Spontaneous Momentum Dissipation and Coexistence of Phases in Holographic Horndeski Theory

We discuss the possible phases dual to the AdS hairy black holes in Horndeski theory. In the probe limit breaking the translational invariance, we study the conductivity and we find a non-trivial structure indicating a collective excitation of the charge carriers as a competing effect of momentum dissipation and the coupling of the scalar field to Einstein tensor. Going beyond the probe limit, we investigate the spontaneous breaking of translational invariance near the critical temperature and discuss the stability of the theory. We consider the backreaction of the charged scalar field to the metric and we construct numerically the hairy black hole solution. To determine the dual phases of a hairy black hole, we compute the conductivity. When the wave number of the scalar field is zero, the DC conductivity is divergent due to the conservation of translational invariance. For nonzero wave parameter with finite DC conductivity, we find two phases in the dual theory. For low temperatures and for positive couplings, as the temperature is lower, the DC conductivity increases therefore the dual theory is in metal phase, while if the coupling is negative we have the opposite behavior and it is dual to an insulating phase. We argue that this behavior of the coupling of the scalar field to Einstein tensor can be attributed to its role as an impurity parameter in the dual theory.


Abstract
We discuss the possible phases dual to the AdS hairy black holes in Horndeski theory. In the probe limit breaking the translational invariance, we study the conductivity and we find a non-trivial structure indicating a collective excitation of the charge carriers as a competing effect of momentum dissipation and the coupling of the scalar field to Einstein tensor. Going beyond the probe limit, we investigate the spontaneous breaking of translational invariance near the critical temperature and discuss the stability of the theory. We consider the backreaction of the charged scalar field to the metric and we construct numerically the hairy black hole solution. To determine the dual phases of a hairy black hole, we compute the conductivity. When the wave number of the scalar field is zero, the DC conductivity is divergent due to the conservation of translational invariance. For nonzero wave parameter with finite DC conductivity, we find two phases in the dual theory. For low temperatures and for positive couplings, as the temperature is lower, the DC conductivity increases therefore the dual theory is in metal phase, while if the coupling is negative we have the opposite behavior and it is dual to an insulating phase. We argue that this behavior of the coupling of the scalar field to Einstein tensor can be attributed to its role as an impurity parameter in the dual theory.
This behaviour of the scalar was used in [18] to holographically simulate the effects of a high concentration of impurities in a material. If there are impurities in a superconductor then the pairing mechanism of forming Cooper pairs is less effective because the quasiparticles are loosing energy because of strong concentration of impurities. Then it was found that as the value of the derivative coupling is increased the critical temperature is decreasing while the condensation gap is decreasing faster than the temperature. Also it was found that the condensation gap for large values of the derivative coupling is not proportional to the frequency of the real part of conductivity which is characteristic of a superconducting state with impurities. Also a holographic superconductor was constructed in [33] with a scalar field coupled to curvature.
Real materials, which are composed of electrons, atoms and so on, except their possible impurities, are usually doped. This has important consequences in the properties of these materials and one of these is that they do not possess spatial translational invariance, and so momentum in these systems is not conserved. An example of such an observable is the conductivity. Under an applied field, the charge carriers of this system will accelerate indefinitely if there is no way for them to dissipate momentum, leading to an infinite DC conductivity. If, however, spatial translational invariance is broken and therefore momentum dissipation is possible, the DC conductivity is finite and the delta function peak at the origin of the AC conductivity spreads out. Therefore, for the holographic theories it is important to incorporate the dissipation of momentum. In the literature there are studies of the transport properties of holographic theories which dissipate momentum. The simplest way is to explicitly break the translational symmetry of the field theory state [34]. Other ways is by coupling to impurities, introducing a parametrically large amount of neutral matter [35,36], or by breaking translation invariance by introducing a mass for the graviton field [37].
In the holographic superconductor practically there is a superconducting state which has a nonvanishing charge condensate, and a normal state which is a perfect conductor. The consequence of this is that even in the normal phase the DC conductivity (ω = 0) is infinite, inspite of the proximity effect in the interface between the superconductor and the normal phase [38]. This is a consequence of the translational invariance of the boundary field theory, because the charge carriers do not dissipate their momentum, and accelerate freely under an applied external electric field. Therefore one is motivated to introduce momentum dissipation into the holographic framework, breaking the translational invariance of the dual field theory.
