p-wave Holographic Superconductors and five-dimensional gauged Supergravity
aa r X i v : . [ h e p - t h ] D ec ICCUB-10-104November, 2010 p -wave Holographic Superconductors andfive-dimensional gauged Supergravity Francesco Aprile , Diego Rodriguez-Gomez and Jorge G. Russo
1) Institute of Cosmos Sciences and Estructura i Constituents de la MateriaFacultat de F´ısica, Universitat de BarcelonaAv. Diagonal 647, 08028 Barcelona, Spain2) Department of Physics,Technion, Haifa, 3200, Israel3) Department of Mathematics and Physics,University of Haifa at Oranim, Tivon, 36006, Israel4) Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA)Pg. Lluis Companys, 23, 08010 Barcelona, Spain
Abstract
We explore five-dimensional N = 4 SU (2) × U (1) and N = 8 SO (6) gauged supergravities asframeworks for condensed matter applications. These theories contain charged (dilatonic) black holesand 2-forms which have non-trivial quantum numbers with respect to U (1) subgroups of SO (6). Aquestion of interest is whether they also contain black holes with two-form hair with the requiredasymptotic to give rise to holographic superconductivity. We first consider the N = 4 case, whichcontains a complex two-form potential A µν which has U (1) charge ±
1. We find that a slight general-ization, where the two-form potential has an arbitrary charge q , leads to a five-dimensional model thatexhibits second-order superconducting transitions of p -wave type where the role of order parameter isplayed by A µν , provided q & .
6. We identify the operator that condenses in the dual CFT, which isclosely related to N = 4 Super Yang-Mills theory with chemical potentials. Similar phase transitionsbetween R-charged black holes and black holes with 2-form hair are found in a generalized version ofthe N = 8 gauged supergravity Lagrangian where the two-forms have charge q & . ontents N = 4 SU (2) × U (1) gaugedsupergravity: Basic setup 4 N = 8 SO (6) gauged Supergravity 26 N = 8 SO (6) gauged Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . 266.2 General STU black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.3.1 Case of one charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.3.2 Case of three equal charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.4 Dual field theory operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 The application of AdS/CFT to the study of condensed matter systems is in rapid evolution. One ofthe most interesting applications has been the investigation of strongly coupled systems which undergoa superconducting phase transition below a critical temperature [1–5]. On the field theory side, theonset of superconductivity is characterized by the condensation of a composite charged operator for lowtemperatures
T < T c . In the dual gravitational description, the superconducting phase transition isrepresented by a transition from a black hole in anti-de Sitter space to a new black hole solution with“hair”, which is thermodynamically preferred below the critical temperature T c .Most of the works have adopted a phenomenological approach, where the gravitational system isconstructed ad hoc and the underlying field theory is unknown. Although these scenarios have led tointeresting qualitative results (see e.g. [6–12] and [13, 14] for reviews and references), understanding theprecise dictionary between gravity and the condensed matter system is obviously important in order tomake further progress, in particular, for eventual applications to real systems. In the AdS/CFT context,having a precise dictionary typically requires a brane construction, so that the field theory undergoing thephase transition can be explicitly constructed while at the same time the dual gravitational backgroundcan be found. In other words, a top-down approach where one would start with some compactification oftype II string or M-theory possibly with some branes present, and consider the dynamics of excitationsaround those solutions. At the linearized level, an example was first given in [15] by Kaluza-Klein2eduction of D = 11 supergravity on a seven-dimensional Sasaki-Einstein space. Explicit examplesof such compactifications leading to systems that exhibit superconducting phase transitions have thenappeared in [16–19] (see [20, 21] for further developments). In holographic superconducting models, the spontaneously broken U (1) symmetry is typically dual toa global U (1) symmetry in the boundary theory. Consider, in particular, the top-down constructions of[16–19]. These consistent truncations of IIB/11d supergravity are closely related to the near horizon regionof branes probing Sasaki-Einstein cones, which are non-compact conical Calabi-Yau spaces. D3 branes(M2 branes in the 11d case) probing such cones yield to examples of the AdS/CFT correspondence whichgenerically preserve 4 supercharges. In this case the dual field theory can be explicitly constructed usingby now standard methods (see e.g. [27]), thus providing the desired microscopic theory. For four unbrokensupercharges, the generic R-symmetry of these field theories is precisely U (1). It is this particular U (1) R the one which is spontaneously broken by the condensation of a scalar representing a breathing mode.As the number of preserved supersymmetries is increased, the R-symmetry of the boundary theoryis enhanced. For the maximal rigid SUSY in four dimensions, namely N = 4, the corresponding R-symmetry is SU (4) ∼ SO (6). Thus, while for N = 1 field theories there is just one single –and uniquelyfixed by superconformal invariance– U (1) R symmetry in the infrared, for N = 4 Super Yang-Mills theorythere are various U (1) generators in the larger non-abelian R-symmetry which might be spontaneouslybroken leading to a superconducting phase transition. Motivated by this observation, in this paper weinvestigate holographic phase transitions within the framework of five-dimensional N = 8 SO (6) gaugedsupergravity [28, 29], N = 4 SU (2) × U (1) gauged supergravity [30] and in related models. N = 4 SU (2) × U (1) gauged supergravity can be derived from a consistent truncation of IIB super-gravity [31] and also from a consistent truncation of eleven dimensional supergravity [32]. N = 8 SO (6)five-dimensional gauged supergravity is expected to arise from a consistent reduction of IIB supergravityon an S , although in this case an explicit construction is presently unknown. As such, these theoriesprovide an interesting arena to study the dynamics associated with the U (1) ∈ SO (6) R-symmetry ofinterest. These theories contain complex two-form fields (whose ten-dimensional origin is the complextwo-form potential of IIB supergravity) which are charged under the U (1) gauge groups. An excitingpossibility is that these complex two-form fields could condense and lead to superconductivity in thedual field theory. We will show that this does not occur, basically because the charge of the two-formis not sufficient large to trigger an instability. Indeed, a slight generalization of the supergravity theory,whereby we allow the U (1) charge q of the two-forms to take generic values, contains phases where theblack hole develops non-trivial two-form hair, thus breaking U (1) symmetry spontaneously. Interestingly,the order parameter transforms as a vector under SO (3) spatial rotations. Therefore the system repre-sents a p -wave superconductor. This is of course expected, since in five dimensions a 2-form potential isHodge dual to a one-form potential. While for the free theory a 2-form is dynamically equivalent to a1-form, this is not the case in the gauged supergravity. Models of holographic p -wave superconductorshave been constructed first in [33, 34] and then different versions have appeared (see e.g. [22, 23, 35, 36]).The model has well-known charged black hole solutions, the STU black holes [37]. These are charac-terized by the ADM mass and charges ( Q , Q , Q ) under the maximal abelian subgroup of the gaugesymmetry, namely U (1) . From a ten-dimensional standpoint, these solutions correspond to spinningblack D3 branes in flat R space, and the three electric charges correspond to the three independentangular momenta in R .In this paper we give a detailed investigation of the different phase transitions that take place as finitecharge densities are turned on for different combinations of such U (1) gauge fields. As the temperature isgradually decreased, it will be seen that new phases where two-form hair grows become thermodynamicallyfavored. The transition to these hairy phases is second order. Their onset can be determined by solvingthe equations of motion of the 2-form in the vicinity of the phase transition. This amounts to studying theemergence of regular zero-modes in the black hole backgrounds, see [1]. The generalization of the model A different approach where one has some control over the dual field theory is based on using D-brane probes instring-theory black brane backgrounds (see e.g. [22, 23]). The identification of the field theory is more direct in the D3 brane case. For SCFT’s dual to M2 branes probing CY the field-theory description is less clear (see e.g. [24] for a discussion). Some examples can be found in [25, 26].
