Building a Holographic Superconductor with a Scalar Field Coupled Kinematically to Einstein Tensor

We study the holographic dual description of a superconductor in which the gravity sector consists of a Maxwell field and a charged scalar field which except its minimal coupling to gravity it is also coupled kinematically to Einstein tensor. As the strength of the new coupling is increased, the critical temperature below which the scalar field condenses is lowering, the condensation gap decreases faster than the temperature, the width of the condensation gap is not proportional to the size of the condensate and at low temperatures the condensation gap tends to zero for the strong coupling. These effects which are the result of the presence of the coupling of the scalar field to the Einstein tensor in the gravity bulk, provide a dual description of impurities concentration in a superconducting state on the boundary.

a gap in the excitation energy spectrum. For even in the absence of a gap, so long as correlations persist there is an ordered state below a certain critical temperature which displays the usual properties of a superconductor.
A holographic realization of impurities was discussed in [10]. Using the AdS/CFT correspondence two different types of impurities were calculated and the Nernst response for the impure theory was specified. A model of a gravity dual of a gapless superconductor was proposed in [11]. The gravity sector was consisting of a charged scalar field which provided the scalar hair of an exact black hole solution [12,13] below a critical temperature T c , while an electromagnetic perturbation of the background determined the conductivity giving rise to a gapless superconductor. It was found that the normal component of the DC conductivity had a milder behaviour than the dual superconductor in the case of a black hole of flat horizon [1] in which n n exhibited a clear gap behaviour. The behaviour it was observed in the boundary conducting theory was attributed to materials with paramagnetic impurities as it was discussed in [9]. Impurities effects were studied in [14], while impurities in the Kondo Model were considered in [15,16].
In this work we extend the model in [1] by introducing a derivative coupling of the scalar field to Einstein tensor. This term belongs to a general class of scalar-tensor gravity theories resulting from the Horndeski Lagrangian [17]. These theories, which were recently rediscovered [18], give second-order field equations and contain as a subset a theory which preserves classical Galilean symmetry [19][20][21]. The derivative coupling of the scalar field to Einstein tensor introduces a new scale in the theory which on short distances allows to find black hole solutions [22][23][24][25][26] with scalar hair just outside the black hole horizon, while if one considers the gravitational collapse of a scalar field coupled to the Einstein tensor then the formation of a black hole takes more time to be formed compared to the collapse of a scalar field minimally coupled to gravity [27]. On large distances the presence of the derivative coupling acts as a friction term in the inflationary period of the cosmological evolution [28][29][30]. Moreover, it was found that at the end of inflation in the preheading period, there is a suppression of heavy particle production as the derivative coupling is increased. This was attributed to the fast decrease of kinetic energy of the scalar field due to its wild oscillations [31].
The above discussion indicates that one of the main effects of the kinematic coupling of a scalar field to Einstein tensor is that gravity influences strongly the propagation of the scalar field compared to a scalar field minimally coupled to gravity. We are going to use this behaviour of the scalar field to holographically simulate the effects of a high concentration of impurities in a material. The presence of impurities in a superconductor is making the pairing mechanism of forming Cooper pairs less effective and this is happening because the quasiparticles are loosing energy because of strong concentration of impurities. This holographic correspondence is supported by our finding. We found that as the value of the derivative coupling is increased the critical temperature is degreasing while the condensation gap ∆ is decreasing faster than the temperature. Also by calculating the perturbation of the scalar potential we 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 1 .
The work is organized as follows. In Section II we review the basic ingredients of the holographic model discussed in [1]. In Section III we extend the holographic model of Section II adding a scalar field coupled to Einstein tensor and we give the equations of motion. In Section III A we solve numerically the equations of motion and we find the condensation gap and the critical temperature for various values of the derivative coupling. In Section III B we study the conductivity while in Section IV are our conclusions.

