Holographic models with anisotropic scaling
aa r X i v : . [ h e p - t h ] A p r NORDITA-2010-29, UUITP-15/10
Holographic models with anisotropic scaling
E J Brynjolfsson , U H Danielsson , L Thorlacius , and T Zingg , University of Iceland, Science Institute, Dunhaga 3, IS-107 Reykjavik, Iceland Institutionen f¨or fysik och astronomi, Uppsala Universitet, Box 803, SE-751 08 Uppsala,Sweden NORDITA, Roslagstullsbacken 23, SE-106 91 Stockholm, SwedenE-mail: [email protected] [email protected] [email protected]@nordita.org
Abstract.
We consider gravity duals to d+1 dimensional quantum critical points withanisotropic scaling. The primary motivation comes from strongly correlated electron systems incondensed matter theory but the main focus of the present paper is on the gravity modelsin their own right. Physics at finite temperature and fixed charge density is described interms of charged black branes. Some exact solutions are known and can be used to obtaina maximally extended spacetime geometry, which has a null curvature singularity inside asingle non-degenerate horizon, but generic black brane solutions in the model can only beobtained numerically. Charged matter gives rise to black branes with hair that are dual to thesuperconducting phase of a holographic superconductor. Our numerical results indicate thatholographic superconductors with anisotropic scaling have vanishing zero temperature entropywhen the back reaction of the hair on the brane geometry is taken into account.
1. Introduction
We consider d + 2 dimensional gravity models that are dual to quantum critical points withanisotropic scaling in d + 1 dimensions, t → λ z t, ~x → λ~x , (1)with z ≥ ~x = ( x , . . . , x d ). The d + 2 dimensional bulk action consists of three parts, S = S Einstein-Maxwell + S Lifshitz + S matter . (2)The first term is the standard action of Einstein-Maxwell gravity with a negative cosmologicalconstant, S Einstein-Maxwell = Z d d +2 x √− g (cid:18) R − − F µν F µν (cid:19) . (3)This is followed by a term involving a massive vector field, S Lifshitz = − Z d d +2 x √− g F µν F µν + c A µ A µ ! , (4)whose sole purpose is to provide backgrounds with anisotropic scaling. This is a modified versionof the holographic model of [1], which was formulated in four-dimensional spacetime and obtainednisotropic scaling by including a pair of coupled two- and three-form field strengths. Uponintegrating out the three-form field strength, the remaining two-form becomes a field strength ofa massive vector [3] and in this form the model is easily extended to general dimensions. Finitetemperature corresponds to having a black hole in the higher-dimensional spacetime and withthe Maxwell gauge field, added in [4], the black hole can carry electric charge, which is dual toa finite charge density in the lower-dimensional theory. Finally, we consider matter in the formof a scalar field, which is charged under the Maxwell gauge field but does not couple directly tothe auxiliary massive vector field, S matter = − Z d d +2 x √− g (cid:16) ( ∂ µ φ ∗ + iqA µ φ ∗ ) ( ∂ µ φ − iqA µ φ ) + m φ ∗ φ (cid:17) . (5)At low temperature there is an instability for charged black holes in the model to grow scalarhair, which corresponds to the superconducting phase of holographic superconductors with z > ∇ µ − iqA µ ) ( ∇ µ − iqA µ ) φ − m φ = 0 , (6)along with the Einstein equations, the Maxwell equations, and field equations for the auxiliarymassive vector, G µν + Λ g µν = T Maxwell µν + T Lifshitz µν + T matter µν , (7) ∇ ν F νµ = j µ matter , (8) ∇ ν F νµ = c A µ . (9)The asymmetric scaling symmetry (1), sometimes referred to as Lifshitz scaling, is realized in ad+2 dimensional spacetime, ds = L − r z dt + r d~x + dr r ! , (10)whose metric is invariant under the transformation t → λ z t, ~x → λ~x, r → rλ . (11)Length dimensions are carried by the characteristic length L while the coordinates ( t, r, ~x ) aredimensionless. The scale invariant Lifshitz geometry (10) is a solution of the equations of motionwhen L is related to the cosmological constant Λ viaΛ = − z + ( d − z + d L , (12)and the mass of the auxiliary vector field is fine-tuned to c = √ z d/L . In this solution theMaxwell field vanishes, A µ = 0, but the massive vector field has the background value A t = s z − z L r z , A x i = A r = 0 . (13) See also [2] for early work on gravitational backgrounds with anisotropic scaling. . Charged black branes