3o arbitrary charge q permits a study of the system in the probe limit where q is large. Nevertheless, sincewe are specially interested in the case of supergravity where q = 1 a full analysis including back-reactionwill be provided.The thermodynamics of STU black holes has been widely studied in [38–40]. The present modelscontain additional degrees of freedom, charged 2-form fields, which will lead to radically different phasetransition dynamics.The organization of this paper is as follows. In section 2, we shall first investigate N = 4 SU (2) × U (1)five-dimensional gauged supergravity (which can be thought as a subsector of the full N = 8 5d gaugedsupergravity). This has the advantage that the Lagrangian contains a single scalar field (and in thissense it is one of the simplest top-down models that one can study). There are three distinct N = 4 SU (2) × U (1) gauged supergravity theories, depending on the values of the SU (2) and U (1) couplingconstants. Here we consider the N = 4 + version, where both coupling constants have the same signand the theory contains AdS vacua (in the second theory the coupling constants have opposite signs andin the third theory the SU (2) coupling constant is taken to zero). After a brief review of the gravitytheory, we shall introduce the relevant black hole solution and review some of its salient thermodynamicalfeatures. In particular, we will see that these black holes have a Hawking temperature which cannot beless than some minimal value. This property further motivates the search for two-form condensation, asa possible avenue that the system can take to go to lower T . In section 3 we give the ansatz for the hairyblack hole solution representing the condensed phase and identify two conserved charges. Section 4 isdevoted to the numerical analysis of the solutions. We first find the critical temperature as a function ofthe U (1) charge of the 2-form field by studying the system in the vicinity of the transition, and then solvethe the full system including back reaction, compute the free energy and describe the different phasesin detail. We also discuss the probe limit and compute the conductivity, showing that the condensedphase exhibits transport properties which are characteristic of superconducting materials. In Section 5we discuss some features of the dual field theory. The close connection of the gravity model with gaugedsupergravity is used to make a concrete proposal for the dual operator that condenses. In Section 6, weconsider the more general case of STU black holes with ( Q , Q , Q ) = 0 and discuss in detail the N = 8gauged supergravity setup. In particular, we consider the case Q = Q = Q and compute the criticaltemperature as a function of the U (1) charge of the two-form fields, and show that condensation requiresa minimum charge which is above the values that one finds in supergravity. Some concluding remarksare given in Section 7. N = 4 SU (2) × U (1) gaugedsupergravity: Basic setup The N = 4 + gauged supergravity in five dimensions [30] has a bosonic sector containing the metric, ascalar, a U (1) vector field B µ , SU (2) Yang-Mills vector fields A aµ , and two 2-forms A αµν , α = 1 ,
2, whichtransform as a charged doublet under U (1) transformations. The action is given by I = − πG N Z (cid:20) R ∗ − X − ∗ dX ∧ dX − X ∗ F (2) ∧ F (2) − X − (cid:16) ∗ G a (2) ∧ G a (2) + ∗ A α (2) ∧ A α (2) (cid:17) + L ǫ αβ A α (2) ∧ dA β (2) − A α (2) ∧ A α (2) ∧ B (1) − G a (2) ∧ G a (2) ∧ B (1) + 4 L ( X + 2 X − ) ∗ (cid:21) . (2.1)The field strengths are G a (2) = dA a (1) + 1 √ L ǫ abc A b (1) ∧ A c (1) , F (2) = dB (1) , (2.2) F (3) ≡ DA (2) = dA (2) − iL A (2) ∧ B (1) , (2.3)4here we introduced complex notation A (2) ≡ A + iA . (2.4)The equations of motion derived from (2.1) are [30, 31], R µν = 3 X − ∂ µ X∂ ν X − L ( X + 2 X − ) g µν + 12 X − ( ¯ A ρ ( µ A ν ) ρ − g µν | A (2) | ) (2.5)+ 12 X ( F ρµ F νρ − g µν F ) + 12 X − ( G a ρµ G aνρ − g µν ( G a (2) ) ) ,d ( X − ∗ dX ) = 13 X ∗ F (2) ∧ F (2) − X − ( ∗ G a (2) ∧ G a (2) + ∗ ¯ A (2) ∧ A (2) ) − L ( X − X − ) ∗ ,d ( X ∗ F ) = − G a (2) ∧ G a (2) −
12 ¯ A (2) ∧ A (2) ,d ( X − ∗ G a (2) ) = √ L X − ǫ abc ∗ G b (2) ∧ A c (1) − G a (2) ∧ F (2) ,X ∗ F (3) = iL A (2) . From the above equations we see that the non-abelian gauge fields can be consistently set to zero. Thus,for our purposes, we can just consider the Lagrangian L = √ g h R − X − ∂ µ X∂ µ X − X F µν F µν + 4 L ( X + 2 X − ) i + L i ǫ µνρσδ ¯ A µν ∂ ρ A σδ − ǫ µνρσδ ¯ A µν A ρσ B δ − √ g X ¯ A µν A µν . (2.6) The charged black hole solution can be obtained as a particular case of the STU black hole carryingthree different U (1) charges Q , Q , Q [37]. In the present case we have Q = Q = 0, Q ≡ Q , and thesolution becomes ds = − fH / dt + H / f dr + H / r L ( dx + dy + dz ) , (2.7) X = H / , B = Q √ mr h + Q − Q √ mr + Q , (2.8) H = 1 + Q r , f = r L + Q L − mr , where r h is the position of the event horizon located at r h + Q = mL r h → r h = 12 (cid:16)p Q + 4 mL − Q (cid:17) . (2.9)Unlike the Reissner-N¨ordstrom solution, this geometry does not have a Cauchy horizon, the causal struc-ture is as in the Schwarzschild black hole. There is a curvature singularity at r = 0.The ADM mass and entropy density for the black hole solution are given by MV = 18 πG N L (cid:0) Q + 32 m (cid:1) , s = A h G N V = r h √ m G N L , (2.10)By demanding regularity in the Wick rotated solution, one shows that the Hawking temperature is givenby T = p Q + 4 mL πL p r h + Q . (2.11)5n the limit m → Q fixed, one has r h → T m =0 = Q πL , (2.12)the classical thermodynamics is no longer reliable because the solution is singular. As explained below,this limit is never reached in studying the field theory thermodynamics, where the choice of a definiteensemble implies fixing either chemical potential or charge density (which are given in terms of { Q, m } ).In particular, note that physical electric charge is proportional to Q √ m , so keeping the charge fixed at m → Q → ∞ . The energy and charge densities of the field theory are given by ǫ = 316 πG N L m , ρ = Q √ m πG N L . (2.13)For the sake of simplicity in the formulas, it is convenient to introduce new variables rescaling by a factor2 G N / ( πL ) as in [41]. One obtainsˆ ǫ = 3 m π L , ˆ s = r h √ m πL , ˆ ρ = Q √ m π L . (2.14)Then the equation (2.9) determining the location of the horizon gives the microcanonical equation ofstate: ˆ ǫ = 32(2 π ) / (cid:0) ˆ s + 4 π ˆ s ˆ ρ (cid:1) / . (2.15)The temperature and chemical potential are then given by T = (cid:18) ∂ ˆ ǫ∂ ˆ s (cid:19) ˆ ρ = 2 / (ˆ s + 2 π ˆ s ˆ ρ ) π / (ˆ s + 4 π ˆ s ˆ ρ ) / , (2.16) µ = (cid:18) ∂ ˆ ǫ∂ ˆ ρ (cid:19) ˆ s = (2 π ) / ˆ s ˆ ρ (ˆ s + 4 π ˆ s ˆ ρ ) / . (2.17)They obey the simple relation, Tµ = ˆ s π ˆ ρ + ˆ ρ ˆ s (2.18)i.e. ˆ s , = π ˆ ρTµ ∓ r − µ π T ! . (2.19)One can check that µ = 1 L Q √ mr h + Q . (2.20)This agrees with the identification derived from the standard rules of AdS/CFT using the asymptotic ofthe electromagnetic potential.In this paper we will work at fixed charge density fixed ˆ ρ , which corresponds to specifying the canonicalensemble. In this case the thermodynamics is dominated by the configuration with minimum Helmholtzfree energy, given by F = ˆ ǫ − T ˆ s . (2.21)6he specific heat at constant charge density is given byˆ C ˆ ρ = T (cid:18) ∂ ˆ s∂T (cid:19) ˆ ρ = 3ˆ s (cid:0) ˆ s + 2 π ˆ ρ (cid:1) (cid:0) ˆ s + 4 π ˆ ρ (cid:1) ˆ s + 10 π ˆ ρ ˆ s − π ˆ ρ , (2.22)The expression T = T (ˆ s, ˆ ρ ) given in (2.16) defines ˆ s ( T, ˆ ρ ), though an explicit formula requires findingthe roots of a six-order polynomial. However, the main features can be exhibited by a simple numericalanalysis. Figure 1 is a plot T vs. ˆ s at ˆ ρ = 1. We see that the temperature has a minimal value, which isgiven by T min = √ √ − / ˆ ρ / π / ≈ .
88 ˆ ρ / . (2.23)It should be stressed that this minimum temperature is unrelated to the minimum temperature (2.12),for the reasons explained earlier (in short, the minimum of a function T of two variables Q, m changesaccording to which combination of variables is kept fixed). In particular, from the expression for thecharge density in (2.14) one sees that the limit m → ρ requires that Q → ∞ in such a waythat the temperature T goes to infinity.For T > T min there are two branches, ˆ s ( T, ˆ ρ ) and ˆ s ( T, ˆ ρ ), the latter being the one with higherentropy density. The physically relevant branch that has less free energy is ˆ s ( T, ˆ ρ ), as can be seen fromfigure 1. This stable branch is energetically (and entropically) favored. Branch 1 is unstable, as alsoexpected from the fact that the specific heat is negative on this branch. - - - Figure 1: (Left) Entropy vs. Temperature at fixed ˆ ρ = 1. (Right) Free energy as a function of thetemperature. The dashed vertical line stands for T min . Branch 2 (red) is the upper branch on the Leftpanel, while branch 1 (blue) is the upper branch on the Right panel.One question of interest is if there is any possible configuration with T < T min and with the sameboundary condition as the black hole. In the case at hand, since we are considering Poincar´e patch
AdS ,the thermal
AdS phase is not available, because putting the Poincar´e
AdS geometry at finite temperatureintroduces a conical singularity. While in global
AdS the thermodynamics of the system does involvetransitions between the black hole geometry and thermal anti-de Sitter space with the same boundarydata [38–40], this is not possible in the present case. It is possible that the system cannot be cooledbelow this temperature because there may be no gravitational configuration with lower temperatures: intrying to cool the system by extracting energy, the system would simply move to the unstable branchwith negative specific heat and increases the temperature again. Another possibility is that the systemmight get to lower temperatures by means of more complicated black hole configurations. In particular,the present model involves other fields, like a charged two-form, thus one would like to see if there couldbe black hole configurations with two-form hair that can get to lower temperatures. In section 4, we will Here we correct an earlier version of this paper where we assumed that at sufficiently low temperatures there would bea phase transition to thermal AdS. We thank the referee for kindly reminding us of this important point. U (1) charge q for q greater than some critical value. In that case, we will find that there is a hairy black hole thatcan get to lower temperatures, up to a new minimum temperature (lower than T min ) that depends onthe charge q . For large q , this new minimum temperature goes to 0.The large temperature behavior is ˆ s ≈ π T + O ( T − ) . (2.24)Hence C ˆ ρ ≈ π T + O ( T − ) . (2.25)The specific heat diverges as T → T min .Consider now the behavior of entropy and specific heat near some given low temperature. In particular,near T = ρ / π / where the free energy changes sign, we findˆ s ≈ π ˆ ρ + 4 π ˆ ρ ( T − T ) + . . .C ˆ ρ ≈ π ˆ ρ T + . . . . (2.26)The temperature behavior of the specific heat reproduces a standard metal behavior, C V ≈ c e T + c ph T (2.27)where the linear and cubic term represent electron and phonon contributions. At large temperatures,the phonon contribution is dominant and, in the present case, can thus be identified with c ph = π (cf.(2.25)). From the linear behavior (2.26) near the temperature T , one identifies c e = 4 π ˆ ρ . Given the existence of a non-zero minimum temperature, an interesting question is what are the possiblegravitational solutions that can contribute to the thermodynamics below this temperature. We look forcharged solutions giving rise to a finite charge density configuration in the field theory. One possibilityis that there is no stable/metastable ground state for