II. HOLOGRAPHIC SUPERCONDUCTOR WITH A SCALAR FIELD MINIMALLY COUPLED TO GRAVITY
In this section we review in brief the holographic model of [1]. To introduce the minimal holographic superconductor model, we consider a Maxwell field and a charged complex scalar field coupled to gravity with the action If we rescale the fields and the coupling constant as then 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. In this limit the Einstein equations admit as a solution the planar Schwarzschild AdS black hole Taking the ansatz ψ = |ψ|, A = φdt where ψ, φ are both functions of r only, we can obtain the equations of motion for ψ, φ It was argued in [2] that there is a critical temperature below which the black hole acquires hair [1,3]. The equations (4) and (5) can be solved numerically by doing integration from the horizon out to the infinity taking under consideration the right boundary conditions. The solutions behave like with where L is the scale of the AdS space and µ and ρ are interpreted as the chemical potential and charge density in the dual field theory respectively. The coefficients ψ − and ψ + according to the AdS/CFT correspondence, correspond to the vacuum expectation values 2 of an operator O dual to the scalar field. We can impose boundary conditions that either ψ − or ψ + vanishes.
Fluctuations of the vector potential A x in the gravity sector gives the conductivity in the dual CFT as a function of frequency. The Maxwell equation for zero spatial momentum and with a time dependence of the form e −iωt reads as The above equation can be solved by imposing ingoing wave boundary conditions at the horizon A x ∝ f −iω/3r0 . On the other hand, the asymptotic behaviour of the Maxwell field at boundary is Then according to AdS/CFT correspondence dual source and expectation value for the current are given by Finally, Ohm's law gives the conductivity

III. HOLOGRAPHIC SUPERCONDUCTOR WITH A SCALAR FIELD KINEMATICALLY COUPLED TO EINSTEIN TENSOR
In this section, we will consider a complex scalar field which excepts its coupling to curvature it is also coupled to Einstein tensor with action where and e, m are the charge and the mass of the scalar field and κ the coupling of the scalar field to Einstein tensor of dimension length squared. For convenience we set The field equations resulting from the action (12) are where, and The Klein-Gordon equation is while the Maxwell equations read A. Solution of equations of motion and phase transition As in [1] we also take the ansatz ψ = |ψ|, A = φdt where ψ, φ are both functions of r only. In the probe limit, under the metric (3), the above equations of motion for the matter fields become Note that if κ = 0 then we get the equations (4) and (5). The new equations depend now on the second derivative of the function f in contrast to the preview case where we had only its first derivative, due to the additional term of G µν in the action (12).
With the same study in the minimal case presented in previous section, we can numerically solve the above equations of motion. We especially study the effect of κ on the critical temperature and strength of condensation in the dual supercondutor. Without loss of generality, we set e = 1 and m 2 = −2, so that we have λ − = 1 while λ + = 2. We will choose ψ − as the source which is set to be vanishing, while ψ + as vacuum expectation values.
The critical temperatures with different κ are summarized in Table I. It shows that as the coupling becomes stronger, the critical temperature is lower, meaning that the phase transition is hard to occur as it is expected. This   effect of coupling is very different from the influence of higher-derivative coupling between the matters studied in [33,34]. As the temperature is decreased, the scalar field can have non-zero solution and the strength of condensation becomes higher. We show this phenomena in FIG. 1. It is obvious that the strength of condensation is suppressed by the derivative coupling. And with enlarging the coupling 2 , the condensation decreases faster than the temperature, so that the condensation gap tends to be zero at low temperatures for the strong coupling. This behaviour from κ is coincident with the effect of paramagnetic impurities on superconductors observed in [6][7][8].
We fit the curve near the critical temperatures. As T → T c , the condensation is continuous and behaves as where C 1 are also listed in Table I. We see that the coefficient C 1 decreases drastically to suppress the condensation when the coupling is increased. Observe that the exponent is always 1/2 which implies that κ does not modify the order of the phase transition and it is always second order. In the next subsection we will study the conductivity caused by the condensation below the critical temperature.