In [39] black hole solutions were constructed which were holographically dual to strongly coupled doped theories with explicitly broken translation invariance. The gravity theory was consisted of and Einstein-Maxwell theory coupled to a complex scalar field with a simple mass term. Black holes dual to metallic and to insulating phases were constructed and a their properties were studied. Charged black brane solutions with translation symmetry breaking were considered in [40] and the conductivity at low temperatures was studied.
A generalization of [35] introducing a scalar field coupled to curvature was presented in [41]. An Einstein-Maxwell theory was studied in which except the scalar fields present in [35], scalar fields coupled to Einstein tensor were also introduced. The holographic DC conductivity of the dual field theory was studied and the effects of the momentum dissipation due to the presence of the derivative coupling were analysed. In [42] the thermoelectric DC conductivities of Horndeski holographic models with momentum dissipation was studied in connection to quantum chaos. It was found that the derivative coupling represents a subleading contribution to the thermoelectric conductivities in the incoherent limit.
To explain the generation of FFLO states [43,44] an interaction term between the Einstein tensor and the scalar field is introduced in a model [45,46] with two U(1) gauge fields and a scalar field coupled to a charged AdS black hole. In the absence of an interaction of the Einstein tensor with the scalar field, the system possesses dominant homogeneous solutions for all allowed values of the spin chemical potential. Then calculating the DC conductivity it was found that the system exhibits a Drude peak as expected. However, in the presence of the interaction term, at low temperatures, the system is shown to possess a critical temperature for a transition to a scalar field with spatial modulation as opposed to the homogeneous solution.
In this work we spontaneously break the translational invariance in an Einstein-Maxwell-scalar gravity theory in which the charged scalar field is also coupled kinetically to Einstein tensor and we study the possible phases generated on the dual boundary theory. In the probe limit we calculated the conductivity and we found that depending on the parameter of the translational symmetry breaking and on the coupling of the scalar field to curvature we have on the dual theory a coexistence of phases on the boundary theory. Then we go beyond the probe limit and considering the fully backreacted problem we constructed numerically a hairy black hole solution. To determine the phases of the dual theory to the hairy black hole, we compute the conductivity. When the wave parameter of the scalar is zero, the DC conductivity is divergent due to no mechanism of momentum relaxation, which is dual to ideal conductor. While for nonzero wave parameter with finite DC conductivity, we found two phases in the dual theory. For low temperatures, we found that for positive couplings, as we lower the temperature the DC conductivity increases therefore the dual theory is in metal phase, while if the coupling is negative we have the opposite behavior and it is dual to an insulating phase. We attributed this behavior of the coupling of the scalar field to Einstein tensor to that this coupling is connected to the amount of impurities present in the theory.
The work is organized as follows. In Section II we set up the problem. In Section III we studied the probe limit of the theory. In Section IV we discussed the stability of the theory and calculated the critical temperature. In Section V we calculated numerically the fully backreacted black hole solution and studied the DC conductivity and finally in Section VI are our conclusions.

II. THE FIELD EQUATIONS IN HORNDESKI THEORY
The action in Horndeski theory, in which a complex scalar field which except its coupling to metric it is also coupled to Einstein tensor reads where and q, m are the charge and the mass of the scalar field and β is the coupling of the scalar field to Einstein tensor of dimension length squared. For convenience we define Subsequently, the field equations resulting from the action (1) are where, and and the Klein-Gordon equation is while the Maxwell equations read We note that the matter action in (1) has a 1/q 2 in front, so the backreaction of the matter fields on the metric is suppressed when q is large and the limit q → ∞ defines the probe limit. When q goes to zero, we have neutral scalar field and it has no coupling with the Maxwell field.