T < T min . The Romans theory, however, has morefields and it is possible that the thermodynamically favored solution is actually a black hole configurationwith some hair, which will be provided by the extra fields of the supergravity Lagrangian. Indeed theLagrangian (2.1) contains the complex 2-form A µν charged under the U (1) symmetry associated with B µ , which can in principle lead to the analogous instability found for the Reissner-Nordstr¨om black holein [4, 5]. In what follows we shall turn on the 2-form field A µν and look for such instabilities. Solutionswith A µν hair can spontaneously break the U (1) global symmetry of the boundary field theory and takethe system to a superconducting phase.To have a more complete understanding on the 2-form dynamics, we consider a slight generalizedmodel with respect to the Romans Lagrangian in which the 2-form has a general U (1) charge q . TheLagrangian is given by L = √ g h R − X − ∂ µ X∂ µ X − X F µν F µν + 4 L ( X + 2 X − ) i + L i ǫ µνρσδ ¯ A µν ∂ ρ A σδ − q ǫ µνρσδ ¯ A µν A ρσ B δ − √ g X ¯ A µν A µν . (3.1)For the value q = 1 we recover the model (2.6). It is important to notice that the STU black holesolutions are also solutions of our generalized model (3.1), since A µν = 0 in the black hole backgroundand therefore q does not participate in the equations.8he generalization to arbitrary charge q has also the advantage of permitting the study of the systemin the probe limit. This is obtained by rescaling B µ → B µ /q , A µν → A µν /q and taking the limit q → ∞ . In this limit the Lagrangian for B µ and A µν decouple from the gravity/dilaton part of theLagrangian and one can therefore study the B µ and A µν system in a fixed Q = 0 (Schwarzschild anti-deSitter) background. This system is of course considerably simpler than the full system that includesthe dynamics coupled to g µν , X . Nonetheless, since we are also interested in the system at small q , inparticular, at q = 1, we will first study the full dynamics including back reaction, deferring the analysisof the probe limit to section 4.5. We are interested in finding new black hole solutions with non-trivial profile for the two-form in ourgeneralized gravity theory given by (3.1). The natural ansatz to consider for the metric is ds = g µν dx µ dx ν = e A ( r ) (cid:16) − h ( r ) dt + dx + dy + b ( r ) dz (cid:17) + e B ( r ) drh ( r ) , (3.2) X = X ( r ) , B (1) = Φ( r ) dt , A (2) = A (2) ( r ) . (3.3)As shown below, turning on a non-trivial A (2) ( r ) necessarily breaks the isotropy in x, y, z . This is thereason for the introduction of the function b ( r ), which will be consistent with the choice of direction for A (2) ( r ) adopted below.Consider the first-order equation for the complex 2-form A (2) ≡ A + iA . Written in components,2 iA µν = L X √ g ǫ µνρσδ ( ∂ ρ A σδ − q iL B ρ A σδ ) . (3.4)The requirement that A (2) is a function only of the radial coordinate leads to the following constraints: A r = 0 , (3.5) A i = i L X √ g ǫ rijk ∂ r A jk , (3.6) A ri = − q X √ g ǫ rijk B A jk . (3.7)Due to the antisymmetry of A (2) we can trade A ij by a 3-vector a k in the three spatial directions, A ij = ǫ ijk a k , i, j, k = 1 , , . (3.8)Next, consider the Einstein equations (2.5). The ansatz (3.2) implies that the Ricci tensor is diagonal,therefore we must choose A ij in such a way that also the stress energy tensor be diagonal. A simplechoice which is in agreement with (3.2) is, ~a = (0 , , a ) . (3.9)Thus A xy = a . The remaining non-zero components of the 2-form, A z and A rz , take the form A z = iL X √ g ∂ r a , A rz = − q X √ g Φ a . (3.10)Finally, the equation of motion for A ij , iA ij = − L X √ g ǫ rijk ( ∂ r A k + q iL B A rk ) , (3.11)9mplies the following second-order equation for aa ′′ + (cid:16) h ′ h + b ′ b − B ′ + 2 X ′ X (cid:17) a ′ + q L e B − A Φ h a − L e B X h a = 0 . (3.12)Having obtained the equation for the order parameter a , we next obtain the remaining equations ofmotion and check the consistency of the ansatz. In the setup (3.2) we have explicitly set the fields B r and B i to zero; therefore the source terms appearing in their equations of motion must vanish. In particular,from the Maxwell’s equations ∂ µ (cid:16) √ g X F µρ (cid:17) = q ǫ ραβσδ ¯ A αβ A σδ , (3.13)one finds the conditions ǫ kαβσδ ¯ A αβ A σδ = 0 , ǫ rαβσδ ¯ A αβ A σδ = 0 . (3.14)The first one is trivially satisfied because of A r = 0; the second condition gives ǫ rαβσδ ¯ A αβ A σδ ∝ i (¯ a i ∂ r a i − a i ∂ r ¯ a i ) = 0 . (3.15)This is easily satisfied by taking a ∈ R (more generally, this is solved by a = C ˜ a with real ˜ a and C is anycomplex number). To complete the analysis of the Maxwell’s equations, we write down the equation ofmotion for the time component of the one-form B (1) ,Φ ′′ + (cid:16) A ′ − B ′ + b ′ b + 4 X ′ X (cid:17) Φ ′ − q e B − A X Φ h a = 0 . (3.16)Now consider the equation for the scalar field X ,1 √ g ∂ µ (cid:16) √ g X − ∂ µ X (cid:17) − X F µν F µν + X − A µν A µν + 43 L ( X − X − ) = 0 . (3.17)We obtain X ′′ + (cid:16) A ′ − B ′ + h ′ h + b ′ b − X ′ X (cid:17) X ′ + X e − A h Φ ′ + e B − A Xh a − L X e − A (cid:16) a ′ − q L e B − A Φ a h (cid:17) + e B h L ( X −
1) = 0 . (3.18)Finally, the Einstein equations read R µν − g µν R = T µν , (3.19)with T µν = √ g h − X − ∂ µ X∂ ν X − X F µρ F ρν − X ¯ A µρ A ρν i (3.20) − √ g g µν h − X − ∂ µ X∂ µ X − X F µν F µν − X ¯ A µν A µν + 4 L ( X + 2 X − ) i . In order to fix the metric equations of motion, we consider the three linear combinations R tt − R xx , R zz − R xx , R rr − R tt − R zz − R xx , (3.21)and the xx equation. We find h ′′ + (cid:16) A ′ − B ′ + b ′ b (cid:17) h ′ = X e − A Φ ′ + L e − A X ha ′ + e B − A X a , (3.22)10 ′′ + (cid:16) A ′ − B ′ + h ′ h + b ′ b (cid:17) A ′ == 43 L e B ( X + 2 X − ) h − X e − A h Φ ′ − e B − A X h a − L X e − A (cid:16) a ′ − q L e B − A Φ a h (cid:17) , (3.23) b ′′ b + (cid:16) A ′ − B ′ + h ′ h − b ′ b (cid:17) b ′ b = L X e − A (cid:16) a ′ − q L e B − A Φ a h (cid:17) + e B − A X a h , (3.24)and a first-order constraint 12 A ′ + 3 A ′ h ′ h + (cid:16) A ′ + h ′ h (cid:17) b ′ b + 12 e B − A X a h ++ X e − A Φ ′ h − L X e − A (cid:16) a ′ − q L e B − A Φ a h (cid:17) − X ′ X − L e B X + 2 X − h = 0 . (3.25)The freedom of radial coordinate redefinitions will be fixed by the choice e B = e − A r , i.e. the samecondition obeyed by the “bald” charged black hole (2.8). The constraint (3.25) can be used to simplifythe r.h.s. of the second order equation for A : A ′′ − (cid:16) B ′ + b ′ b (cid:17) A ′ + X ′ X − h ′ h b ′ b + L e − A X a ′ + e B − A a X h = 0 . (3.26) The equations of motions (3.12), (3.16), (3.18) and (3.22)-(3.25) can be obtained from the EffectiveLagrangian, L eff = h √ be A − B h A ′ + 3 A ′ h ′ h + 3 A ′ b ′ b + 12 h ′ h b ′ b − X ′ X i + 12 X √ be A − B Φ ′ + 4 L √ be A + B ( X + 2 X − ) − L X e − B h √ ba ′ + q X e − A + B √ bh Φ a − √ b X e B a . (3.27)We will now search for symmetries under scaling transformations, t → λ t t , ( x, y ) → λ ( x,y ) ( x, y ) , z → λ z z , (3.28)with λ = 1 + δλ , together with the associated infinitesimal transformations for the fields, h → (1 + ǫ h ) h , A → A + ǫ A , B → B + ǫ B , Φ → (1 + ǫ Φ )Φ , b → (1 + ǫ b ) b , a → (1 + ǫ a ) a . (3.29)Because V ( X ) is not an homogeneous polynomial the dilaton X must scale trivially, X → X . We nowdemand that L eff and the background are invariant under the above scaling transformations: δ L eff = δds = δA (2) = δB (1) = 0 . (3.30)This leads to an algebraic system admitting a two-parameter family of solutions. Choosing ǫ A and ǫ b asindependent parameters, we find ǫ h = − ǫ A − ǫ b , ǫ B = − ǫ A − ǫ b ,ǫ Φ = 3 ǫ A − ǫ b , ǫ a = 2 ǫ A ,δλ t = 3 ǫ A + ǫ b , δλ z = − ǫ A − ǫ b , δλ ( x,y ) = − ǫ A . (3.31)11he transformations with parameters (3.31) represent two scaling symmetries of the Lagrangian (3.27)(before eliminating B by a choice of radial coordinate). The two scaling symmetries are summarized intable 3.2, where ~x = ( x, y, z ) and the charges α are assigned following the rule f → λ α f , for a genericfield or variable f . Symm. t ~x r e A h X Φ a I -1 -1 1 1 0 0 1 2II 1 -1 0 1 -4 0 -1 2Table 1: Weights for the scaling symmetries of the effective Lagrangian (3.27).Using the Noether procedure we find the two associated conserved charges, Q = h √ be A − B (cid:18) h ′ h − b ′ b (cid:19) − X L h √ be − B a ′ a − √ bX e A − B Φ ′ Φ ,Q = h √ be A − B (cid:18) h ′ h − b ′ b (cid:19) − √ bX e A − B Φ ′ Φ . (3.32)It is easy to check that by differentiating these equations one obtains a linear combination of the dif-ferential equations of the system given in section 3.1; in other words, the charges Q , Q represent twointegrals of the equations of motion. Note the combination Q ≡ Q − Q = 2 h √ be A − B b ′ b − X L h √ be − B a ′ a , (3.33)which exhibits the fact that a non-trivial a ( r ) turns on the metric component b ( r ). The numerical problem involves the resolution of six coupled second-order differential equations from theblack hole horizon up to infinity, where boundary conditions need to be fixed by the “shooting” method,which means that one needs to satisfy boundary requirements both at infinity and at the horizon. Thestrategy is a slight generalization of the procedure explained in detail in [4,5], which we here review, withemphasis on the new aspects we encountered that are inherent to the present system.
