Holographic phase transitions from higgsed, non abelian charged black holes

We find solutions of a gravity-Yang-Mills-Higgs theory in four dimensions that represent asymptotic anti-de Sitter charged black holes with partial/full gauge symmetry breaking. We then apply the AdS/CFT correspondence to study the strong coupling regime of a $2+1$ quantum field theory at temperature $T$ and finite chemical potential, which undergoes transitions to phases exhibiting the condensation of a composite charged vector operator below a critical temperature $T_c$, presumably describing $p+ip/p$-wave superconductors. In the case of $p+ip$-wave superconductors the transitions are always of second order. But for $p$-wave superconductors we determine the existence of a critical value $\alpha_c$ of the gravitational coupling (for fixed Higgs v.e.v. parameter $\hat m_W$) beyond which the transitions become of first order. As a by-product, we show that the $p$-wave phase is energetically favored over the $p+ip$ one, for any values of the parameters. We also find the ground state solutions corresponding to zero temperature. Such states are described by domain wall geometries that interpolate between $AdS_4$ spaces with different light velocities, and for a given $\hat m_{W}$, they exist below a critical value of the coupling. The behavior of the order parameter as function of the gravitational coupling near the critical coupling suggests the presence of second order quantum phase transitions. We finally study the dependence of the solution on the Higgs coupling, and find the existence of a critical value beyond which no condensed solution is present.


Introduction
In recent years the application of AdS/CFT or more generally gauge/gravity correspondence [1] [2] [3] to the study of condensed matter physics has attracted a lot of attention, providing in particular gravitational descriptions of systems exhibiting superconductor/superfluid phases [4] [5]. Since in condensed matter physics we are typically dealing with systems at finite charge density and temperature, in the context of the AdS/CFT correspondence the dual gravity descriptions should be given in terms of gravitational models with a negative cosmological constant which admit charged black holes as vacuum solutions. In fact, a charged black hole naturally introduces a charge density/chemical potential and temperature in the quantum field theory (QFT) defined on the boundary using the gauge/gravity correspondence. This set-up allows in particular to study phase transitions and construct phase diagrams in parameter space.
The simplest model is provided by an Einstein-Maxwell theory coupled to a charged scalar field. This is in the framework of the AdS/CFT correspondence dual to a scalar operator which carries the charge of a (global) U(1) symmetry. It has been shown that a charged black hole solution, interpreted as the uncondensed phase, becomes unstable and develops scalar hair at low temperature breaking the U(1) symmetry near the black hole horizon [6] [7]. This phenomenon in general may be interpreted as a second order phase transition between conductor and superconductor phases, interpretation that is supported by analyzing the behavior of the conductivity in these phases [5]. There were also studied vortex like solutions that describe type II holographic superconductors [8] [9] and more recently, spatially anisotropic, abelian models of superconductors [10].
Soon after these "s-wave" holographic superconductor models were introduced, holographic superconductors models with vector hair, known as p-wave holographic superconductors, were explored first in [11] and [12]. The simplest example of p-wave holographic superconductors may be provided by an Einstein-Yang-Mills theory with SU(2) gauge group and no scalar fields, where the electromagnetic gauge symmetry is identified with an U(1) subgroup of SU (2). The other components of the SU(2) gauge field play the role of charged fields dual to some vector operators whose non-zero expectation values break the U(1) symmetry leading to a phase transition in the dual field theory.
More recently solutions to gravity-matter field equations where both scalar and vector order parameters are present were considered,; they describe systems where competition/coexistence of different phases are present [13]- [16].
Regular, self-gravitating dyonic solutions of the Einstein-Yang-Mills-Higgs (EYMH) equations asymptotic to global AdS space, were constructed time ago in [17] and [18]. They were extended to dyonic black hole solutions in [19] and [20], where they were interpreted as describing a so-called p + ip-wave superconductor (isotropic) system at finite temperature in the condensed phase. The purpose of the present work is to find more general black hole solutions of EYMH in asymptotically AdS 4 space with finite mass and electric charge density, to interpret them via the gauge/gravity duality as describing phases of a strongly coupled field theory, and construct the corresponding phase diagrams. 1 . We start by revisiting the analysis of [19] [20],verifying the existence of second order phase transitions all along the parameter space. It was found (see for example [21]) that some holographic systems pass from a second order phase transition as a function of the temperature in the non back-reaction limit to a first order when the gravitational coupling exceeds a certain value. Such a phenomenon occurs in holographic superfluids when the velocity is high enough [22] [23], and it was measured in certain types of superconductors [24] [25] [26]. We have found this kind of behaviour in our system in the anisotropic case, finding condensed solutions and construct the phase diagram. Furthermore we compute free energies and find that for any set of values of the free parameters that determines the solutions, the anisotropic phase is energetically favored over the isotropic phase, as conjectured in other contexts [27] [28] [29]. Third, we analyzed the zero temperature case, and for low enough gravitational coupling we find solutions which spontaneously break the U(1) symmetry and have zero entropy, and so describe the true ground state of the system. For gravitational couplings higher than a critical value the solution disappears, which is interpreted as a quantum phase transition.
The paper is organized as follows. In section 2 we present the model, writing the translational invariant ansatz for the fields and the equations of motion that reduces to a a nonlinear system of coupled ordinary differential equations. In section 3 we present generalities of the systems to be studied at non-zero temperature, in particular the analysis of the holographic map to be used. In section 4 we present the numerical results, including computations of free energies. Section 5 is devoted to the study of the T = 0 case and the description of the ground state of the superconductor, including the presence of quantum phase transitions as function of the gravitational coupling. A summary and discussion of the results is given in section 6. Finally two appendices are added, one containing the boundary expansions of the fields and other containing the equations of motion and free energy in other parameterization commonly used in the literature.

