The signature scaling property underlying quantum critical theory in dimensions is

 t→λzt,→x→λ→x (1.1)

where is the dynamical critical exponent; corresponds to conformal invariance, whereas implies an anisotropic scaling invariance. Recently a form of gauge-gravity duality was proposed for [4], in which the geometrical dual is obtained from the (asymptotic) metric

 ds2=l2(−dt2r2z+dr2r2+dx2+dy2r2) (1.2)

which is called Lifshitz spacetime and obviously satisfies

 t→λzt,r→λr,→x→λ→x. (1.3)

When the metric (1.2) is that of (asymptotic) AdS spacetime and (1.1) recovers the conformal symmetry of the CFT. When , (1.1) restores the scaling symmetry of quantum critical theories. The more general anisotropic scaling symmetry, , submerged in the gravity theory and the field theory, is the foundation for a Lifshitz spacetime/Quantum Critical Theory (Lifshitz/QCT) correspondence.

One issue associated with this approach is how to obtain non-trivial spacetimes that asymptote to the anisotropic metric (1.2). Two approaches have been considered to this end. First, it is obvious from the Einstein equations that an anisotropic energy-momentum tensor could support an anisotropic geometry; for example a massive vector field with the appropriate asymptotic behaviour can suffice. An alternate approach involves adding higher curvature terms into the Einstein action [5]; by appropriately tuning the different gravitational constants, metrics asymptotic to (1.2) can be obtained. In this paper, we follow the first approach, investigating the Einstein action coupled to a massive vector field in dimensions.

Related to Lifshitz field theory, an interesting feature attracting much recent attention is associated with renormalization group flow. In Lifshitz theory, the action is

 SLif=12∫dτd2x((∂τϕ)2−κ(∇2ϕ)2), (1.4)

which has an anisotropic scaling invariance. It is known that action describes strongly correlated electron systems; its fixed points seem to flow to a non-Abelian gauge theory by perturbing the action with a term . From the perspective of holographic duality, we expect holographic renormalization flow from a UV-Lifshitz fixed point to an AdS fixed point under the relevant perturbation, a result obtained numerically in [4]. To obtain a metric asymptotic to (1.2), a Proca field is necessary [4]; its essential physics for in (3+1) dimensions is that of a marginally relevant operator in the quantum critical theory, which induces a flow from the theory to a relativistic infrared fixed point. Advancing this study further, the effect of such marginally relevant operators at finite temperature was recently explored, with the renormalization flow for UV Lifshitz to IR AdS described and the physics explored in the quantum critical regime [6].

In this paper, we consider -dimensional Lifshitz spacetime and -dimensional Quantum Critical Theory(QCT), and study their holographic duality. While QCT is well described in a dimensional context, more general theories of physics including the standard model and gravity are implemented in a higher-dimensional context. The success of the AdS/CFT correspondence therefore provides motivation to understand the extent to which the broader notions of Lifshitz/QCT duality are applicable in higher dimensions, and what different behaviour emerges. Motivated by these interests, we especially focus on the marginally relevant operators in the QCT extended to higher dimensions, with the goal of understanding their behaviour from the perspective of holographic duality, where these operators correspond to the deformed Lifshitz spacetime solutions.

Previous work in this subject has concentrated on the (2+1)-dimensional case [6]. Here we demonstrate that renormalization group flow from Lifshitz spacetime in the UV to AdS spacetime in the IR generalizes to any dimensionality in the marginally relevant case yielding deformations of the pure Lifshitz spacetime. From a thermodynamic perspective, we find that physical quantities such as the ratios (entropy density to temperature), (free energy density to ), and (energy density over ) exhibit progressively weaker dependence on temperature at sub-leading order in as dimensionality increases. We also find that the maximal flux of the vector field near the horizon grows linearly with increasing dimension.

In section 2, the action, equations of motion, and basic setup are introduced, along with an ansatz for which all constants are fine-tuned and normalized for both Lifshitz and AdS spacetime. In section 3, asymptotic solutions consistent with a marginally relevant operator are derived by bringing in a dynamically generated momentum scale (assumed very small), which deforms Lifshitz spacetime in the high energy regime. In section 4, we carry out holographic renormalization, rendering the action finite by constructing proper counterterms. In section 5, we numerically match our asymptotic near-Lifshitz solutions with black hole solutions near the horizon. We then describe the renormalization group flow, and compute physical quantities such as the entropy density , the free energy density , and the energy density for and .