III. SIGNATURE OF BREAKING THE TRANSLATION SYMMETRY IN THE PROBE LIMIT
We firstly focus on the probe limit in the above setup, in which the Einstein equations admit the planar Schwarzschild AdS black hole solution By setting Ψ = Ψ(z) and A = φ(z)dt in [18] a holographic superconductor was built in the probe limit in the background of the black hole (12). In this section we will break the translational invariance leading to momentum dissipation and calculate the conductivity in the probe limit. To this end, we consider the ansatz for the matter fields as where τ indicates the strength of the breaking of the translational invariance. Then under the metric (12), the equations of motion for ϕ(r) and φ(r) become φ Because of the breaking of the translational invariance in both the scalar field and vector potential equations we have an extra term multiplied by τ . We note that when τ = 0 and β = 0, the system recovers our previous work [18]. And when τ = 0 and β = 0, this model goes back to the minimal holographic superconductor of [5].
Near the boundary, the matter fields behave as and ∆ is where µ and ρ are interpreted as the chemical potential and charge density in the dual field theory, respectively. The coefficients ϕ − and ϕ + according to the AdS/CFT correspondence, correspond to the vacuum expectation values x of an operator O dual to the scalar field. We can impose boundary conditions that either ϕ − or ϕ + vanishes. The equations (14) and (15) can be solved numerically by doing integration from the horizon to the infinity by taking regular condition near the horizon. Then we extract the data near the boundary. As we decrease the temperature to a critical value T c , there exists a phase transition from normal black to superconducting black hole. We choose the boundary condition ϕ − = 0. So the solutions correspond to vanishing ϕ + for normal phase and non-vanishing ϕ + for superconducting phase, respectively. We show the critical temperature T c in Fig 1. We can see that as the derivative coupling β and the momentum dissipation parameter τ are increasing, the critical temperature is decreasing, indicating that the system is harder to enter the superconducting phase. This behaviour can also be seen in Fig. 2 which shows the condensation gap at the position x = 0. For a fixed value of the derivative coupling β, as the temperature of the system decreases, the strength of the condensation gap is enhanced as the dissipation parameter τ is increased.
To compute the conductivity in the dual CFT as a function of frequency, we need to solve the Maxwell equation for fluctuations of the vector potential A x . The Maxwell equation at zero spatial momentum and with a time dependence of the form e −iωt gives We will solve the perturbed Maxwell equation with ingoing wave boundary conditions at the horizon, i.e., A x ∝ f −iω/3r h . The asymptotic behaviour of the Maxwell field at large radius is x r + · · · . Then, according to AdS/CFT dictionary, the dual source and expectation value for the current are given by x , respectively. Thus, the conductivity is read as We first fix the derivative coupling β = 0.01 and we vary the dissipative parameter τ . We see in Fig. 3 (left panel) that when τ = 0 we do not have any peak. As the τ is increasing, we have a clear formation of peaks. In the right panel of Fig. 3 we see that as we go beyond the critical temperature T c in the supercondacting phase there is a clear formation of peaks. Finally in Fig. 4 we fix the dissipative parameter and we vary the derivative coupling. We see that larger values of β give a clear peak at larger ω.
This behaviour is interesting showing that the AC conductivity shows a non-trivial structure indicating a collective excitation of the charge carriers. Similar behaviour was observed in [47] in which the translational symmetry is broken by massive graviton effects. A study of competition of various phases was carried out in [48] using as control parameters the temperature and a doping-like parameter. This coexistence of phases can be seen more clearly if you are away from the critical temperature where due to the proximity effect there is a leakage of Cooper pairs to the normal phase. There are two competing effects. The first one is the momentum dissipation which restricts the kinetic properties of the charge carries and in the same time the derivative coupling which acts as a doping parameter magnifying the effect of the momentum dissipation. To have a better understanding of these effects in the next sections we will study the conductivity in the fully backreacting theory.