The model
We consider a gravity-Yang-Mills-Higgs system in a 1 + 3 dimensional space-time with Minkowski signature (−+++). We take SU(2) as the gauge group, with generators satisfying the algebra (ǫ 012 ≡ +1) [X a , X b ] = ǫ abc X c ; a, b, c = 0, 1, 2 (2.1) and the scalar field in the adjoint representation, H = H a X a . The full action to be considered is, where κ, e and λ are the gravitational, gauge and scalar couplings respectively, L is the AdS scale related to the negative cosmological constant through Λ = −3/L 2 , and H 0 > 0 defines the vacuum expectation value of the Higgs field (and so the boundary condition at infinity, see below in (2.14)). As it is well-known, the Gibbons-Hawking term S (GH) is necessary to have a well defined variational principle [30], where K ≡ ∇ a n a is the trace of the extrinsic curvature, and h and n the induced metric and normal vector on ∂M. The counter-term action S (ct) will be discussed in Section 4. The field strength F a M N and the covariant derivative D M acting on the Higgs triplet H a are defined as, Let us consider coordinates (x µ , y) and an ansatz preserving translational invariance in the coordinates {x µ , µ = 0, 1, 2}, In what follows it will be convenient to introduce the dimensionless coupling constants, The gravity equations of motion (e.o.m.) derived from (2.3) result, while that the matter e.o.m. are, where we have defined, In this paper we will consider the "BPS limit" λ 0 = 0, but conserving the crucial Higgs vacuum value H 0 > 0.