In order to study finite temperature effects in the dual strongly coupled field theory we lookfor static black brane solutions of the equations of motion (6) - (9) which are asymptotic to theLifshitz fixed point solution given by (10) and (13). From now on we set L = 1 and consider ametric of the form ds = − r z f ( r ) dt + r d~x + g ( r ) r dr . (14)An asymptotically Lifshitz black brane with a non-degenerate horizon has a simple zero of both f ( r ) and g ( r ) − at the horizon, which we take to be at r = r , and f ( r ) , g ( r ) → r → ∞ .It is straightforward to generalize this ansatz to include black holes with a spherical horizon ortopological black holes with a hyperbolic horizon but it is the flat horizon case (14) that is ofdirect interest for the gravitational dual description of strongly coupled d + 1 dimensional fieldtheories. In a static electrically charged black brane background the Maxwell gauge field andthe massive vector can be taken to be of the form A µ = r z f ( r ) { α ( r ) , , . . . , } ; A µ = s z − z r z f ( r ) { a ( r ) , , . . . , } . (15)with α ( r ) → a ( r ) → r → ∞ . For static configurations the equations of motion can be expressed as a first order system ofordinary differential equations. Introducing a scale invariant radial variable u = log( r/r ) andwriting ˙ ≡ ddu , the scalar field equation becomes˙ φ = χ , (16)˙ χ + ˙ ff − ˙ gg ! χ = − ( d + z ) χ + (cid:16) m − q α (cid:17) g φ , (17)while the Maxwell equations and the equations of motion for the massive vector reduce to,˙ α + ˙ ff α = − z α + g β , (18)˙ β = − d β + q g φ α , (19)˙ a + ˙ ff a = − z a + z g b , (20)˙ b = − d b + d g a . (21)The functions χ , β , and b are defined via (16), (18), and (20), respectively. The Einsteinequations can also be written in first order form,˙ gg + ˙ ff = ( z − (cid:16) g a − (cid:17) + 12 d (cid:16) χ + q g α φ (cid:17) , (22)2 d ˙ ff = d (1 − d − z ) + χ g " ( z − (cid:16) da − zb (cid:17) − β
2+ 12 (cid:16) q α − m (cid:17) φ + z + ( d − z + d (cid:21) . (23)The field equations (16) - (23) are manifestly invariant under the scaling (11) and the Lifshitzfixed point solution is given by f = g = a = b = 1 and α = β = φ = χ = 0.n the absence of charged matter, the Maxwell equation (19) integrates to β ( u ) = ˜ ρ e − du .The integration constant ˜ ρ = ρ/r d is proportional to the electric charge per unit d -volume of theblack brane, which in turn corresponds to the charge density in the dual field theoretic system.For given d and z ≥
1, the remaining field equations then have a one parameter family of blackbrane solutions, labelled by ˜ ρ . A neutral black brane without scalar hair has ˜ ρ = 0 while theextremal limit is given by ˜ ρ → ± p z + ( d − z + d ). A non-vanishing charged scalar fieldchanges this picture as discussed below.Equation (18) can easily be solved in the Lifshitz geometry (10) without matter, giving α ( u ) = ( ˜ µ e − zu + z − d ) ˜ ρ e − du if z = d , ˜ µ e − du + ˜ ρ u e − du if z = d , (24)where the integration constant ˜ µ = µ/r z corresponds to having non-vanishing chemical potentialin the dual system. In general, we are not working with the Lifshitz background but withsolutions that are only asymptotically Lifshitz as u → ∞ . However, as long as the value of z isn’t too high, the asymptotic behavior of the gauge potential carries over from the Lifshitzbackground to the more general case, and one can read off the charge density and chemicalpotential in the dual