IV. SPONTANEOUS BREAKING OF TRANSLATIONAL INVARIANCE NEAR THE CRITICALITY
In this section we will discuss the critical temperature below which the black hole solution will develop hair and the way we break spontaneously the translational invariance near the criticality. Our aim is to go beyond the probe limit and find a fully backreacted hairy black hole solution. In general the resulted hairy solution could have two possible sources of instabilities, a negative mass for the scalar field and from the propagating modes of the scalar field coming from its kinetic energy. In the context of the AdS/CFT correspondence the first one is known as the  Breitenlohner-Freedman (BF) bound [50,51]. However, this bound corresponds to a conformally coupled scalar in the background of a Schwarzschild AdS black hole, and arises in contexts in which the AdS 4 /CF T 3 correspondence is embedded into string theory and M theory. We should note that our Lagrangian resulted from the action (1) is not clear how it arises from string or M theory. On the other hand even if the BF bound is satisfied it does not guarantee the nonlinear stability of hairy black holes under general boundary conditions and potentials as it was discussed in [12]. For the other source of possible instabilities in [20,22] fully backreacted black hole solutions were found in the presence of the derivative coupling β. However, these solutions exists only in the case of a negative sign of the derivative coupling while if the coupling constant β is positive, then the system of Einstein-Maxwell-Klein-Gordon equations is unstable and no solutions were found. In [32] a very small window of positive β was shown to be allowed. For positive derivative coupling the stability of the Galileon black holes was investigated but there is no any conclusive result. For example in [52,53] the black hole quasinormal modes in a scalar-tensor theory with the scalar field coupled to the Einstein tensor were calculated. In the following we will calculate the BF bound for the action (1) and in the next section we will investigate the effects of the derivative coupling β of both signs.
We consider the following ansatz for the metric, Maxwell and scalar field respectively where the functions f, V 1 , V 2 , φ, ϕ are all to be determined. In the normal phase, we have Ψ = 0 and the usual Reissner-Nordström black hole is a solution to the field equations where The Hawking temperature is The near horizon extremal solution is the usual AdS 2 × R 2 where the extremal horizon is located at r h = µ 2 √ 3 and the AdS 2 radius is L 2 2 = L 2 6 . From the Klein-Gordon equation (10) in the background of (20), we see that the scalar Ψ gets an effective momentum dependent mass Further considering the expression of (20), we deduce To fulfill the above condition, we require which obviously decreases the effective mass. In order to find the critical temperature, we have to solve the Klein-Gordon equation (10) with the ansatz Ψ = e ikx ϕ(r) in the background of the Reissner-Nordström black hole. We have to look for a normalizable solution of the field ϕ where the source of ϕ is zero but the vacuum expectation value is not. Since the boundary condition of ϕ is the same as shown in (16), we will find solutions with ϕ − = 0 and ϕ + = 0. We then integrate the equation numerically shooting from the horizon and we search for a normalizable solution. In particular, we fixed the value of the derivative coupling to β = −1/4, and find out the critical temperature T c (β) corresponding to different momentum. The results are shown in Fig. 5. We see that the lowest critical temperature is very small, but certainly happens at a non-zero value of momentum. As k becomes larger, T c increases monotonously, which means that the maximum critical temperature may go to be infinity. The critical temperature does not have the bell-shaped behavior as it was expected [54]. However, to find a finite critical temperature and produce a bell-shaped diagram, a method of regularization was proposed in [55] where another term with higher derivatives of the scalar field coupled to Einstein tensor was added into the action (1). Then by solving the radial Klein-Gordon equation by setting Ψ(r, x) = ϕ(r) cos(kx), the authors concluded that when β µ 2 = β µ 4 = 0, the maximum critical temperature T c is found at k = 0, which shows that the homogeneous solution is dominant. Turning on the higher-derivative interaction terms, the T c of the system depends on the coupling constants β, β , and the homogeneous solution no longer dominates. Then it was found that for β large enough while β = 0, the asymptotic (k 2 /µ 2 → ∞) transition temperature will be higher than that of the homogeneous solution. In this case, the transition temperature monotonically increases as we increase the wavenumber k having the same behaviour we found in Fig. 5. As they switched on β > 0, the transition temperature attains a maximum value at a finite k as it was shown in the Fig. 2 of [55]. Thus the second higher-derivative coupling acts as a UV cutoff on the wavenumber, The physical explanation of this behaviour is that the presence of the first derivative coupling to Einstein tensor β, is the encoding of the electric field's back reaction near the horizon and it is the cause of spontaneous generation of spatial modulation, while the presence of the second higher derivative coupling β can be understood as stabilizing the inhomogeneous modes introduced by the first derivative coupling.
In the next section we will find numerically a fully backreacted hairy black hole solution of the charged scalar field coupled to Einstein tensor with the coupling β, assuming that near the critical temperature T ≈ T c the effects of the cutoff are negligible and set β = 0. Then, trusting our linearization below the critical temperature because it will not rely on a gradient expansion but on an order parameter proportional to (T − T c ) 1/2 , we will calculate the conductivity.