Boundary conditions
We will search for charged black hole solutions which present a horizon at y = y h where f (y h ) = 0. The associated Bekenstein-Hawking temperature of the black hole is given by, The ansatz (and e.o.m.) are invariant under the scale transformations, They allow to fix some normalization imposing the b.c., A(y), c(y) y→∞ −→ 1, in such a way that the x µ 's are identified with the minkowskian coordinates of the boundary QFT, and (2.10) with its temperature. Furthermore, when y h = 0 there exists another scaling symmetry 2 , (2.12) which allows to fix y h = 1. Since now on we will fix the position of the horizon in this way, having in mind that we have to consider only scale invariants quantities.
In [19] and [20] solutions to (2.7)-(2.8) with a horizon and asymptotically AdS 4 were studied. More specifically, there were found solutions with K 1 = K 2 = K and the following boundary conditions; near the horizon y → 1 + , while on the boundary y → ∞, where consistency with the e.o.m. and finiteness of K(y) fixes κ 1 to be, For more about the b.c. at the boundary, we refer the reader to the Appendix A. We will take the b.c. (2.13)-(2.14) in this paper except in section 5 where the b.c. on the horizon will have to be modified. The bulk theory is invariant under the gauge group SU(2); however the b.c. on the Higgs field, H(y) y→∞ −→ 1 , breaks this invariance to the U(1) generated by X 0 . With respect to this gauge subgroup the electric charge density of a solution is defined as usual by, As we show in section 4, at fixed couplings (α,m W ) a general solution to (2.7)-(2.8) with the b.c. (2.13)-(2.14) is determined by J 0 , which is related to the U(1) chemical potential by, From (2.16) and (2.17) the standard asymptotic expansion follows, In what follows we will work with the dimensionless, scale invariant temperature, where a 0 and f 1 are defined in (2.13). A solution is determined by the three free parameters (α,m W , J 0 ), and so the temperature (through the coefficients a 0 , f 1 ) results a function of them 3 .
In the analytic solution to the equations (2.7)-(2.8) that preserves the U(1) X 0 symmetry matter fields take the form, where we imposed smooth behavior of the gauge field at the horizon which yields the condition J(1) = 0, see the last line in (2.13), and then fixed In what the metric functions concern, they correspond to the AdS Reissner-Nordström (AdS-RN) black hole, with temperature, 3 Solutions at T > 0: superconducting state When the "magnetic part" of the gauge field is non-trivial, i.e. K i (y) = 0 for some i = 1, 2, the solution breaks not only the U(1) X 0 invariance, but also the rotation invariance in the (x 1 , x 2 )-plane. According to the AdS/CFT dictionary this hair is interpreted as a spontaneous breaking of a global U(1) symmetry present in the boundary QFT, whose currents take an expectation value, Giving that the order parameter is dual to (a component of) the gauge field we are presumably modeling a p-wave superconductor [12]. The normal state of the superconductor is described by the AdS-RN solution (2.20)-(2.21); such solution is energetically favored until a critical temperature T c is reached; when T < T c the non symmetric, hairy solution gives rise to a superconductor phase. We remark that with the b.c. on the Higgs field we are breaking explicitly the gauge group from SU(2) to U(1) X 0 ; this yields a mass for the "W" gauge bosons, The problem is thus the following: can we find under this condition a solution with K i (y) = 0 that breaks spontaneously the U(1) X 0 ? In the boundary QFT this is then interpreted as the breaking of a global U(1) symmetry as it happens in superfluids and superconductors with weakly coupled photons. From here we identify T c with the critical temperature of the phase transition in the QFT. We will consider two cases.
• The isotropic case: K(y) ≡ K 1 (y) = K 2 (y) Although both gauge and rotational symmetries are broken by a hairy solution, a configuration (2.5) with K 1 (y) = K 2 (y) preserves the diagonal subgroup, . This configuration give rise to an energy-momentum tensor isotropic in the x 1 -x 2 plane; therefore the metric function c(y) must be a constant, even when back-reaction is taken into account.
This kind of configurations were first studied in [19]- [20], using relaxation methods.
We will re-obtain these solutions here for later use by using shooting methods.
• The anisotropic case: K(y) ≡ K 1 (y) ; K 2 (y) = 0 As stated above, a configuration with K 1 = 0 preserves the U(1) X 0 and spatial rotations. When K 1 (y) develops a non zero value the gauge symmetry U(1) X 0 breaks, and the condensate K 1 (y) X 0 dx 0 choose a direction x 1 as special. Then if we take into account back-reaction effects the system cannot support the condition g 11 = g 22 [12]. Due to this fact T x 1 x 1 = T x 2 x 2 and the function c(y) can not be a constant; in conclusion the system will be in an anisotropic phase.
In both cases the vacuum expectation value in the d = 3 field theory of the current operator O K , dual to the function K associated with the magnetic field in the bulk, follows from the identification, O K ∼ K 1 with K 1 defined in (2.14); K 1 = K 1 (T ) can be taken as the order parameter that describes the phase transition of the system. As discussed for different models [7]- [20] one can interpret this result by stating that a condensate is formed above a black hole horizon because of a balance of gravitational and electrostatic forces. From the asymptotic behavior in (2.14) we get the dimension From numerical solutions we conclude that a finite temperature continuous symmetry breaking transition takes place so that the system condenses at a critical temperature T c , as can be seen from the behavior of K 1 (T ) for T ≈ T c in figures 3, 4 and 5. Furthermore, we compare the free energies corresponding to both phases in Figure 8 and 9, finding that the anisotropic phase is favored, see [27] [28] [29] for related results.