field theory from the leading two terms in the expansion of α at large u .Calculations in this paper refer to fixed ρ , corresponding to a fixed density of charge carriers inthe dual system, but one can also work at fixed chemical potential. Black brane solutions at generic z > f ( u ) = √ u ( f + f u + . . . ) , g ( u ) = 1 √ u ( g + g u + . . . ) , (25)near u = 0. The appropriate near-horizon behavior of the remaining field variables can beworked out and then inserted into the equations of motion to generate initial value data for thenumerical integration. For any given values of d and z , we obtain a three parameter family ofinitial values, where for instance φ (0), β (0), and b (0) can be taken as the independent parameters.The condition that the metric and massive vector approach the Lifshitz fixed point solution (10)and (13) sufficiently rapidly as u → ∞ restricts the solutions further [5, 6, 7]. As a result, b (0) is fixed for given φ (0) and β (0) and one has a two-parameter family of solutions. Thismeans in particular that, in the absence of scalar hair, there is a unique (up to overall scale)asymptotically Lifshitz charged black brane solution for given d , z and ˜ ρ .The Hawking temperature is determined by the near-horizon behavior of the black branemetric, T H = r z π f g . (26)The full numerical solution of the field equations is required, however, to relate the coefficients f and g to the charge density ρ and other physical variables of the dual field theory. At z ≥ d non-linear effects give rise to additional terms in (24) with a falloff in between that of the chargedensity and chemical potential terms. .3. Conserved charge under radial evolution A conserved charge under radial evolution was found in [6] for electrically neutral black braneswith d = 2 and arbitrary dynamical critical exponent z . Such a conserved charge is useful formatching solutions across the bulk geometry from the near-horizon region to the asymptotic large u region and also provides a check on numerical solutions. The charge found in [6] generalizesto charged black branes with scalar hair in general spatial dimensions, D = r z + d e ( z + d ) u f (cid:20) g ( χ − d ( d +1)) − d ( z − ab − d αβ + g (cid:20) ( z − d a − z b ) + z +( d − z + d −
12 ( β + m φ − q α φ ) (cid:21)(cid:21) . (27)It is straightforward to check that ddu D = 0 when the field equations (16) - (23) are satisfied.The conserved charge is related to thermodynamic state variables of the dual system in a simpleway. Inserting a perturbative near-horizon expansion of the fields, one finds D = 2 r d + z f g = 32 πS T, (28)where T is the temperature (26) and S = r d / As always, it is useful to have explicit analytic solutions to work with. Although Lifshitz blackbranes at z > d an isolated z = 2 d exact solution can be found [4, 8], b = 1 , f = 1 g = a = 1 − e − du , f α = ±√ e − du (1 − e − du ) , (29)with φ = 0 and ˜ ρ = ±√ d . It is straightforward to continue the exact black brane metric insidethe horizon and obtain the globally extended geometry [4]. Define a tortoise coordinate u ∗ by u ∗ = 12 d r d log (cid:16) − e − du (cid:17) , (30)and then transform the ( t, u ) variables to a pair of null coordinates V = exp h d r d ( u ∗ + t ) i , U = − exp h d r d ( u ∗ − t ) i . (31)In the new coordinate system the metric is given by ds = − dU dVd (1 + U V ) + r d~x (1 + U V ) /d , (32)and is manifestly non-singular at the horizon, which is located at U V = 0. There is a nullcurvature singularity at
U V → ∞ , which corresponds to r → r → ∞ corresponds to U V → −
1. By a further transformation The corresponding exact solutions for d = 2, z = 4 black holes with a spherical horizon and topological blackholes with a hyperbolic horizon were also found in [4]. In the limit of vanishing electric charge these black holesolutions reduce to the previously discovered z = 4 black hole solution of [6]. Sfrag replacements r = r = ∞ r = r Figure 1.