A. The hairy black hole with full backreaction
To find the black hole solution with full backreaction, we consider the ansatz of the fields as then the horizon is located at z = 1 and the asymptotical boundary is at z → 0 . With this ansatz, the coupled Einstein-Maxwell-scalar field equations (6), (10) and (11) become a set of coupled differential equations which are given in the Appendix A.
To solve numerically the highly coupled system, we need to analyze the asymptotical solution of the scalar field ϕ in the Klein-Gordon equation (10) or its equivalent equation (16) with the coordinate transformation z = r h /r. In order to simplify the asymptotical behavior of the scalar field, we will choose so that ϕ| z→0 = ϕ − z + ϕ + z 2 , and we choose ϕ + to be renormalized. Furthermore, as pointed out in [56,57], it is convenient to define so, we will obtain numerically the functions u(z), ψ(z), a(z), V 1 (z) and V 2 (z) with appropriate boundary conditions. The boundary conditions at radial infinity (z → 0) read and we take Λ = −3. We impose regularity condition for all the fields at the horizon. The boundary conditions for the numerics we imposed, are similar to the boundary conditions which were discussed in details in [39]. In our case we have an additional β coupling term, but this will not modify the boundary conditions after we propose the relation (28). Thus, for fixed m 2 , our theory is also specified by three dimensionless parameters T /µ, λ/µ and k/µ which are similar to the ones in [39]. Profiles of the fields are shown in Fig. 6 -Fig. 9 for a choice of parameters. These figures show that we have found hairy black hole solutions with k = 0 (Fig. 6, Fig. 7) or k = 0 (Fig. 8, Fig. 9) having the translational symmetry preserved or broken. We observe that in both cases the found hairy black hole solutions have the same behavior and the scalar field is regular on the horizon. Then in the following subsection, we shall compute the conductivity of the hairy black hole in both cases.

B. Conductivity
Having the fully backreacted solution we will calculate the conductivity and compare the results with the conductivities we found in the probe limit in section III. To calculate the conductivity at the linear level we consider the following perturbations We will consider two cases k = 0 and k = 0 depending on having momentum relaxation or not.

k = 0
In this case, the equation of the scalar field perturbation decoupled from the other equations. The coupled differential equations for the electromagnetic and gravitational perturbations are which control the conductivity. Eliminating h tx from the above equation, we get When z → 0, the asymptotic behavior of the perturbation is derived to be a x (z) = a 0 x + za 1 x + O(z 2 ). Then, according to holographic dictionary, the conductivity is given by Near the black hole horizon z → 1, we impose purely ingoing boundary conditions with u(1) = −U (1) = 4πT . Then, using the results from the last subsection, we solve equation (34) at the boundary and calculate the conductivity using the relation (35). The conductivity σ(ω) with k = 0 is shown in Fig. 10. In this case, the hairy solution is homogeneous in spatial directions and the translational symmetry is hold. So the conductivity behaves similarly as that in RN black hole, the DC conductivity has a delta function at zero frequency due to the infinity of imaginary part and the real part of σ approaches to 1 at large frequency limit. Observe that we have the same behavior of the conductivity for both signs of the derivative coupling β.

k = 0
Now, we turn to the case with k = 0 in which the coupled system is more complicated because of the highly coupled three perturbed equations, which are listed in the Appendix B. To read off the conductivity, we analyze the boundary conditions of the perturbed fields. The asymptotic behavior of the fields are h tx (z) = h 0 tx + O(z), a x (z) = a 0 x + za 1 x + O(z 2 ) and δφ(z) = δφ 0 + zδφ 1 + O(z 2 ) and the expression of the conductivity is also given in (35). The behavior of the fields near the horizon is with u(1) = −U (1) = 4πT which are all regular. The conductivity for different couplings β is shown in Fig. 11 with fixed k/µ = 1/ √ 2, λ/µ = 0.5 and T /µ = 0.1. We see with nonzero k, the DC conductivity σ(0) is finite which according to the holographic dictionary our hairy black hole solution in the bulk is dual to a material with momentum relaxation on the boundary. If β = 0 we recover the results of the Q-Lattice [39] in which the translational invariance is explicitly broken. We observe that as the derivative coupling β is increased in positive values the electric conductivity σ(0) becomes lower while if β is negative, σ(0) is enhanced. It is interesting to see how the DC conductivity varies with the temperature for various values of the coupling β. We expect in the dual theory at low temperatures T << µ, the DC conductivity to increase as the temperature is lower and the material to enter a metal phase, while as the temperature is increased the material to enter an insulating phase with the DC conductivity to decrease.