Numerical Solutions
We analyzed numerically equations (2.7)-(2.8) and found solutions that satisfy the required b.c. (2.13)-(2.14) in a wide region of the parameter space, that lead to the phase diagram in Figure 1. Such solutions in the anisotropic case are shown in Figure 2.
Before presenting the results, we think is worth to spend a few words on the method used. As discussed in the appendix, after fixing some normalization and asking for finiteness the solution near the boundary admits the expansions in equation (A.1), and is determined by six constants, (F 1 , C 1 , J 0 , J 1 , K 1 , H 1 ). However the b.c. on the horizon impose five conditions. The first two come from the definition of the horizon and the regularity of the gauge field, They essentially fix the mass (∼ F 1 ) and the charge density (∼ J 1 ) of the black hole. The remaining three conditions fix (C 1 , K 1 , H 1 ) and are obtained from an analysis of the (singular) behavior of the e.o.m. near the horizon, Therefore the only additional free parameter that determines the solution is J 0 , i.e. the chemical potential (2.17). In practice we integrate the system from the horizon, where according to (4.1)-(4.2) the free parameters are, as defined in (2.13). These parameters are selected in such a way that the solution matches the conditions on the boundary (2.14), Figure 1: It is showed a typical phase diagram in the anisotropic case, at fixedm W . The analogous diagram in the isotropic case can be found in [20].
We have varied the temperature by moving J 0 at fixed parameters α andm W , and obtained the generic phase diagram (for a givenm W ) showed in figure 1. In regions I and II the system is in the condensed phase while that it is in the uncondensed or normal phase in region III. Below a certain value α c of the gravitational coupling the system experiments a second order phase transition at low temperatures. The transition becomes of first order for α > α c from region II to region III .
In figure 2 the fields are shown as functions of the coordinate y, at fixed J 0 andm W and for different α's. Forα ≈ 0.50964 a second horizon appears, as displayed from the curves corresponding to f (y)/y 2 . The uncondensed and condensed phases are separated by a critical straight line (see figure 1) on which the formation of the second horizon takes place, for a given gauge boson massm W .
In figure 3 the order parameter K 1 in the isotropic case is plotted as a function of the temperature for different values ofm W , at fixed gravitational coupling α = 0.7. In this case the transition is of second order independently of α, in agreement with [11]. Figures 4 and 5 shows the order parameter K 1 as function of T in the anisotropic case from two perspectives: at fixed α = 0.7 and varyingm W in figure 4, and at fixedm W = 0.4 and varying α in figure 5. From figure 4 it is seen that for α c ≈ 0.53 K 1 becomes multivalued, fact that signals the passage from second to first order phase transitions [21], as corroborated from the free energy computations of the next subsection. This phenomenon has been found recently in p-wave superfluids by studying the role of the back-reaction in the phase transitions [21] (for experimental results on first order phase transitions in superconductors, see [24][25] [26]). By comparing figures 3 and 5 it is observed that the temperature at which the order parameter becomes zero is the same in both cases, and that the critical temperature decreases whenm W increases, what can be interpreted as the presence of the Higgs field hinders the condensation. Furthermore, we have checked that near T c and for weak gravitational couplings α < α c , K 1 behaves like (T c − T ) 1 2 , indicating a second order phase transition with mean field exponent 1 2 as usually happens in holographic descriptions of critical systems in the limit of large number degrees of freedom.

The free energy
According to the AdS/CFT correspondence, the free energy of the QFT is given by, From (2.3) and using the e.o.m. the bulk contribution to the free energy density can be written as, The Gibbons-Hawking contribution is, Here we have introduced y ∞ to regularize the expressions since they present divergent terms. To this end we introduce a counter-term action [32] [33], which give rise to the following contribution to the free energy density, The total free energy density of the system f is then given by, We remark that in order to analyze the results the right thing to do is to work with the scale invariant free energy density,f ≡ κ 2 L 2 µ 3 f (4.11) Figures 6 and 7 show the evolution of the free energy density (4.11) with the mass of the gauge boson for two different values of α, in the isotropic and anisotropic cases respectively. Figure 6 displays the continuity off at the critical temperature (where the free energy density of the uncondensed phase intersects the curve of the condensed phase) for both values of α, for any m W , fact that indicates the second order character of the phase transition, as the behavior of K 1 in figure 3 suggested. In figure 7 instead it is observed the discontinuity in the first derivative of the free energy density at the critical temperature for α = 0.7 > α c = 0.53, for anym W , signaling a first order phase transition. In both cases the critical temperature decreases with growingm W , in agreement with the analysis of the order parameter behavior made above.
In figures 8 and 9 the free energy densities of the isotropic and anisotropic phases are compared, for two values α < α c and α > α c respectively. From them one see that the free energy density of the anisotropic phase is where the free energy of the uncondensed phase crosses that of the condensed one lesser than that in the isotropic phase; no matter the region where the value of α is, i.e. if first or second order phase transitions take place, the anisotropic phase is energetically favored over the isotropic one.    Figure 9: The free energy densities for the isotropic (black line) and anisotropic (blue) cases are plotted as function of the temperature for different values ofm W , at fixed α = 0.7. The red curve represents the free energy density of the uncondensed phase.