We look for black hole solutions with regular event horizons. The location r = r h of the horizon is definedby the simple zero of h lying at larger r . The Hawking temperature associated with black holes of theform (3.2) is then given by T Hawk = 14 π e A ( r h ) h ′ ( r h ) r h . (4.1)Regularity of the horizon requires that h ′ ( r h ) is a non-zero finite quantity.As a warm-up example, let us first consider the numerical derivation of the bald black hole (2.8),which has a = 0. In this case we can consistently set the metric component b = 1. We fix the horizoncoordinate at r h = 1 by means of symmetry I. Then the remaining four equations (3.16), (3.18), (3.22),(3.26), being second order, are completely specified by eight boundary values at r h . Not all of them arefree parameters: in order to have a fully regular solution, one must require that Φ( r h ) = 0 and that X isregular at the horizon. This condition gives X ′ ( r h ) = (cid:20) e B (1 − X ) h ′ − X e − A h ′ Φ ′ (cid:21) r = r h . (4.2)12urthermore, one integration constant is eliminated by the energy constraint and another integrationconstant can be fixed by symmetry II. Therefore, based on the horizon boundary conditions, there existsa three-parameter family of solutions. This means that two additional constraints have to be imposedin order to match the analytic expression of the bald black hole, which, at fixed charge density, dependson a single parameter –which can be taken to be the temperature. Let us now consider the asymptoticbehavior. From the fact that the metric approaches the anti-de Sitter solution, we obtain A = A ∞ + log rL + Q A r + . . . (4.3) h = h ∞ − mL r + . . . (4.4)Φ = µL − π ˆ ρL r + . . . (4.5) X = 1 + Q X r + C X r log rL + . . . (4.6)We now require that C X = 0 and Q X = 2 Q A . The first requirement removes logarithmic terms; thesecond requirement fixes the dilaton charge to a special value (which allows one to find an analytic solu-tion, see [37]). These conditions translate into two non-linear relations on the three horizon parameters,leaving only one free parameter. Then numerical integration of the differential equations reproduces thebald black hole solution (2.8).It is now straightforward to apply the same procedure for black holes with hair. For the hairy blackhole ansatz, the condition (4.2) gets modified by the addition of the term − e B − A a / (6 Xh ′ ) evaluatedat r h . Now consider the second-order equations of motion (3.12), (3.24) for a and b . We are adding fourmore boundary values at r h to the previous discussion. The regularity conditions a ′ ( r h ) = (cid:20) e B aX h ′ (cid:21) r = r h , b ′ ( r h ) = (cid:20) e B − A a X h ′ (cid:21) r = r h , (4.7)and the rescaling b → λb , reduce these four parameters to a single one. An additional requirement comesfrom the physics we aim to describe, namely a background that represents spontaneous U (1) symmetrybreaking in the dual field theory. This proceeds as usual: the 2-form A (2) is dual to the operator O that is expected to condense (for a discussion on the dual operator, see section 5). Therefore, from theasymptotic behavior of a , a = O r + O r + . . . , (4.8)the coefficient of the non-normalizable term, O , is interpreted, in the field theory, as a source term for theoperator O , whereas the coefficient of the normalizable term, O , gives the value of the condensate. Then,demanding spontaneous (rather than explicit) symmetry breaking of the global U (1) in the field theoryamounts to imposing O = 0. This condition fixes the additional parameter we found in considering theequations (3.12), (3.24).In the case that O is different from zero, the value of the condensate is affected by logarithmicdivergences, originating from a term n r log r in the asymptotic behavior that is not shown explicitly in(4.8). These logarithmic terms disappear upon our choice O = 0.Summarizing, we obtain a one-parameter family of superconducting black holes, provided there are solutions satisfying this condition O , which, as we will see, it is not always the case. In particular, For theRomans Lagrangian (2.1), we have studied the coupled system of six differential equations numerically Here one may also use the conserved charges discussed above to write a first order equation for b . This leads to equivalentresults. O and therefore does not represent U (1)spontaneous symmetry breaking. The underlying reason will be understood in the next subsection: thecharge q = 1 of the A µν field is not sufficiently large to drive to an instability.Some useful information can be obtained by computing the conserved charges both at the horizonand at infinity. Because h vanishes at the horizon, we find that Q = 0. Now computing Q at infinityand using that O = 0, we find that b = 1 + O (1 /r ). Now consider the calculation of Q . At the horizonwe find Q = 4 π T H ˆ s , where ˆ s is the entropy density normalized as in section 2. Now computing Q atinfinity we find Q = 4 π ( ˆ ǫ − µ ˆ ρ ). Thus charge conservation implies the thermodynamic relationˆ ǫ = 34 ( T ˆ s + µ ˆ ρ ) . (4.9)In [42], in a different context, it was noticed that these type of relations follow from the assumption ofa traceless field-theory energy-momentum tensor. In the present, four-dimensional case, this assumptionimplies ˆ ǫ = 3ˆ p , where ˆ p is the pressure. Now one uses the relation − ˆ p = ˆ ǫ − T H ˆ s − µ ˆ ρ , thereby, strikingly,the relation (4.9) follows. Consider the theory given by the Lagrangian (3.1), where the charge q is taken as a real parameter. Beforepresenting the full analysis including back-reaction, we first obtain the curve of critical temperature asfunction of the q parameter.The idea is based on the following observation. In second (or higher) order phase transitions, the orderparameter approaches zero near the critical temperature. In the gravity solution, O → a →
0, which in turn implies that the hairy black hole approaches the bald black hole (2.8). Therefore,in the vicinity of a continuous phase transition, we just need to study the a equation (3.12) in the baldblack hole background (2.8). This has X = H / , b = 1 . (4.10)While this method is numerically very accurate for the determination of the critical temperature, itis however inappropriate to detect possible first-order phase transitions, where a is not small near thetransition. The complete picture that covers the case of first-order transitions as well will be clear uponsolving the full system including back reaction.Substituting the bald black hole solution in (3.12) we find a ′′ + r − Q r + 3 ¯ mr ˜ f a ′ + 1˜ f ( r + Q ) (cid:16) ¯ mq Q ( r − r h ) r ˜ f ( r h + Q ) − r (cid:17) a = 0 , (4.11)with ˜ f ≡ r + Q − ¯ mr , m ≡ ¯ m/L . (4.12)By a further rescaling ¯ m = r h ˜ m , Q = r h ˜ Q and introducing a new variable z = r h /r , the dependence on r h drops out from the equation. In addition, we note that ˜ m = 1 + ˜ Q by virtue of the horizon equation.The final equation depends only on the parameters ˜ Q and q . Define p ( z ) = z a ( z ), so that, at small z (large r ), p has the expansion p ( z ) = r h O + O z r h + n z r h log z + .... The differential equation becomes p ′′ ( z ) + F ( z ) p ′ ( z ) + G ( z ) p ( z ) = 0 , (4.13)14 ( z ) ≡ (cid:16) ˜ Q + 1 (cid:17) z − ˜ Q z + zz ( z − (cid:16)(cid:16) ˜ Q + 1 (cid:17) z + 1 (cid:17) G ( z ) ≡ q ˜ Q z (cid:0) z − (cid:1) − (cid:16) ˜ Q + 1 (cid:17) z (cid:16)(cid:16) ˜ Q + 1 (cid:17) z + 1 (cid:17) (cid:16)(cid:0) z − (cid:1) ˜ Q + 3 (cid:0) z + 1 (cid:1) ˜ Q + 3 (cid:17)(cid:16) ˜ Q + 1 (cid:17) z ( z − (cid:16) ˜ Q z + 1 (cid:17) (cid:16)(cid:16) ˜ Q + 1 (cid:17) z + 1 (cid:17) Regularity near the horizon (located at z = 1) implies the boundary condition p ′ (1) = (cid:16) Q + 6 ˜ Q + 3 (cid:17) p (1)2 (cid:16) ˜ Q + 3 ˜ Q + 2 (cid:17) . (4.14)As explained in section 4.1, we must look for solutions which have O = p ( z = 0) = 0. Solutions with O = 0 have no logarithmic terms in the expansion at infinity, in particular, n = 0. In general, onehas O = O ( ˜ Q, q ) hence O = 0 gives ˜ Q = ˜ Q ( q ). The critical temperature is then obtained from theHawking temperature (2.11) which, expressed in terms of the new variables, reads T = (cid:16) ˜ Q + 2 (cid:17) r h πL q ˜ Q + 1 = (cid:16) ˜ Q + 2 (cid:17) ˆ ρ / πL ˜ Q / ( ˜ Q + 1) / . (4.15)Thus we have T = T ( q ) = T (cid:0) ˜ Q ( q ) (cid:1) . Figure 2 shows the temperature as a function of the A µν -charge q obtained from the numerical resolution of the differential equation (4.13) (we set ˆ ρ = 1). We see thatsolutions with O = 0 exist for q above some critical value, q cr ≈ . q > q cr ≈ .
