One of the first applications of gauge/gravity duality [1–3] was the building of a holographic superconductor and the study of its properties. The basic mechanism of generating a holographic superconductor is the formation in the gravity sector of a hairy black hole below a critical temperature and the generation of a condensate of the charged scalar field through its coupling to a Maxwell field of the gravity sector [4]. Then it was shown [5] that a holographic superconductor can be built on a boundary of the gravity sector first on the probe limit, where the charge of the scalar field is much smaller than the charge of the black hole, and then beyond the probe limit, in which the backreaction of the scalar field to the spacetime metric was considered [6]. Considering fluctuations of the vector potential, the frequency dependent conductivity was calculated, and it was shown that it develops a gap determined by the condensate. The formation of the gap is due to a pairing mechanism which is due to strong bounding of Cooper pairs. This strong pairing mechanism resulted to 2Δ≈8.4Tc, where Δ is the condensation gap, which has to be compared to the BCS prediction 2Δ≈3.54Tc, which is much lower in real materials, due to impurities (for a review see [7,8]).

The presence of impurities plays an important role in the superconducting materials chancing the properties of superconductors [9]. A detailed investigation of the effects of paramagnetic impurities in superconductors was carried out in [10], while holographic impurities were discussed in [11]. A model of a gravity dual of a gapless superconductor was proposed in [12]. Below a critical temperature it was shown that a black hole solution [13,14] acquired scalar hair, the DC conductivity was calculated and it was found that it had a milder behavior than the dual superconductor in the case of a black hole of flat horizon [5] which exhibited a clear gap behavior. This behavior was attributed to materials with paramagnetic impurities as it was discussed in [10]. Impurities effects were studied in [15], while impurities in the Kondo model were considered in [16,17].

A holographic model of a superconductor was discussed in [18]. In this model the gravity sector consists of a Maxwell field and a charged scalar field which except its minimal coupling to gravity it is also coupled kinetically to Einstein tensor. This term belongs to a general class of scalar-tensor gravity theories resulting from the Horndeski Lagrangian [19]. The main effect of the presence of this kinetic coupling is that gravity influences strongly the propagation of the scalar field compared to a scalar field minimally coupled to gravity. It allows us to find hairy black hole solutions [20–27], the gravitational collapse of a scalar field with a derivative coupling to curvature takes more time for a black hole formation compared to the collapse of minimally coupled scalar field [28] and acts as a friction term in the inflationary period of the cosmological evolution [29–32]. Moreover, it was found that at the end of inflation, there is a suppression of heavy particle production because of the fast decrease of kinetic energy of the scalar field coupled to curvature [33].

This behavior 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 [34] with a scalar field coupled to curvature, and the holographic entanglement entropy of a Horndeski black hole was calculated in [35].

The real materials 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 the momentum in the inner structure of these systems is not conserved. An example of such an observable is the conductivity. Under applied fields, the charge carriers of these systems can accelerate indefinitely if there is no mechanism for momentum dissipation, which leads the DC conductivity to be infinite. However, once the spatial translational invariance is broken which makes the momentum dissipation possible, then the peak at the origin of the AC conductivity spreads out such that the DC conductivity is finite. Therefore, for the holographic theories it is important to incorporate the momentum dissipation. In the literature, there are plenty of studies on the transport properties of holographic theories which dissipate momentum. The simplest way is to explicitly break the translational symmetry of the field theory state [36]. Other ways are by coupling to impurities, introducing a large amount of neutral scalar fields [37,38], a special gauge-axion [39], or by breaking translation invariance by introducing a mass for the graviton field [40].

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, in spite of the proximity effect in the interface between the superconductor and the normal phase [41]. 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 [42] 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 [43] and the conductivity at low temperatures was studied.

A generalization of [37] introducing a scalar field coupled to curvature was presented in [44]. An Einstein-Maxwell theory was studied in which except the scalar fields present in [37], 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 analyzed. In [45] 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 [46,47] an interaction term between the Einstein tensor and the scalar field is introduced in a model [48,49] 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. In the procedure, the system first is with the scalar field spatially dependent and the scalar potential has a constant chemical potential. Then we perturb the system around the critical temperature. We find that the scalar develops an x-dependence backreacted solution, which implies the spontaneously generated inhomogeneous phase of the system.