## 2 Einstein Gravity with a Massive Vector fields in (n+1) dimensional spacetime

The action for gravity in -dimensional spacetime coupled to a massive vector field is described by

 S=∫dn+1x√−g(12κn+12[R+2~Λ]−1gv2[14H2+γ2B2]) (2.1)

where in which is the dimensional gravitational constant, and and is the (n+1) dimensional coupling constant of the vector field. The equations of the motion are

 1κn+12(Rμν−12gμνR−~Λgμν)=1gv2(HμρHρν−14gμνH2)+γgv2(BμBν−12gμνB2), (2.2)

and

 ∇μHμν−γBν=0 (2.3)

where is the squared mass of the vector field. For the action to yield solutions asymptotic to those having the scaling symmetry (1.3), we require the spacetime metric

 ds2=l2(−dt2r2z+dr2r2+dx2+dy2+⋯r2) (2.4)

to be a solution to the field equations, where is arbitrary. Note that in these coordinates corresponds to the boundary of the spacetime.

The vector potential yielding a stress-energy supporting this metric is given by

 B=gvlκn+1qrzdt. (2.5)

These ansatz and boundary conditions fine-tune the cosmological constant to be

 ~Λ=(z−1)2+n(z−2)+n22l2, (2.6)

and the squared mass and the charge of the vector field to be

 γ=(n−1)zl2,q2=z−1z. (2.7)

Regardless of the dimensionality of the spacetime, setting in (2.4) yields solution

where the vector potential vanishes. As the cosmological constant has been already fixed due to the Lifshitz boundary condition we introduce a scaling constant, , into the AdS metric and adjust its value to be

 a=n(n−1)(z−1)2+n(z−2)+n2. (2.9)

Once we fix the cosmological constant (2.6) with space dimension and dynamical critical exponent , then those values determine the scaling constant for the AdS spacetime metric.

In order to describe the renormalization group flow which involves breaking the anisotropy of the spacetime by running from the UV Lifshitz to the IR AdS, we employ the ansatz

 ds2 = l2(−f(r)dt2+dr2r2+p(r)(dx2+dy2+⋯)), (2.10) B = gvlκn+1h(r)dt (2.11)

so for the Lifshitz spacetime

 Lifshitz : f=1r2zp=1r2,h=√z−1√z1rz, (2.12)

whereas for the spacetime

With (2.10) and (2.11), the equations of motion yield three independent non-linear ODEs for

 2χ+z(4n−6)h(r)2f(r)−rf′(r)f(r)+r2f′(r)22f(r)2−(3n−5)r2f′(r)p′(r)2f(r)p(r)−(n−2)2r2p′(r)22p(r)2−r2f′′(r)f(r)=0, −2zh(r)2f(r)−rp′(r)p(r)+r2f′(r)p′(r)2f(r)p(r)+r2p′(r)22p(r)2−r2p′′(r)p(r)=0, χ+(n−1)zh(r)2f(r)−r2h′(r)2f(r)−(n−1)r2f′(r)p′(r)2f(r)p(r)−(n−2)(n−1)r2p′(r)24p(r)2=0, (2.14)

where . We shall rewrite the equations of the motion with the new variables

 p(r)=e∫rq(s)sds,f(r)=e∫rm(s)sds,h(r)=k(r)√f(r). (2.15)

These variables have the added benefit of turning the second order differential equations into first order and postponing the determination of rescaling ambiguities on , , and .