407 there aretwo values of ˜ Q that solve p ( ˜ Q, q ) = 0 and hence two values for the temperature T ( q ) ≤ T ( q ). Weanticipate the result that the set of physically relevant critical temperatures will be the higher branch T = T ( q ). More features of this curve will be discussed in the next subsection. T = T min - charge T e m p e r a t u r e Figure 2: Critical temperatures as a function of the A µν -charge q . The theory (3.1) contains different black hole solutions. The uncondensed a = 0 phase is described asusual by the bald solution (2.8). When the temperature is below the critical value T ( q ) (upper branchin fig. 2), one or more hairy black hole solutions appear, depending on the value of the parameter q . We For higher q , new branches appear, but they are not physically relevant as the corresponding critical temperatures are,again, smaller than T ( q ). ρ = 1. T T T C O (a) T T T C O (b) Figure 3: (a) and (b) show O ( T ) for q = 12 and q = 8, respectively. Only the red part (upper branch tothe right of vertical dashed line) of the plots has a physical relevance: in that range of temperatures thesystem will be in the condensed phase. The solid vertical line corresponds to the T min of the bald blackhole. Since T C < T min , the system can reach temperatures lower than T min through the hairy black holeconfiguration.Figure 3a shows the condensate O as a function of the temperature for q = 12. Note that for T C ≈ . < T < T there are two black hole solutions with A µν hair. Figure 3b is the similar plot for q = 8. The existence of a range of temperatures with two black hole solutions is a feature which is commonfor q &
6, as can be seen from figure 4. Figure 4 also illustrates the fact that the maximum of O ( T ) isgoing to zero when q is reaching q cr . Therefore the hair of the black hole solutions continuously disappearas q → q cr . Figures 3, 4 also show that the condensed phase gets extended to lower temperatures as q is increased. A natural guess is that in the limit q → ∞ the condensed phase extends all the way downto T = 0. This guess will be supported by the study of the probe limit in section 4.5. For finite q , T C represents the minimum temperature that the system can reach. This is a similar situation to the oneexplained in section 2: if one attempts to cool the system by extracting energy, the system will be pushedto the unstable branch, resulting into an increase of temperature (unless a more complicated gravitationalconfiguration with T < T C exists that could provide a new equilibrium configuration). O Figure 4: Condensate O ( T ) for the values q = 6, q = 5 . q = 5 .
45 (from top to bottom).16 .4 Free energy and Phase Diagram
In the canonical ensemble the configuration that dominates the thermodynamics is the one with leastHelmholtz free energy F . We have compared the free energies of two different phases corresponding tothe bald black hole and the hairy black hole. For T > T ( q ) there is no hairy black hole solution and thethermodynamically relevant solution is the bald black hole (2.8). The hairy black hole solution appearsat T < T ( q ). We have numerically studied the free energy of the hairy black holes for different values ofthe 2-form charge q . The calculation proceeds as follows. From the formula F = ˆ ǫ − T ˆ s , we see that calculating the free energy of the hairy solutions requires a combination of asymptotic andhorizon quantities. We first obtain the value of m from the expansion (4.4); then the energy can bededuced using (2.14) and the entropy from the standard definition in terms of the area of the horizon.The temperature is computed by using (4.1).We begin by discussing what happens for a fixed value of q . From fig. 4 we see that for q > .
65 thereis a region of temperatures where there are two black holes at the same temperature. In general, we findthat the lower branch solution has a free energy which is always higher than the free energy of the upperbranch solution.Figure 5 shows the free energy for the bald black hole (upper, blue curve) and for the hairy black hole(lower, red curve) for q = 12. In the range of temperatures T C ≤ T ≤ T , the hairy black hole solutionhas less free energy with respect to the uncondensed black hole configuration, therefore the system staysin the condensed phase. The transition at T = T from the uncondensed to the condensed phase issecond-order, since the first derivative of the free energy is continuous. T C T - - -
20 Temperature F r ee - E n e r gy Figure 5: Free energies for the bald (blue, upper curve) and hairy (red, lower curve) black holes at q = 12.The main result of this section is summarized by the phase diagram shown in Figures 6(a),(b). Recallthat the minimum temperature (2.23) of the bald black hole is T min ≈ . ≈ .
88 and that T ( q ), T ( q ) denote the lower and upper branches in figure 2. There are four special points in this diagram,summarized in table 2:1. The minimum charge at which the hairy black hole solution exists is q = q ≡ q cr ≈ . T ≈ . q ≈ .
65 such that for q > q there are two hairy black hole solutions with thesame T in a range of temperatures.3. A q = q ≈ .
85 at which T C coincides with T min . For q < q one has T min < T C ; for q > q onehas T min > T C . 17. The minimum temperature on the branch T ( q ). This is T ≈ . q = q ≈ . T min . q T q < q cr ≡ q , the system stays all the way down to T min in the uncondensed phase described by thebald black hole.- For q < q < q , the system undergoes a second-order phase transition from the uncondensed to thecondensed (superconducting) phase at T ( q ) and when the temperature reaches T ( q ) there is anothersecond-order phase transition back to the uncondensed phase (an example is q = 5 .
45 in fig. 4). It staysthere until T min .- For q < q < q , the system undergoes a second-order phase transition from the uncondensed to thecondensed (superconducting) phase at T ( q ), where it stays until the temperature reaches the minimumtemperature T C of the hairy solution. In the range T min < T < T C , the bald black hole is the onlyremaining solution and the system should undergo a transition back to the uncondensed phase.- For q > q , the system undergoes a second-order phase transition from the uncondensed to the condensed(superconducting) phase at T ( q ) and stays there until it reaches the minimum temperature T C (belowwhich there is probably no equilibrium configuration). Condensed PhaseUncondensed Phase - charge T e m p e r a t u r e (a) q q q q Condensed PhaseUncondensed Phase - charge T e m p e r a t u r e (b) Figure 6: (a) Phase Diagram for the system (3.1). (b) Zoom-in of the same diagram. The dashed lineafter the point 2 is the remaining part of the curve in figure 2, that has now become physically irrelevantas does not represent any separation between phases. The dotted horizontal line represents the minimaltemperature of the bald black hole.In the field theory description, the dilatonic black hole represents a metallic (uncondensed) phase.This is supported by the temperature behavior of the entropy and specific heat as discussed in section2.2. The hairy black hole represents a superconducting phase.18 .5 Probe limit
As pointed out in section 3, the introduction of the charge parameter q has the additional bonus ofgranting a probe limit, whereby the relevant dynamics for condensation is encoded in a decoupled non-gravitational sector. To that matter, let us consider our stress-energy tensor for the complete system T µν = √ g h − X − ∂ µ X∂ ν X − X F µρ F ρν − X ¯ A µρ A ρν i (4.16) − √ g g µν h − X − ∂ µ X∂ µ X − X F µν F µν − X ¯ A µν A µν + 4 L ( X + 2 X − ) i . Then we consider a limit in which we re-scale B µ → q − B µ , A µν → q − A µν . (4.17)The { B µ , A µν } part of the action scales homogeneously as q − . In the q → ∞ limit the B µ and A µν donot back-react on the geometry and the gravitational sector gives rise to a Schwarzschild-Anti de Sittergeometry (where X = 1). The { B µ , A µν } sector can then be studied independently. L = −√ g F µν F µν − √ g A µν A µν + L i ǫ µνρσδ ¯ A µν ∂ ρ A σδ − ǫ µνρσδ ¯ A µν A ρσ B δ , (4.18)in the background of the Schwarzschild-AdS black hole, which reads ds = − f dt + dr f + r L d~x , f = r L − M r . (4.19)This geometry has a horizon at r h = √ M L . The horizon radius is related to the temperature as T = r h π L . (4.20)We can read off the equations of motion from the generic ones in (3.12) and (3.16). Introducing a newvariable z = r h r and redefining B = Φ = r h L ϕ a = r h L pz . (4.21)the differential equations to solve become z ϕ ′′ − ϕ ′ − z − z p ϕ = 0 , (4.22) p ′′ z − z (1 − z )1 − z p ′ + 3 z (1 − z ) + ϕ (1 − z ) p = 0 , (4.23)where the primes now denote derivatives with respect to z . As discussed above, the boundary asymptoticsfor the fields are ϕ → L µr h − π ˆ ρ L r h z + · · · p → L O r h + L O r h z + · · · (4.24)As discussed in section 4.1, one must impose boundary conditions where O = 0. Boundary conditionsare implemented by solving the differential equations (4.23) by a standard shooting method. From thesolution one obtains O = O ( T ). Figure 7 shows the curve corresponding to the condensate O ≡ h O i as a function of temperature. The main new feature with respect to the finite q case is that at q → ∞ the condensate curve extends all the way down to T = 0. This confirms the tendency previously inferred19 .3 0.4 0.5 0.6 0.7010203040506070 T X O \ Figure 7: Condensate as a function of the temperature in the probe approximation.from figures 3 and 4, where it is seen that, as q increases, the condensed phase gets extended to lowertemperatures.Figure 7 is also in numerical agreement with figure 3, previously computed including the back reaction.For comparison, the temperature in figure 7 must be rescaled by q / , owing to the rescaling of ϕ by 1 /q ,which produces a rescaling of ˆ ρ/r h in the asymptotic expansion (4.24). We recall that the temperatureis obtained from the numerical solution by reading the value of r h from the second term in (4.24), whichwill be thus rescaled by a factor q / . Therefore, at large q , T ( q ) = q / T probe , T probe ≈ . T /q / as a function of q (obtained from the upper branch in fig. 2) is shown infigure 8. - charge T q Figure 8: Critical temperature T /q / as a function of q .The universality class of the transition can be inspected by computing the critical exponent β in h O i ∼ ( T c − T ) β . Figure 9 is a logarithmic plot of h O i vs. T , which can be accurately fitted by a straightline. We find the critical exponent β ≈ .