We observe these changes of phases as we vary the coupling β. In Fig. 12 we fix β = 0.1 and show the conductivities for different low enough temperatures. We see that the dual theory is in a metal phase because DC conductivity increases as the temperature decreases. In Fig. 13 for β = −0.2, the DC conductivity decreases as we decreases T and the dual theory is in an insulating phase. The Q-Lattice model in [39] also supports two phases of a doped material in the boundary theory. However, these two phases are dual to two different black hole solutions in the bulk. In our case the coupling β plays the role of the doping parameter.
In Fig. 14 and Fig. 15 we show the conductivity for different momentum dissipation numbers k for positive and negative coupling β and for fixed low temperature. They show a competing effect between k and the coupling β. In the metal phase Fig. 14 shows that if the momentum dissipation is large the conductivity is low. This can be understood from the fact that the charge carriers for large momentum dissipation have more chances of finding impurities, because of the presence of a non-zero coupling β, so they loose energy and for this reason the conductivity is low. In the insulating phase Fig. 15 shows that even for large momentum dissipation the conductivity is low, because the charge carriers do not have the freedom to travel.
In [18] it was shown that in the probe limit the bulk theory is dual to a material in a metal phase and the positive coupling β signifies the amount of impurities in the material. In the fully backreacted theory we study in this work if the coupling β is positive we still have impurities but their concentration is not high enough to prevent conductivity. However, if β is negative then the friction on the velocities of the charge carriers is so high that the system enters the insulating phase. A change of sign of the coupling β in the gravity sector corresponds to a change of sign of the kinetic energy of the scalar field coupled to Einstein tensor. It is interesting to note that in the boundary theory due to proximity effect [58] there is a reflection of the charge carriers on the interface between the normal and the superconducting phases of the material. This is known as the Andreev Reflection effect [59]. Note also, this only happens in very low temperatures which means that there is no thermal energy to influence the transition. This behavior of the coupling β is a very interesting effect, it is a pure gravity effect because the derivative coupling is showing how strong is the coupling of matter to curvature. We set k/µ = 1/ √ 2, λ/µ = 0.5.

VI. CONCLUSIONS
In this paper, we analyzed the possible dual phases in Horndeski theory with a coupling between the scalar field and the Einstein tensor. Extending the holographic study in [18], we considered the simple inhomogeneous matter field in the probe limit, and studied the holographic superconducting phase transition and the conductivities of the dual boundary theory. The AC conductivity shows a non-trivial structure indicating a collective excitation of the charge carriers as a result of the breaking of translation invariance and the presence of the coupling of the scalar field to the Einstein tensor.
We then discussed possible instabilities in the theory and we analyzed the spontaneous breaking of translation invariance near the critical temperature. We studied the fully backreacted system of Einstein-Maxwell-scalar equations and numerically found the hairy black hole solution. We computed the conductivity of the dual theory and studied the generated phase. For the zero wave number of the scalar field, the DC conductivity was divergent as expected and the system is dual to ideal conductor. For nonzero wave number, the DC conductivity was finite having momentum dissipation on the dual boundary theory. At low temperatures, we found that for positive coupling the DC conductivity increases as the temperature is lower, indicating that its dual phase is a metal. For negative coupling we found the DC conductivity to decrease as the temperature is lower indicating that the dual phase is an insulator.
Our results are interesting and deserve a further study. We found that there is a change of a phase in the boundary theory as we change the value and the sign of the coupling of a charged scalar field to the Einstein tensor. This coupling shows the way matter is coupled to curvature and it is a pure gravity effect. On the dual theory our results show that the variation of this coupling influences the kinetic properties of the charge carriers. In a way this coupling parameterizes the amount of impurities present in a material on the boundary. On the other hand, a change on the sign of the kinetic energy of the scalar field allows the transition from one phase to an another in the boundary theory.