For further simplification, we introduce a new variable

 x(r)=(4χ+4(n−1)zk(r)2−2(n−1)m(r)q(r)−(n−2)(n−1)q(r)2)12. (2.16)

Putting (2.15) and (2.16) into (2) gives

 rx′(r) = −2(n−1)zk(r)−(n−1)2q(r)x(r), rq′(r) = χ(n−1)−zk(r)2−n4q(r)2−14(n−1)x(r)2, rk′(r) = −χ(n−1)k(r)q(r)−zk(r)3q(r)+(n−2)4k(r)q(r)−x(r)2+14(n−1)k(r)x(r)2q(r). (2.17)

In terms of the new variables , , and , Lifshitz spacetime is described by

 Lifshitz : q=−2x=2√z−1√z,k=√z−1√z, (2.18)

So far we have worked with a general value of in dimensions. We are interested in studying the effects of marginal operators, which have scaling dimension in Lifshitz spacetime, because of the different scaling of the time coordinate. While it has been shown that the linearized equation of motion for the scalar part of constant perturbations in a Lifshitz background in (3+1) dimensionals [7] (and the gravitational field has solutions that are marginal for general ), the vector field only admits a single degenerate solution at the special value of , where the vector operator also becomes marginal. Applying this analysis to dimensions [11], the condition for having a single degenerate solution for the vector field is , and the operators with this value are considered to be marginal. Henceforth we deal with the case satisfying .

## 3 Asymptotic Behaviour

We consider the spacetime slightly thermally heated and so slightly deformed from the pure Lifshitz case, restricting our considerations to for which the massive vector field becomes marginal. Under these assumptions, the general form of the solutions near the boundary is

 k(r) =√z−1√z(1+1(z−1)2log(rΛ)+(z−1)(−3z+2(z−1)3λ)+2(1−3z)log(−log(rΛ))2z(z−1)4log2(rΛ)+⋯) +(rΛ)2zlog2(rΛ)(β(1+2(3z−1)log(−log(rΛ))z(z−1)2log(rΛ)+⋯)+α(1log(rΛ)+ (2z2−4z+1)−2(z−1)4(2z−1)λ+2(6z2−5z+1)log(−log(rΛ))2z(z−1)2(2z−1)log(rΛ)+⋯)), (3.1) q(r) =−2(1−1(z−1)log(rΛ)−z+2(z−1)4λ−2(3z−1)log(−log(rΛ))2z(z−1)3log2(rΛ)+⋯) −2√z−1√z2z−1(rΛ)2zlog2(rΛ)(β(1+−z(4z2−7z+2)+2(2z−1)(3z−1)log(−log(rΛ))z(z−1)2(2z−1)log(rΛ)+⋯) +α(1log(rΛ)−(2z2−4z+1)+2(z−1)4λ−2(3z−1)log(−log(rΛ))2z(z−1)2log2(rΛ)+⋯)), (3.2) x(r) =2√z−1√z(1+z(z−1)2log(rΛ)+(z−1)4λ+(1−3z)log(−log(rΛ))(z−1)4log2(rΛ)+⋯) −2z22z−1(rΛ)2zlog2(rΛ)(β(1+−z(4z2−5z+1)+2(6z2−5z+1)log(−log(rΛ))z(z−1)2(2z−1)log(rΛ)+⋯) +α(1log(rΛ)−(2z−1)2+2(z−1)4λ−2(3z−1)log(−log(rΛ))2z(z−1)2log2(rΛ)+⋯)), (3.3)

where is a momentum scale, generating a marginally relevant mode, whereas is an energy scale with the spatial dimension. As the solution recovers the pure Lifshitz spacetime. The parameters and describe other modes of the solution, and is nothing but a ’gauge choice’ [6]. In other words is related to defining the scale , and the solution transforms as

 F(Λr;α,β;λ)=F(eλ′/zΛr;e−2λ′(α−λ′β),e−2λ′β;λ+λ′) (3.4)

where stands for the and functions. This is easily verified by noting that the solutions and with can be obtained by setting , and replacing , , and with , , and respectively, and then re-expanding the solutions under the assumption . Here we fix this ambiguity by setting .

As we are interested in the high energy regime, we expand by introducing an arbitrary scale and write

 log(rΛ)=log(rμ)−logμΛ. (3.5)

In the high energy regime where we have

 ∣∣∣1logμΛ∣∣∣,∣∣∣log(rμ)logμΛ∣∣∣⩽1. (3.6)

Upon expansion, equations (3.1) (3.3) become

 k(r)=√z−1√z(1+1(z−1)2log(μΛ)−3z(z−1)+2z(z−1)2log(rμ)+2(3z−1)log(−log(μΛ))2z(z−1)4log2(μΛ)+⋯), (3.7) q(r)=−2(1−1(z−1)log(μΛ)+−z+2z(z−1)2log(rμ)+2(3z−1)log(−log(μΛ))2z(z−1)3log2(μΛ)+⋯), (3.8) x(r)=2√z−1√z(1+z(z−1)2log(μΛ)−z(z−1)2log(rμ)+(3z−1)log(−log(μΛ))(z−1)4log2(μΛ)+⋯). (3.9)