The work is organized as follows. In Sec. II we set up the problem. In Sec. III we studied the probe limit of the theory. In Sec. IV we discussed the stability of the theory and calculated the critical temperature. In Sec. V we calculated numerically the fully backreacted black hole solution and studied the DC conductivity and finally in Sec. VI are our conclusions.

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 I=∫d4x-g[R-2Λ16πG-14FμνFμν-(gμν(DμΨ)*DνΨ+m2|Ψ|2+βGμν(DμΨ)*DνΨ)],(1)where Dμ=∇μ-iqAμ(2)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 Φμν≡DμΨ(DνΨ)*,(3)Φ≡gμνΦμν,(4)Cμν≡gμν+βGμν.(5)Subsequently, the field equations resulting from the action (1) are Gμν+Λgμν=8πTμν,Tμν=Tμν(Ψ)+Tμν(EM)+βΘμν,(6)where, Tμν(Ψ)=Φμν+Φνμ-gμν(gabΦab+m2|Ψ|2),(7)Tμν(EM)=FμαFνα-14gμνFαβFαβ,(8)and Θμν=-gμνRabΦab+Rνa(Φμa+Φaμ)+Rμa(Φaν+Φνa)-12R(Φμν+Φνμ)-GμνΦ-12∇a∇μ(Φaν+Φνa)-12∇a∇ν(Φμa+Φaμ)+12□(Φμν+Φμν)+12gμν∇a∇b(Φab+Φba)+12(∇μ∇ν+∇ν∇μ)Φ-gμν□Φ,(9)and the Klein-Gordon equation is (∂μ-iqAμ)[-gCμν(∂ν-iqAν)Ψ]=-gm2Ψ,(10)while the Maxwell equations read ∇νFμν+Cμν[2q2Aν|Ψ|2+iq(Ψ*∇νΨ-Ψ∇νΨ*)]=0.(11)We note that the matter action in (1) has a 1/q2 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.

We first focus on the probe limit in the above setup, in which the Einstein equations admit the planar Schwarzschild AdS black hole solution ds2=-f(r)dt2+dr2f(r)+r2(dx2+dy2),f(r)=r2-rh3r.(12)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 Ψ(r,x)=φ(r)e-iτx,Aμ=ϕ(r)dt,(13)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 [1+β(fr2+f′r)]φ′′+[2r+f′f+β(3f′r2+f′2rf+f′′r)]φ′+[q2ϕ2f2(1+β(fr2+f′r))-m2f]φ-τ2φr2f[1+β(f′r+f′′2)]=0,(14)ϕ′′+2rϕ′-τ2r2fϕ-2q2φ2f[1+β(fr2+f′r)]ϕ=0.(15)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 φ|r→∞=φ-rΔ-+φ+rΔ+,ϕ=μ-ρr(16)and Δ is Δ±=32±9+4m2+27β21+3β,(17)where μ and ρ are interpreted as the chemical potential and charge density in the dual field theory, respectively.

According to the AdS/CFT correspondence, either of the coefficients φ- and φ+ may correspond to the source of an operator O dual to the scalar field with the other dual to the vacuum expectation values ⟨O-⟩=2φ-e-iτx, ⟨O+⟩=2φ+e-iτx, depending on the mass window and the choice of quantization. In details, from (17), the condition m2≥-94(1+3β) is needed for the relevance of the operator. When-54(1+3β)≥m2≥-94(1+3β), both φ- and φ+ are normalizable and we can choose both modes as source while the other as vacuum expectation values. When m2≥-54(1+3β), only φ+ is normalizable, so we can only treat φ+ as vacuum expectation value of the operator while φ- as the source. Moreover, we shall give two comments about the dual theory. First, β modifies the mass windows which are different from that in Einstein case [50]. It is obvious that β=-1/3 leads the dual theory to be ill-defined and the bulk theory at β=-1/3 may have no good holographic dual description in the boundary. Second, by redefining m2→(1+3β)m2, the dual theory described above seems to reduce the theory dual to Einstein scalar theory. This question is worthy to further study because with the redefinition, the action (1) from the bulk side does not apparently have similarity as the Einstein-scalar theory.