Solutions at T = 0: anisotropic case
In this section we will address the problem of quantum phase transitions in three dimensional p-wave superconductors (anisotropic case), i.e. transitions at T = 0, that to our knowledge was not considered before in the literature (see however [34] [35] [36] [37] for related studies in other settings).
It is known that when a charged AdS black hole is driven to a state of zero temperature it becomes extremal, but its entropy is different from zero and then it can not describe the ground state of the superconductor that we are presumably modeling holographically. To reach our goal the radius of the black hole needs to become null, to agree with the third law of thermodynamics and really describe the quantum ground state [27] [38]. So we must impose that y h = 0, i.e. the coordinate y ∈ [0, ∞). A very important thing from a technical point of view is that while the asymptotic behavior of the fields is as in (2.14), the expansions near the horizon drastically change with respect to the T > 0 case. At leading order the (non analytical) behavior of the fields for y → 0 + is,  Figure 10 shows the fields for different values of the gravitational coupling. Such solutions describe the quantum ground state of the superconductor in the condensed phase. It is worth to note that, for fixedm W and J 0 , the value ofα does not coincide with the value of the gravitational coupling for the system when T > 0. In the example showed (m W = 1.6, J 0 = 6) we obtainα| T >0 = 0.5094 andα| T =0 = 0.3018, see figure 2 and 10 respectively. Aboveα the only solution that exists is AdS space.

Conclusions and outlook
In this paper we have investigated four dimensional solutions of black holes with non-abelian, SU(2) hair introduced by Yang-Mills gauge bosons and a non trivial Higgs field in the adjoint representation, which triggers the breaking of the gauge symmetry to a U(1) subgroup under which the black hole is charged.
In the spirit of the AdS/CFT correspondence, the symmetric solution given by the AdS-RN black hole when the temperature is positive and AdS when T = 0, describes the uncondensed phase of the dual three dimensional QFT. A solution with non-abelian hair generically breaks the U(1) gauge symmetry together with the rotational symmetry, and is interpreted as describing a condensed phase of the QFT. The order parameter is the coefficient of the leading order term of the magnetic component of the gauge field, and thus the systems described are generically termed p-wave superfluids/superconductors. We have considered two cases. The isotropic case that describes p + ip-wave superconductors where the diagonal subgroup of U(1) gauge × SO(2) rot is preserved, and the anisotropic case where no symmetry is preserved. In both cases we get phase transitions at critical temperatures that decrease when the gravitational coupling grows and in the case of anstropic superconductors the phase transitions become of first order for large gravitational couplings [21].
We also found solutions that describe the zero entropy ground state of the p-wave superconductor, showing the existence of phase transitions from the normal phase (described by AdS space) to this condensed phase, that is present below a certain value of the gravitational couplingα. The occurrence of AdS space near the horizon (with the the same scale as the AdS in the boundary but different light velocity) presumably indicates that there is an emergent scale invariance in the T = 0 limit [21] [22].
A very relevant fact illustrated in figures 8 and 9 is that below the critical temperature in all the parameter space we found that the free energy density of the anisotropic solution is lower than that of the isotropic one, indicating that the p-wave superconductor phase is more stable that the one corresponding to the p + ip-wave superconductor.
It is not difficult to see from the e.o.m. that a configuration where a v.e.v. (3.1) is present necessary implies a non-trivial Higgs field; however the v.e.v. of its dual scalar operator O(x), O(x) ∼ H 1 (6.1) does not indicate any spontaneous breaking in view of the presence of the source H 0 . This situation differs from that in reference [13] [16], where the source is put to zero and systems that present competition between phases representing s-wave and p-wave superconductors are considered 5 . We stress that, although they share some similar characteristics, the presence of the Higgs fields with the non trivial b.c. | H(∞)| = H 0 > 0, introduces a scale that makes our systems different from those considered in precedence since [12] [34], in which the scalars were not present. First from the obvious fact that the system is larger and more complex; in particular we have three free parameters (α,m W , J 0 ) and, among other things, the dimension (3.3) of the order parameter remains arbitrary. Instead, in EYM systems where the Higgs field is not present, the temperature for example is a function of just one parameter α ≡ κ e L [21]. Second and more important, from the QFT that the systems we have considered are presumed to be dual from the holographic point of view. This fact can be elucidate by studying the transport properties of the system, i.e. the conductivities. The system of couple second order differential equations to solve results, even when back-reaction effects are not considered, a very intricate one. We hope to report results in this direction in future work [39].

Acknowledgments
We would like to thank Nicolás Grandi and Ignacio Salazar Landea for encouragement and continuous support, and Borut Bajc and Jorge Russo for careful reading of the manuscript and useful comments. This work was supported in part by CONICET, Argentina.