Using these solutions for and , we employ the change of variables (2.15) and (2.16) in reverse to obtain the original form of the solutions

 f(ρ) =F20(rΛ)2z(−log(rΛ))2zz−1(1−(7z−4)+2(3z−1)log(−log(rΛ))(z−1)3log(rΛ)−(23z4−142z3+152z2−57z+6)4z(z−1)6log2(rΛ) +(3z−1)2(5z−2)log(−log(rΛ))+(3z−1)3log2(−log(rΛ))z(z−1)6log2(rΛ)+⋯) (3.10) p(ρ) =P20(−log(rΛ))2z−1(rΛ)2(1+(5z−2)+2(3z−1)log(−log(rΛ))z(z−1)3log(rΛ)+(31z4−64z3+106z2−69z+14)4z2(z−1)6log2(rΛ) +(3z3+26z2−21z+4)log(−log(rΛ))+(3z−1)2(z−3)log2(−log(rΛ))z2(z−1)6log2(rΛ)+⋯) (3.11)

where and are constants. Furthermore, in the high energy regime the same expansion for eq. (3.10) –(3.11) yields

 f(ρ)=1r2z(1+7z−4+2z(z−1)2log(rμ)+2(3z−1)log(log(μΛ))(z−1)3log(μΛ)+⋯), (3.12) p(ρ)=1r2(1−5z−2+2z(z−1)2log(rμ)+2(3z−1)log(log(μΛ))z(z−1)3log(μΛ)+⋯) (3.13)

upon rescaling the and coordinates to

 t→(Λlog1z−1(μΛ))zF0t,x→(Λlogz−1(μΛ))21P0x. (3.14)

## 4 Holographic Renormalization

In this section we investigate thermodynamic quantities such as free energy density or energy density at an asymptotic boundary of the deformed Lifshitz spacetime. We begin with the definition of the free energy density

 F=−TlogZ=TSϵ(g∗) (4.1)

where and are respectively the Euclidean action and the metric, and is the partition function. Upon carrying out a variation of the on-shell action, boundary terms arise, and to cancel these out a Gibbons-Hawking boundary term is added into the action. After Euclideanization, the action and the metric can be explicitly written as

 Sϵ =∫dn+1x√g(12κn+12[R+2~Λ]−1g2v[14H2+γ2B2])+1κn+12∫dnx√γK, (4.2) dsϵ2 =l2(f(r)dτ2+dr2r2+p(r)(dx2+dy2+⋯)), (4.3)

where indicates the Euclidean version of the quantities.

Calculating the free energy density (the free energy per unit -dimensional spatial volume), the Einstein-Hilbert action and Gibbon-Hawking term yield

 FEH=−ln−12κn+12limr→0r√f(r)p′(r)p(r)n−32, (4.4) (4.5)

where is the induced metric on the boundary and is the extrinsic curvature defined as in which is the normal vector on the boundary surface. The free energy is . However for the marginally relevant modes both (4.4) and (4.5) are divergent as the boundary () is approached. We incorporate boundary counterterms [7, 8, 9, 10] into the action as a remedy for this problem. We construct these counterterms as a power series in [6], so as to satisfy covariance at the boundary, obtaining

 FC.T. = 12lκn+12limr→0√γ2∑j=0Cj(−κn+12g2vB2−(z−1)z)j (4.6) = (4.7)

where we have used instead of , since these must vanish for the pure Lifshitz case. The coefficients are not constants but rather a series of the logarithmic functions, with at least three needed to eliminate divergences.

The final expression for the free energy density is

 F = FEH+FGH+FC.T. (4.8) = ln−12κn+12limr→0√f(r)p(r)n−12((n−2)rp′(r)p(r)+rf′(r)f(r)+2∑j=0Cj(k(r)2−(z−1)z)j).