Global geometry of an asymptoticallyLifshitz black hole. The null singularity at r = 0is indicated by the jagged lines.to new null variables P, Q , defined through V = tan πP , U = tan πQ , the global geometry canbe represented by a simple diagram shown in Figure 1. Each point in the diagram representsan entire d -volume parametrized by ~x . The global diagram in Figure 1 differs from standardCarter-Penrose conformal diagrams in that the boundary at r → ∞ is not conformally flat.This is a consequence of the scaling asymmetry between t and ~x and is readily apparent in theglobally extended metric (32).When z = 1 the auxiliary massive vector field A µ can be consistently set to zero and thefield equations then reduce to those of Einstein-Maxwell gravity with a negative cosmologicalconstant. In this case, there is a well known exact solution, the AdS-Reissner-Nordstrm blackbrane, which has a timelike curvature singularity inside an inner and an outer horizon. Theinterior geometry of the exact z = 2 d black brane is markedly different with a null curvaturesingularity at r = 0 and no smooth inner horizon.
3. Holographic superconductors with asymmetric scaling
We now consider charged black branes with hair. Static spherically symmetric solutions of thescalar field equation (6) have the asymptotic form φ ( u ) → c − ( e − ∆ − u + . . . ) + c + ( e − ∆ + u + . . . ),with ∆ ± = d + z ± s(cid:18) d + z (cid:19) + m , (33)while the asymptotic behavior of the electromagnetic field field strength is β ( u ) ≈ ρr d e − du + . . . . (34)Working at fixed charge density in the dual field theory, we read the radial location of the horizonoff from the asymptotic behavior of β ( u ) and then the temperature can be obtained from thenumerical solution for f ( u ) and g ( u ) using (26).In the following we set the scalar mass squared to m = − (cid:16) d + z (cid:17) , which is inside therange where there is a choice of two boundary theories [9]. This choice leads to convenientvalues, ∆ ± = d + z ± , for the dimensions of the operators O ± that are dual to the scalar fieldin each of the two boundary theories. Non-linear descendants of the leading scalar field modesare suppressed by O ( e − u ) at u → ∞ and this choice of mass squared ensures that the firstdescendant of ψ − falls off faster than ψ + .In order to study holographic superconductivity, we first select some value for d , z and theelectric charge q carried by the scalar field and then generate numerical black brane solutions for range of initial values β (0) and φ (0). We then investigate the asymptotic large u behavior ofthe scalar hair in the numerical solutions. A superconducting condensate corresponds to either c + = 0 , hO − i = c − = 0 , or c − = 0 , hO + i = c + = 0 , (35)depending on which of the two boundary theories is being considered [10, 11]. We look for acurve in the β (0) vs. ψ (0) plane of initial values at the horizon, for which the correspondingblack brane solution has vanishing c + ( c − ), and tabulate the value of c − ( c + ) along this curve.The temperature is found from the same numerical solutions via (26) and (34). Figure 2 showsa plot of c − vs. T obtained by this procedure for d = 2, z = 2, and q = 1. The results areexpressed in terms of dimensionless ratios that are insensitive to the overall scale (set by thecharge density ρ , which is held fixed at some finite value throughout).These results demonstrate that a superconducting condensate can form in systems withanisotropic scaling but we have not touched on a number of interesting topics including theelectric conductivity and magnetic properties of these holographic superconductors. (cid:144) T c - (cid:144) T c D - (cid:144) PSfrag replacements r = 0 r = ∞ r = r Figure 2.
Scalar field condensate in aholographic superconductor at z = d = 2. (cid:144) T c (cid:144) S c PSfrag replacements r = 0 r = ∞ r = r Figure 3.
Entropy as a function oftemperature in the superconducting phase ofthe holographic superconductor in Figure 2.