49. Modulo numerical errors (coming mostly from the estimateof the critical temperature in the fit), this indicates that our phase transition has mean field criticalexponent, as expected. One interesting feature of the probe limit is that it pushes to zero the undetermined, possible unstable, Non-mean field behavior can be accommodated in phenomenological models of holographic superconductors by meansof non-analytic terms [8–10]. Such terms are not expected in classical
Lagrangians originating from string/M theorycompactifications, though they might effectively be induced by quantum corrections (see [10] for a discussion). - - - - H - TT c L Log X O \ Figure 9: Fit of the logarithmic plot of h O i vs. T by a straight line of slope ∼ . T phase by replacing it with the hairy BH. In fact, this was our first hope, which is indeed realizedin this large q limit. Many universal features of superconductors just follow from the fact that these materials exhibit a spon-taneous breakdown of U (1) gauge invariance. One of these features is vanishing electrical DC resistance.Holographically, this phenomenon occurs in a manner which is entirely analogous to the field theory coun-terpart, namely the U (1) breaking turns on a new term in the Maxwell equations playing the role of acurrent –the London current. As long as this term is non-vanishing, one expects infinite DC conductivity,as there is a finite current even for an infinitesimal electric field. In the present case, the simplest way tosee the emergence of this term is by considering a time-dependent perturbation of the form B z = b z ( r ) e iωt + b ∗ z ( r ) e − iωt . (4.25)Turning on any other component like B x or B y , will lead to a complicated system of coupled equations.This is a reflection of the fact that the A µν background has broken isotropy and z is a preferred direction.We will compute the conductivities in the probe limit described by the Lagrangian (4.18) in thefixed Schwarzschild-Anti de Sitter geometry (where X = 1). The conductivity, arising from the retardedcurrent-current correlator, can be computed by studying fluctuations of the gauge field around the con-densed solution discussed in the previous subsection. In the complete system, where gauge field andtwo-form backreact on the geometry, off-diagonal metric components in the Minkowski would be sourced.However, in the probe limit at hand, a consistent solution is in fact given by the same ansatz (3.8), (3.10)for A µν , with the addition of a small A r component, which is then given by A r = − q √ g (cid:0) b z ( r ) e iωt + b ∗ z ( r ) e − iωt (cid:1) a ( r ) . (4.26)For the remaining components we have A i = A ri = A zi = 0, i = x, y . All other components of (3.4)are satisfied identically, except for the xy component, which gives (4.11) plus an O ( B z ) correction thatwe neglect. The z component of the Maxwell equation (3.13) (particularized for Schwarzchild-AdS) thenreads b ′′ z + (cid:18) A ′ − B ′ + h ′ h (cid:19) b ′ z + ω e B − A h b z = q e B − A h a b z . (4.27)The r.h.s represents the holographic analog of the London current.21ince we are interested in computing conductivities, the equation (4.31) has to be solved imposingcausal boundary conditions at the horizon. This requires ingoing wave conditions, which set, close to thehorizon b z → (1 − z ) − i ˜ ω X k =0 b k (1 − z ) k , (4.28)where we have introduced z = r h /r as in the previous subsection and ˜ ω is determined from the equation(see below). Then, the conductivity can be extracted from the asymptotic behavior of the gauge fieldfluctuation. As discussed in [6], the asymptotic behavior of the gauge field fluctuation in five dimensionsinvolves a logarithmic term which has to be reabsorbed by adding a suitable counterterm. Genericallywe have b z → b (0) z + b (1) z r h z − b (0) z ω r H z log z . (4.29)Then, following [6], the conductivity is given by σ = − i b (1) z b (0) z ω + i ω . (4.30)Provided b (1) z is not zero, the imaginary part of the conductivity has a pole, Im( σ ) ∼ /ω . By standardrelations of complex analysis, this pole is associated with a delta function δ ( ω ) in the real part of theconductivity (see [4,9,33] for discussions). Note that the London term in (4.27) is crucial for the emergenceof the delta function. Without this term, the ω → b (1) z = 0. In the presence of this term, b (1) z cannot vanish, leading to Re σ ∼ δ ( ω ) and thus DCsuperconductivity.Let us now explicitly compute the conductivity. In this case, we can consider (4.27) particularized tothe Schwarzschild-Anti de Sitter background. We get b ′′ − (1 + 3 z ) z (1 − z ) b ′ + ˜ ω − (1 − z ) p (1 − z ) b = 0 . (4.31)where ˜ ω = L r h = ωπ T . Implementing the boundary conditions as described above, we now numericallycompute the frequency-dependent conductivity. The results are shown in figures 10 and 11. The imaginarypart of the conductivity shown in figure 11 exhibits the 1 /ω behavior associated with a δ ( ω ) behavior inthe real part (the delta function is not seen in figure 10 due to numerical reasons). There are basicallythree regimes for the frequency:a) ω < h O i = O (ˆ ρ ). Here the conductivity is strongly affected by the presence of the condensate andexhibits the expected gap in Re( σ ) (related to the energy which is required to break the condensate).b) h O i ≪ ω ≪ T . Here the contribution from the condensate p in (4.31) can be neglected. One is thenbasically computing the conductivity of Schwarzchild-AdS black hole.c) ω ≫ T . Here the frequency is above all relevant scales and the conductivity approaches the conductivityof AdS , with a linearly increasing behavior.It is interesting to note the difference with Schwarzchild-AdS (discussed e.g. in [33]), where theconductivity becomes constant at high frequencies, rather than linearly increasing (and the temperaturedoes not play any role). In contrast, in the case of Schwarzchild-AdS , the conductivity is linearlyincreasing at sufficiently large frequencies. A detailed discussion can be found in [6]. A similar linearbehavior also appeared in [23] in some p -wave four-dimensional holographic superconductor models basedon probe D-branes. 22 .0 0.5 1.0 1.5 2.00246810 Ω X O \ Re @ Σ D Figure 10: AC conductivity for TT c = { . , . , . , . } (from bottom to top). - - Ω X O \ Im @ Σ D Figure 11: Imaginary part of the conductivity for TT c = { . , . , . , . } (from top to bottom). We have so far discussed a modification of N = 4 gauged supergravity in five dimensions in which atwo-form field undergoes condensation as temperature is lowered. Now we would like to turn to itsimplications for the putative dual field theory. Even though our model cannot be obtained –or at leastnot in any obvious way– as a consistent truncation of ten-dimensional IIB supergravity, the backgroundswe are considering asymptote to AdS . Therefore they are, on general grounds, subject to the generalprinciples of AdS/CFT. In particular, we would like to determine the operator which is triggering thephase transition by its condensation. To that matter, let us consider the part of the action involving the2-form, I ⊃ − πG N Z (cid:20) − X − ∗ A α (2) ∧ A α (2) + L ǫ αβ A α (2) ∧ dA β (2) − q A α (2) ∧ A α (2) ∧ B (1) (cid:21) . (5.1)Dropping momentarily the interaction term with B (1) , we have a first order action for a two-form potentialwith an effective mass given by X − . Note that, because the Lagrangian is first order, the parameterappearing in the Lagrangian is directly the mass of the field. In turn, the X field has a potential given23y V = 4 L (cid:16) X + 2 X − (cid:17) . (5.2)The minimum of this potential –which the field must approach at large r – is located at X = 1. Therefore,in the asymptotic region, the two-form behaves as a two-form field with mass = 1, governed by a firstorder Lagrangian in AdS . We can then borrow the results in [43], where it was found that a mass m two-form field in AdS with first order Lagrangian couples to a boundary operator of dimension∆ = 2 + m . (5.3)Particularizing this expression to our case, we see that the A (2) fluctuation is dual to a dimension 3operator which transforms as an antisymmetric tensor under the Lorentz group. This dual operatorwill live on the boundary strongly coupled CFT, which generically is expected to contain a gauge sectortogether with a matter sector. Since our dual operator must have two antisymmetric Lorentz indices, itmust contain either two derivatives or else an insertion of a boundary gauge field strength. The first casewould then require at least two scalars (or fermions), which would necessarily involve a dimension largerthan 3. It is then natural to guess that the dual operator contains F µν . In order to have a non-vanishingtrace, it is natural to conjecture that the dual operator is of the form O µν ∼ Φ F µν , being Φ a certainscalar field of dimension 1 , such that ∆ O = 3. Besides, the bulk 2-form potential is charged with charge q under the U (1) field B (1) , which suggests that the dual theory contains a global symmetry under whichΦ has charge q , so that the quantum numbers of our proposed O µν would match the expectations fromthe bulk physics.It is perhaps instructive to consider the case q = 1 as test of our proposal. At q = 1 our modifiedmodel becomes N = 4 SU (2) × U (1) Romans gauged SUGRA, which is a IIB consistent truncation.From the 10d point of view, the gauge field B (1) is turned on by angular momentum, and the solutioncan be thought as a stack of D3 branes in R spinning in a U (1) ⊂ SO (6). This global symmetry isactually the R-symmetry of the theory, so that the background is dual to N = 4 SYM at finite chargedensity in a U (1) subgroup of the R-symmetry. The two-form is charged under this U (1) R subgroup,and naturally corresponds to Φ F + µν , where Φ is a complex scalar of the N = 4 SYM theory so thatthe operator indeed has charge q = 1 under the relevant U (1) and dimension 3. We can understand thisidentification by slightly moving in the Coulomb branch of the theory higgsing the gauge group from SU ( N c ) to SU ( N c − × U (1). At large N c we can think of the system as a probe D N c − ∼ N c . The action for such a probe D3 is S = − T Z e − φ r det (cid:16) P [ g ] + F (cid:17) + T Z X P [ C n ] ∧ e F , (5.4)where F = P [ B ] + 2 πl s F . Let us re-write the DBI determinant as r det (cid:16) P [ g ] + F (cid:17) = p det P [ g ] r(cid:16) P [ g ] − F (cid:17) . (5.5)Then, expanding to second order the DBI we find a term with S DBI ⊃ T Z F ∧ ⋆ F , (5.6)where the Hodge-star is taken with respect to the pull-back metric. In particular, this contains S DBI ⊃ T πl s Z P [ B ] ∧ ⋆F . (5.7) Note that the inserted operator Φ has dimension 1, and thus corresponds to a free field. Despite the lack of explicitembedding into IIB and the lack of SUSY, it seems reasonable to expect that indeed 3 is the minimal dimension that canbe attained.