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 Tc, 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 nonvanishing φ+ for superconducting phase, respectively. We show the critical temperature Tc 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 behavior 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 Ax. The Maxwell equation at zero spatial momentum and with a time dependence of the form e-iωt gives Ax′′+f′fAx′+[ω2f2-2φ2f(1+β(f′′2+f′r))]Ax-2qτφ2r2(1+β(f′r+f′′2))=0.(18)

We will solve the perturbed Maxwell equation with ingoing wave boundary conditions at the horizon, i.e., Ax∝f-iω/3rh. The asymptotic behavior of the Maxwell field at large radius is Ax=Ax(0)+Ax(1)r+⋯. Then, according to AdS/CFT dictionary, the dual source and expectation value for the current are given by Ax=Ax(0) and ⟨Jx⟩=Ax(1), respectively. Thus, the conductivity is read as σ(ω)=-iAx(1)ωAx(0).(19)

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 Tc 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 behavior is interesting showing that the AC conductivity shows a nontrivial structure indicating a collective excitation of the charge carriers. Similar behavior was observed in [51] in which the translational symmetry is broken by massive graviton effects, and also in holographic models that behave effectively as prototypes of Mott insulators [52]. A study of competition of various phases was carried out in [53,54] 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.

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 [55,56]. 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 AdS4/CFT3 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 [33] 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 [57,58] 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 ds2=-f(r)dt2+dr2f(r)+e2V1(r)dx2+e2V2(r)dy2,A=ϕ(r)dt,Ψ=eikxφ(r),(20)where the functions f, V1, V2, 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 V1(r)=V2(r)=logr,f(r)=r2(1-(rh2+μ24)rhr3+rh2μ24r4),ϕ(r)=μ(1-rhr).(21)The Hawking temperature is T=3rh4π-μ216πrh.(22)The near horizon extremal solution is the usual AdS2×R2 where the extremal horizon is located at rh=μ23 and the AdS2 radius is L22=L26.

From the Klein-Gordon equation (10) in the background of (20), we see that the scalar Ψ gets an effective momentum dependent mass meff2(r)=m2+k2Cxx=m2+k2(1r2+β(2f′+rf′′)2r3).(23)Further considering the expression of (20), we obtain that an instability in the near horizon region appears when meff2(rh)L22=L26(m2+12(6β+1)k2μ2)<mBF2=-14(1+3β).(24)Moreover, to have a well-defined theory, β>-1/3 is required to satisfy mBF2<0. Thus, to fulfill the above conditions, the range of β is -13<β<-16,(25)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=eikxφ(r) (1+β(fr2+f′r))φ′′+(2r+f′f+β(3f′r2+f′2rf+f′′r))φ′-(m2f+k2r2f+βk2(f′r3f+f′′2r2f))φ=0(26)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 Tc(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 nonzero value of momentum. As k becomes larger, Tc 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 [59].

However, to find a finite critical temperature and produce a bell-shaped diagram, a method of regularization was proposed in [60] where another term with higher derivatives of the scalar field coupled to Einstein tensor -β′|DμGμνDνΨ|2(27)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 βμ2=β′μ4=0, the maximum critical temperature Tc is found at k=0, which shows that the homogeneous solution is dominant. Turning on the higher-derivative interaction terms, the Tc 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 (k2/μ2→∞) transition temperature will be higher than that of the homogeneous solution. In this case, the transition temperature monotonically increases as we increase the wave number k having the same behavior 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 [60]. Thus the second higher-derivative coupling acts as a UV cutoff on the wave number,

The physical explanation of this behavior is that the presence of the first derivative coupling to Einstein tensor β, is the encoding of the electric field’s backreaction 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≈Tc 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-Tc)1/2, we will calculate the conductivity.