4. Zero temperature entropy vs. scalar hair
In the absence of charged matter, the charged black branes in our model have an extremal limitgiven by ˜ ρ → ± p z + ( d − z + d ). It then follows from ˜ ρ = ρ/r d that the radial location ofthe horizon r has a finite value in the extremal limit for fixed charge density ρ . This in turnmeans that the entropy density S = r d / d = 2, z = 2 holographicsuperconductor as was considered in the previous section is shown in Figure 3. Both the entropydensity and the temperature are normalized to their values at the onset of condensation, S c and T c respectively.Although the numerical calculations break down before absolute zero is reached, the numericaldata strongly suggest that S vanishes in the T → z , which is consistent with a non-degenerate ground state in the dual field theory. Thecorresponding result for conformal systems with z = 1 was established in [12]. . Summary We have presented an overview of the construction of charged black brane solutions ingravity models that realize the anisotropic scaling symmetry that is characteristic of manyinteresting quantum critical points. The motivation for the study of these models comesfrom condensed matter theory, in particular from two- and three-dimensional systems involvingstrongly correlated electrons. The relevance of gravitational models to real world condensedmatter systems remains highly speculative, but the gravitational approach continues to produceeffects that are intriguingly similar to what is seen in experiments. A recent example from ourown work [13] involves non-Fermi-liquid behavior in the specific heat of anisotropic black branesof the type considered in the present paper, which turns out to be qualitatively similar to themeasured specific in certain heavy fermion metals near a quantum phase transition [14, 15].While much of the work on gravitational modeling of strongly coupled field theories to dateinvolves asymptotically AdS spacetime and an underlying conformal symmetry, it was crucialto the success of this particular application to have a non-trivial dynamical critical exponent z >
1. This provides impetus for further study of gravity models with anisotropic scaling.
Acknowledgments
This work was supported in part by the G¨oran Gustafsson foundation, the Swedish ResearchCouncil (VR), the Icelandic Research Fund, the University of Iceland Research Fund, and theEimskip Research Fund at the University of Iceland.
References [1] Kachru S, Liu X, and Mulligan M 2008 Gravity duals of Lifshitz-like fixed points
Phys. Rev. D Preprint arXiv:0808.1725 [hep-th])[2] Koroteev P and Libanov M 2008 On existence of self-tuning solutions in static braneworlds withoutsingularities
J. High Energy Phys.
JHEP02(2008)104 (
Preprint arXiv:0712.1136 [hep-th])[3] Taylor M 2008 Non-relativistic holography (
Preprint arXiv:0812.0530 [hep-th])[4] Brynjolfsson E J, Danielsson U H, Thorlacius L and Zingg T 2010 Holographic superconductors with Lifshitzscaling
J. Phys. A: Math. Theor. Preprint arXiv:0908.2611 [hep-th])[5] Danielsson U H and Thorlacius L 2009 Black holes in asymptotically Lifshitz spacetime,”
J. High EnergyPhys.
JHEP03(2008)070 (
Preprint arXiv:0812.5088 [hep-th])[6] Bertoldi G, Burrington B A and Peet A (2009) Black holes in asymptotically Lifshitz spacetimes witharbitrary critical exponent
Phys. Rev. D Preprint arXiv:0905.3183 [hep-th])[7] Ross S F and Saremi O (2009) Holographic stress tensor for non-relativistic theories
J. High Energy Phys.
JHEP09(2009)009 (
Preprint arXiv:0907.1846 [hep-th])[8] Pang D W 2010 On charged Lifshitz black holes
J. High Energy Phys.
JHEP10(2010)116 (
Preprint arXiv:0911.2777 [hep-th])[9] Klebanov I R and Witten E 1999 AdS/CFT correspondence and symmetry breaking
Nucl. Phys. B
Preprint hep-th/9905104)[10] Gubser S S 2008 Breaking an Abelian gauge symmetry near a black hole horizon
Phys. Rev. D Preprint arXiv:0801.2977 [hep-th])[11] Hartnoll S A, Herzog C P and Horowitz G T 2008 Building a Holographic Superconductor
Phys. Rev. Lett.
Preprint arXiv:0803.3295 [hep-th])[12] Horowitz G T and Roberts M M 2009 Zero temperature limit of holographic superconductors
J. High EnergyPhys.
JHEP11(2009)015 (
Preprint arXiv:0908.3677 [hep-th])[13] Brynjolfsson E J, Danielsson U H, Thorlacius L and Zingg T 2010 Black hole thermodynamics and heavyfermion metals (
Preprint arXiv:1003.5361 [hep-th])[14] L¨ohneysen H v, Pietrus T, Portisch G, Schlager H G, Schr¨oder A, Sieck M, and Trappmann T 1994 Non-Fermiliquid behavior in a heavy-fermion alloy at a magnetic instability
Phys. Rev. Lett. Rev. Mod. Phys.
797 (Addendum-ibid. 200678