24n the other hand, from the Wess-Zumino part of the D3-brane action (5.4), we find a term with S W Z ⊃ T πl s Z P [ C ] ∧ F . (5.8)Thus, altogether we have S D ⊃ T π l s Z P [ C ] ∧ F + P [ B ] ∧ ⋆F . (5.9)If F is a self-dual two-form field, it follows that S D ⊃ T π l s Z (cid:16) P [ C ] + P [ B ] (cid:17) ∧ F + . (5.10)At this point it is useful to recall the truncation ansatz in [31], where the starting point for the reductionon the 5-sphere P x i = 1 is the following coordinate system x = sin ξ cos τ ; x = sin ξ sin τ ; x = cos ξ cos α ; x = cos ξ sin α cos α ; x = cos ξ sin α sin α cos α ; x = cos ξ sin α sin α sin α . In terms of these angles, the 10-dimensional B , C potentials are written in terms of A (2) as B = − sin ξ sin τ Re A (2) + sin ξ cos τ Im A (2) ; C = − sin ξ cos τ Re A (2) − sin ξ sin τ Im A (2) . (5.11)Going back to the Cartesian coordinates we obtain B = − x Re A (2) + x Im A (2) ; C = − x Re A (2) − x Im A (2) . (5.12)Therefore, we have P [ C + B ] = − ( x + x )Re A (2) + ( x − x )Im A (2) . (5.13)Let us now define Φ = e i π ( x + i x ). This field naturally corresponds to one of the three complexifiedscalars of N = 4 SYM, and it is then charged under a U (1) subgroup inside the maximal torus of SO (6).Conversely, we have Φ = e i π ( x + i x ) = sin ξ e i π e iτ (5.14)which explicitly shows that the U (1) symmetry under which Φ has charge 1 corresponds to rotations inthe τ direction from the ten-dimensional perspective. Furthermore, the action of the probe brane cannow be written as S D ⊃ T π l s Z (cid:16) P [ C ] + P [ B ] (cid:17) ∧ F + = T √ π l s Z A (2) ∧ Φ F + + c . c . (5.15)This equation explicitly shows how A (2) couples to the proposed operator. Note that this operator isnon-vanishing already at the abelian level, and thus there is no need to consider the non-abelian extensionof the D3 brane action. Nonetheless, repeating the same exercise for the non-abelian case, we obtain thestraightforward extension of the above coupling, Tr (cid:16) Φ F + (cid:17) .It should be stressed that it is actually the self-dual part of the gauge field strength what enters thedual operator. The reason is that N = 4 gauged SUGRA fixes the axiodilaton to zero. This bulk fieldcouples to the boundary operator F µν F µν . Since the axiodilaton is not turned on, it must be the self-dualpart of F the relevant one entering in the operator dual to the complex 2-form. Indeed, the proposedoperator actually agrees with the results in [44] (see eq. (A.7)), in support of our conjecture.25t is interesting to compare with the picture of [16], where R-charged black holes undergoing a phasetransition through the condensation of a chiral primary operator in IIB theory were considered. ThroughAdS/CFT, these systems can be thought of as finite-temperature versions of D3 branes probing the tipof CY cones which are U (1) fibrations over a Sasaki-Einstein space. The U (1) fiber corresponds to theR-symmetry of the theory, and by means of a chemical potential on such U (1) R charge, condensation isachieved. Interestingly, the operator condensing is also dimension 3. However, in this case corresponds toa chiral primary operator, and thus it has R = 2. Indeed, it was argued to correspond to the superpotentialof the theory. It is a subtle question whether this is the leading instability, since, depending on the theoryunder consideration, there might be R-charged chiral operators with lower R-charge. In our case, dueto the vector-like nature of the condensing operator, the dual operator should contain, as discussedabove, either two derivatives or an insertion of the field strength. The case with two derivatives wouldcorrespond to an operator of dimension at least 2. In order to have a gauge invariant operator withnon-vanishing trace we would need at least two scalars, thus leading to a dimension greater than 3.Thus, we see that the considered operator is the lowest dimensional one which can trigger the vector-likecondensation. N = 8 SO (6) gauged Super-gravity N = 8 SO (6) gauged Supergravity Ungauged 5d SUGRA contains 42 scalars parameterizing the symmetric space E /U Sp (8). Theycan be compactly grouped in the 27-bein V abAB , being { A, B } E indices and { a, b } U Sp (8) indices.When turning to the gauged version, the natural SO (6) gauged subgroup is embedded in a maximal SL (6 , R ) × SL (2 , R ) subgroup of E . Since the fundamental representation of E breaks under SL (6 , R ) × SL (2 , R ) as → ( ˜15 , ) + ( , ), the A, B indices are split into I = 1 · · · α = 1 , N = 8 SUGRA comprises both one-form and two-form gauge potentials. The two-formfields are in the ( , ) and therefore have indices A Iαµν . In turn, the one-form fields carry indices B µ, IJ in the ( ˜15 , ). From [28, 29] we can read off the relevant Lagrangian for the N = 8 two-forms A Iαµν coupled to the SO (6) gauge field B µ, IJ L ⊃ − ǫ µνρστ ǫ αβ η IJ A Iαµν ∂ ρ A Jβστ + 12 ǫ µνρστ ǫ αβ B µ, IJ A Iανρ A Jβστ − A Iαµν A µν, Jβ M Iα, Jβ (6.1)where the mass matrix is given in terms of the scalar vierbein as M Iα, Jβ = V abIα V Jβ, ab (6.2)In the SL (6 , R ) × SL (2 , R ) subsector of the scalar manifold, the vielbein simplifies into [29] V abIα = 12 √ (cid:16) Γ Kγ (cid:17) ab S KI ˜ S γα , Γ Kσ = (cid:16) Γ K , i Γ K Γ (cid:17) , σ = 1 , , (6.3)where S ∈ SL (6 , R ) and ˜ S ∈ SL (2 , R ). The matrices Γ i are the SO (7) gamma matrices satisfying { Γ i , Γ j } = 2 δ ij for i, j = 0 · · ·
6. Latin indices are raised and lowered with Ω ab = − Ω ab = − i (cid:16) Γ (cid:17) ab . We stress that this U (1) R is different from the one we turned on in N = 4 SU (2) × U (1) supergravity. In the contextof generalized STU black holes it would correspond to q = q = q ; see section 6. One might also wonder why an operator made out of two fermions with the suitable Dirac matrix does not do the job.The reason is that the combination ¯ΨΓ µν Ψ would be neutral under the U (1) R , and thus would require yet another scalar,thus making the dimension higher than 3. Note that there is a factor of 1 /
26t is now a straightforward exercise to show that M Iα, Jβ = V abIα V Jβ, ab = − M IJ ˜ M αβ (6.4)being M and ˜ M the SL (6 , R ) and SL (2 , R ) metrics respectively. Explicitly M IJ = S KI S KJ , ˜ M αβ = ˜ S γα ˜ S γβ . (6.5)We will be interested in the scalar sector which is singlet under the SL (2 , R ), and thus we canjust take ˜ M αβ = δ αβ . Furthermore, we would like to consider a breaking of the gauge group down to SO (2) × SO (2) × SO (2). This symmetry breaking pattern can be encoded in the scalar matrix MM IJ = diag (cid:16) X , X , X , X , X , X (cid:17) (6.6)Since this is an SL (6 , R ) matrix, its determinant must be one, which implies X X X = 1 . (6.7)Let us now consider the gauge field kinetic term. This can be written as L ⊃ − F µν, ab F µν, ab = − F µν, IJ F µνKL V IJ, ab V KL, cd Ω ac Ω bd . (6.8)Following [29], the ( ,
1) part of the scalar vielbein is written as V I, ab = 18 (cid:16) Γ KL (cid:17) ab U IJKL , U
IJKL = 2 S [ I [ K S J ] L ] . (6.9)After some algebra, the kinetic term for the gauge fields becomes L ⊃ − F µν, IJ F µνKL M IL M JK . (6.10)Taking into account our diagonal form for M we have L ⊃ − X F µν, F µν − X F µν, F µν − X F µν, F µν . (6.11)This exhibits in an explicit manner the symmetry breaking pattern, which leaves an unbroken U (1) gauge group whose three gauge potentials are {B , B , B } . Defining B µ, = 12 B µ , B µ, = 12 B µ , B µ, = 12 B µ , (6.12)the kinetic term for the gauge fields becomes L ⊃ − X X i F iµν F µν i , F iµν = ∂ [ µ B iν ] . (6.13)Finally, let us consider the Lagrangian for the two-forms. We consider the following subsector: A i − µν = A i µν , A i − µν = −A i µν , i = 1 , , . (6.14)It is convenient to define the complexified 2-forms A i (2) = A i − + i A i − . (6.15)Then, the Lagrangian becomes L ⊃ i n ǫ µνρστ ¯ A i (2) µν ∂ ρ A i (2) στ − i ǫ µνρστ B iµ A i (2) νρ ¯ A i (2) στ − i √ g X i A i (2) µν ¯ A i µν (2) o . (6.16)27 .2 General STU black holes N = 8 gauged Supergravity is expected to arise as a consistent truncation from ten-dimensional typeIIB theory (even though this truncation has not been explicitly found yet). Under the light of the ten-dimensional interpretation, the above symmetry breaking pattern stands for spinning branes in flat R .The maximal torus in SO (6) is U (1) , therefore there are at most three independent angular momenta.From the 5d perspective, the gauge fields B i are precisely associated with the U (1) symmetries generatedby three possible spins. In particular, electric charge under these gauge fields uplifts to 10d as angularmomenta in the corresponding Cartan plane of rotation.The family of black hole solutions with three angular momenta arising from spinning D3 branes areprecisely the STU black holes of which a particular one-charge case was given in section 2. The generalsolution for three different charges that is naturally embedded in N = 8 SO (6) gauged Supergravityis [37] ds = − f H − / dt + H / f − dr + H / r L d~x ,B i = Q i √ mr h + Q i − Q i √ mr + Q i , X i = H − i H / , (6.17) f = r L H − mr , H = H H H , H i = 1 + Q i r . For Q = Q , Q = Q = 0 this solution becomes precisely the solution (2.8) discussed in previous sections,upon defining X = X − / . In the following, we will be mostly interested in the case Q = Q = Q = Q (with no loss of generality we can assume Q ≥ r → r h r so that the horizonis located at r h = 1 and define Q = r h ˜ Q and m = r h ˜ m . Then, the horizon equation f = 0 implies˜ m = (1 + ˜ Q ) . In terms of the new variables the Hawking temperature takes the form T = (2 − ˜ Q ) r h q Q πL . (6.18)On the other hand, the three electric field potentials become equal. For large r , they behave as B ∼ − ˜ Q q Q r h L + ˜ Q (1 + ˜ Q ) / r h L r + · · · (6.19)Thus, using the usual AdS/CFT prescription, we can read off the charge densityˆ ρ = ˜ Q (1 + ˜ Q ) / π L r h . (6.20)Thus T = (2 − ˜ Q )(2 π ) / L ˜ Q / ˆ ρ / . (6.21)The temperature vanishes as ˜ Q → √