To obtain the energy density, we use the definition of the boundary stress tensor to the case in which additional non-vanishing boundary fields are present [12]. From the boundary stress tensor, we obtain the charge via variation of the on-shell action with respect to the boundary fields; this process in our case produces

 δS=√γ2τabδγab+JaδBa. (4.9)

Here, however we are dealing not with scalar matter fields but with massive vector fields, and so the usual charge defined by

 Q=−∫dn−2x√σξakbτab (4.10)

where is the spatial volume element, is a boundary Killing fields, and is the unit normal vector to the boundary Cauchy surface, is not conserved. The boundary stress tensor must therefore be redefined so as to fix the matter fields in the boundary. Employing the vielbein frame defined by

 γab=η^a^be^aae^bb,η=diag(±1,1,1,⋯) (4.11)

where

We find that the variation of the free energy density retains its original form, but that is replaced with , where

 δS=√γTa^aδe^aa+J^aδB^a, (4.13)

with

 (4.14)

The energy density is then given by

 E=√γτtt+JtBt (4.15)

and the pressure is

 P=−√γτxx. (4.16)

Computing the distinct components of , we find

 τab =2√γδSδγab=1κn+12(Kγab−Kab) (4.17)

and

 J^t = √f(r)δSδBt, (4.18) = ln−2gvκn+1limr→0√f(r)p(r)n−12(r(k(r)√f(r))′√f(r)+k(r)2∑j=0jCj(k(r)2−(z−1)z)j−1), = ln−2gvκn+1limr→0√f(r)p(r)n−12(−12x(r)+k(r)2∑j=0jCj(k(r)2−(z−1)z)j−1),

and other component of become zero. Putting these together into (4.15) yields

 E=ln−12κn+12limr→0√f(r)p(r)n−12((n−1)rp′(r)p(r)−x(r)k(r)+2∑j=0Cj(k(r)2−(z−1)z)j) (4.19)

and

 P=−F. (4.20)

Imposing finiteness of physical quantities of (4.8), (4.18), and (4.19), the coefficients of the counter terms are found to be

 C0= 2(2z−1)−2z2(2z−1)(z−1)3log2(rΛ)+(4z4+2z3−3z2−2z+1)(z−1)5(2z−1)2log3(rΛ) +4z(3z−1)log(−log(rΛ))(z−1)5(2z−1)log3(rΛ)+⋯, (4.21) C1= z+2z3(2z−1)(z−1)2log(rΛ)−z(14z3−z2−10z+3)+4z2(2z−1)(3z−1)log(−log(rΛ))2(2z−1)2(z−1)4log2(rΛ) −1log3(rΛ)(z2(34z4−54z3+72z2−46z+9)+8a(2z−1)2(z−1)52z(2z−1)2(z−1)6 −(6z4+25z3−45z2+21z−3)log(−log(rΛ))(2z−1)2(z−1)6−2z(3z−1)2log2(−log(rΛ))(z−1)6(2z−1))+⋯, (4.22) C2= z2(1−3z)4(2z−1)(z−1)+z2(15z2−14z+3)4(2z−1)2(z−1)3log(rΛ)+alog2(rΛ)−z(3z−1)2(5z−3)log(−log(rΛ))4(2z−1)2(z−1)5log2(rΛ) (4.23)

where the first two are infinite series in that include powers of such that the order of the terms do not exceed the order of the terms. It is sufficient for to retain terms up to second order in . Note that there exists an ambiguity in these expressions. This ambiguity does not affect numerical evaluation of the free energy density and the energy density that we shall later compute, though it does affect , reflecting the reaction of the system to changes in the boundary Proca field. Our counter term construction (4) – (4.23) is minimal; additional terms such as or would also yield solutions.

Applying (4) – (4.23) into (4.8), (4.18) and (4.19), the physical quantities become

 F=ln−1κn+12√z√z−1(2z−1)(zα−(2z3−2z2−2z+1)(2z−1)(z−1)4β), (4.24) E=−ln−1κn+12√z√z−1(2z−1)(zα+(2z3−4z2+4z−1)(2z−1)(z−1)4β), (4.25) J^t=1gvln−2κn+1(z(20z5+18z4−22z3−23z2+24z−5)2(z−1)4(2z−1)3+4(z−1)az)β. (4.26)

Since the pure Lifshitz solution does not depend on