2. In terms of the original variables this corresponds to Q =2 L √ m/ (3 √ Q > √ Q is restricted to be in the interval ˜ Q ∈ [0 , √ Q corresponds to r h = r π / L ˆ ρ / . (6.22)It is worth stressing that this value is not zero. Therefore, the horizon has a finite area in this limit.28 .3 Condensation N = 8 gauged SUGRA contains two-forms which could potentially condense, leading to superconductingtransitions of the same nature as in section 4, between R-charged black holes and black holes with 2-formhair. Thus it provides an interesting framework to study these type of transitions in the more generalcontext of a three-charge STU black hole. Just as we did earlier, we shall consider a more general modelwhere the 2-form has electric charge q (with q = 1 being the case of N = 8 SUGRA) and search for azero mode in the bald black hole background with suitable properties. In this manner we shall determinethe critical temperature as a function of the electric charge of the 2-form field.In order to fix our conventions, it is useful to look at the scalar potential and compare it with thepotential used in the derivation of the STU black hole [37]. As shown in [29], the scalar potential for N = 8 gauged SUGRA in the SL (6 , R ) sector can be written as V = 12 (cid:16) W ab − W abcd (cid:17) , (6.23)where W ab = W cacb and W abcd = 18 ǫ αβ η IJ (cid:16) Γ Lσ (cid:17) he (cid:16) Γ Kγ (cid:17) fg S KI S LJ ˜ S γα ˜ S σβ . (6.24)After some algebra, one can verify that the potential can be compactly written in terms of M as V = 12 (cid:16) ( η IJ M IJ ) − η IJ M JK η KS M SI (cid:17) . (6.25)Particularizing for the explicit expression for M in (6.6), we have V = 4 X i =1 , , X i , X X X = 1 , (6.26)which coincides with the standard expression for the scalar potential for STU black holes (see e.g. [37,41]),thus fixing our conventions. As a warm up, let us start by considering the case of a single charge Q , which corresponds to the N = 4gauged SUGRA studied in previous sections. Choosing with no loss of generality the rotation plane alongthe 1 − A is X . The potential in turn becomes V = 4 (cid:16) X + 2 p X (cid:17) , (6.27)which agrees with (2.1) with the identification X = X − ( X = X = X ). Using the expression forthe X i fields in the STU black hole (6.17) we have X = H / . One also checks that the part of theLagrangian containing the 2-form agrees with (2.1). In this case the gauge sector only comprises thegauge potential B , whose kinetic term reads − X F µν F µν , (6.28)in agreement with (2.1). 29 .3.2 Case of three equal charges Let us now turn to the case of interest Q = Q = Q = Q . In this case the condition X X X = 1implies that X = X = X = 1. As in previous sections, it is instructive to consider a more generalLagrangian where the 2-form has electric charge q . The three two-forms A i (2) have the same Lagrangian, L ⊃ i n ǫ µνρστ ¯ A (2) µν ∂ ρ A (2) στ − iq ǫ µνρστ B µ A (2) νρ ¯ A (2) στ − i √ g A (2) µν ¯ A µν (2) o (6.29)We turn on, say A , and adopt the same ansatz for the 2-form components as in section 3.1. We findthe equation a ′′ + (cid:16) h ′ h + b ′ b − B ′ (cid:17) a ′ + q L e B − A Φ h a − L h e B a = 0 . (6.30)Defining again Q = r h ˜ Q , ¯ m = r h ˜ m , z = r h /r , and p ( z ) = z a ( z ), the relevant equation is now p ′′ ( z ) + F ( z ) p ′ ( z ) + G ( z ) p ( z ) = 0 , (6.31)with F = − (cid:16) Q z + 3 (1 + ˜ Q ) z − ˜ Q z (1 + 3 ˜ Q + 5 ˜ Q + ˜ Q ) − ˜ Q z (cid:17) z (1 − z ) (1 + ˜ Q z ) (1 + (1 + 3 ˜ Q ) z − ˜ Q z ) ,G = ˜ f (1 − z ) (1 + ˜ Q z ) (1 + (1 + 3 ˜ Q ) z − ˜ Q z ) , (6.32)and˜ f = z (cid:16) z ˜ Q − (cid:16) Q + 1 (cid:17) z − (cid:17) (cid:16) ˜ Q (cid:16)(cid:16) ˜ Q z + (cid:16) ˜ Q + 5 ˜ Q + 3 (cid:17) z − Q − (cid:17) ˜ Q + z − (cid:17) − (cid:17) + q ˜ Q (cid:16) Q (cid:17) (cid:0) − z (cid:1) (cid:16) z ˜ Q (cid:17) . (6.33)The large r (small z ) asymptotic of the two-form is controlled by the mass parameter, and it is the sameas in the case studied in section 4, p ( z ) → r h O + O z r h + · · · (6.34)Thus, the normalizable zero-mode must satisfy p ( z = 0) = 0. In order to search for such a zero mode,we can numerically solve (6.32) as a function of q and ˜ Q ∈ [0 , √
2] and determine ˜ Q = ˜ Q ( q ) from thecondition p ( z = 0) = 0. Using the formula (6.21) for the temperature we then find T = T ( q ). The resultsare shown in figure 12.The figure shows that there are hairy black holes provided q > q min ≈ .
8. This excludes the case of N = 8 SO (6) gauged Supergravity where q = 1. In other words, the charge of the 2-form fields of N = 8 SO (6) SUGRA is not large enough to drive to an instability. The modified theory with arbitrary U (1)charge q has critical temperatures ranging from 0 to infinity.For the minimal charge q min ≈ . Q → √ q = q min has a normalizable zero-mode at zerotemperature. Therefore, the theory has a quantum critical point. It would be interesting to studyits properties (for s -wave holographic superconductors, studies of quantum critical points have recentlyappeared in [12]). 30 - charge T e m p e r a t u r e Figure 12: Critical temperature as a function of the 2-form U (1) charge q . We have seen that the theory described by the generalized Lagrangian inspired in N = 8 gauged SUGRA,where the 2-form field has electric charge q , undergoes a superconducting phase transition provided q & .
8. We now turn to a holographic interpretation along the lines of the N = 4 case discussed insection 5.In the three-charge case the mass of the 2-form becomes a constant equal to one. Following [43], weconclude that it corresponds to a dimension three operator transforming as an antisymmetric Lorentztensor.The natural operators which are dual to A i (2) are O iµν = Tr (cid:2) Φ i F + µν (cid:3) (6.35)They all have the same charge under the diagonal U (1) subgroup in U (1) ∈ SO (6). In N = 4 SYMthis operator has R -charge equal to one. In our modified theory it must have R-charge q in order tomatch quantum numbers of the two-form fluctuation. Like in the N = 4 case discussed in section 5, thisidentification can be motivated by considering the non-abelian theory Born-Infeld theory with chemicalpotentials representing the rotating D3 branes. In this paper we have investigated holographic p -wave superconductors emerging from the condensationof bulk 2-form fields in models which are a slight modification of N = 4 , U (1) charge q . The gaugedsupergravity setup is particularly attractive given the explicit knowledge of the dual field theory givenin terms of Super Yang-Mills theory with chemical potentials. We found that condensation requires, atleast for some specific ans¨atze, a minimal value of the charge q which is, unfortunately, above the valueof the U (1) charge in supergravity. The model with general charge q describes an alternative holographicrealization of p -wave superconductivity that complements previous approaches [22, 23, 33–36].There are a number of very interesting open problems: • One can similarly consider N = 8 SO (8) four-dimensional gauged supergravity as a framework forthe study of p -wave superconducting models in three dimensions based on condensation of chargedvector fields. It would be interesting to see if also in this case condensation requires minimal chargeswhich are above the gauged supergravity values.31 One may also study condensation of charged vector fields in the context of the N = 4 and N = 8five-dimensional gauged supergravities discussed in this paper. In particular, in the N = 4 case onecould consider STU black holes charged with respect to the U (1) ⊂ SU (2) and look for condensationof the W vector bosons. • It would be interesting to elucidate more detailed condensed matter aspects of the present models.In particular, to clear up the anisotropic properties of the conductivity.
Acknowledgements
We would like to thank Oren Bergman, Diederik Roest and Oscar Varela for discussions. We also thank ananonymous referee for correcting an important point in an earlier version of this paper. F.A. is supportedby a MEC FPU Grant No.AP2008-04553. D.R-G. is supported by the Israel Science Foundation undergrant no. 392/09. He also acknowledges support from the Spanish Ministry of Science through theresearch grant no. FPA2009-07122 and Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042). J.R. acknowledges support by MCYT Research Grant No. FPA 2007-66665 and Generalitat deCatalunya under project 2009SGR502. 32 eferences [1] S. S. Gubser, “Breaking an Abelian gauge symmetry near a black hole horizon,” Phys. Rev. D ,065034 (2008) [arXiv:0801.2977 [hep-th]].[2] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, “Building a Holographic Superconductor,” Phys.Rev. Lett. , 031601 (2008) [arXiv:0803.3295 [hep-th]].[3] S. S. 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