B. Conductivity
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/3r0 . 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 A x = A (0) x , respectively. Thus, similar to the equation (11), the conductivity is also read as 2 There is an instability in our numerical solution of the equations of motion for large or small values of the derivative coupling when the temperature is far way from Tc. Specially, it is difficult to reach a low temperature T /Tc ∼ 0.1 when κ > 0.5. For this reason in FIG. 1 we restricted the curves to the values of κ in the range −0.01 κ 0.5. However, when we are near Tc our numerics work better, so we could fit relation (25) for larger values of κ. So in order to study the same shift from the critical temperature in the following study we focus on the range −0.01 κ 0.5.
where P denotes the Cauchy principal value, the divergence of the imaginary part at zero frequency indicates a delta function of real part at ω = 0. At very low temperature, the energy gap in the real part of the conductivity measured by the critical temperature (left part of FIG. 2) decreases as the derivative coupling becomes stronger, which agrees well with the behaviour of the condensation as it is depicted in FIG. 1 in the last subsection. In order to show how the conductivity opens a gap as the temperature decreases, in FIG. 3, we show the frequencydependent conductivity with κ = 0.01 at different temperatures. In the left plot, the horizontal line, which is frequency independent, corresponds to temperatures at or above the critical value and there is no condensate. As we lower the temperature in the subsequent curves, we see that a gap ω g opens in the real part of the conductivity and the gap becomes wider and deeper as the temperature becomes smaller. In the right plot, we plot the conductivity for temperature lower than T c related the left plot by rescaling the frequency by the condensate. It is obvious that the curves tend to a limit in which the width of the gap is proportional to the size of the condensate. i.e., ω g ≃ < O + >. The above features are similar with those in minimal coupling disclosed in [1] .
Furthermore, the frequency-dependent conductivity with κ = 0.5 at different temperatures is presented in FIG. 4. Similarly to the cases with κ = 0.01 and minimal coupling, lower temperature gives us wider energy gap. By comparing the left plots of FIG. 3 and FIG. 4, we see that at the same scaled temperature T /T c , the gap is narrower for larger coupling, which is consistent with the results shown in FIG. 1 and FIG. 2. However, if being scaled by the condensation, see the right plot for κ = 0.5, the behavior ω g ≃ < O + > at low T /T c is violated and it is ω g ≃ 20 < O + > for T /T c ≃ 0.1. The coefficient 20 is not out of expect. Because if we see carefully the condensation plot FIG. 1 and FIG.  2, we will find < O + > ≃ 0.2T c and ω g ≃ 4T c , so we get the same relation as before ω g ≃ 20 < O + >.  For the systems far away from the critical temperature (T /T c ≃ 0.10), by careful study, we summarize the values of ωg √ <O+> with some κ in Table II. It is obvious that as |κ| shifts from the minimal coupling, the relation ω g ≃ < O + > is sharply violated, which is one of the main features of a superconducting state with impurities. Note that this effect of possible impurity parameter κ is different from that studied in [14] where it can not introduce this kind of violation.
We can now explore the behaviour of conductivity at very low frequency. When T < T c , the real part of the conductivity present a delta function at zero frequency and the imaginary part has a pole, which is attributed to the KK relations (28). More specifically, as ω → 0, the imaginary part behaves as Im(σ) ∼ n s /ω, and according to Kramers-Kronig relations, the real part has the form Re(σ) ∼ πn s δ(ω). Here the coefficient n s of the delta function is defined as the superfluid density. By fitting data near the critical temperature, we find that with various couplings, the superfluid density has the behaviour which means that n s will vanish linearly as T goes to T c . This is consistent with that happens in the minimal coupling. Also we find that the coefficient C 2 is not very sensitive to the coupling and the value oscillates around 24 with ±2 shifting for the samples of κ. This insensitivity to the changes of the coupling is reasonable because in the region very close to the critical temperature, the system is marginally shifted from the normal state, so that varying κ has very small effect on the behaviour of the solutions. Meanwhile, as proposed in [1], the non-superconducting density can be defined as n n = lim ω→0 Re(σ). Far from the critical temperature (with T /T c ∼ 0.1), we fit and obtain that n n decays as where Ω g can be explained as the energy gap for charged excitation at the corresponding temperature. The values of Ω g close to T /T c ∼ 0.1 are summarized in Table III. Rough comparison with the numerical energy gap ω g /T c shown in the left plot of FIG.2 gives us that ω g ≃ 2Ω g , where the factor 2 suggests that the gaped charged quasiparticles are formed in pairs as addressed in [1]. Analyzing the data in the table, we see that κ = 0.5 corresponds to ω g ≃ 2Ω g ≃ 4T c which is very near the prediction ω g ≃ 3.54T c in BCS theory. In this tendency, we believe that BCS prediction may be fulfilled by larger coupling once the numerics are controlable. Thus, in this sense, the derivative coupling somehow mimic the effect of the impurities in a real material.

IV. CONCLUSIONS
We studied a holographic description of a superconductor in which the gravity sector consists of a Maxwell field and a charged scalar field which except its usual minimal coupling to gravity it is also coupled to Einstein tensor. Solving the equations of motion numerically in the probe limit, we found that as the strength of the new coupling is increased, the critical temperature below which the scalar field condenses is lowering, the condensation gap decreases faster than the temperature, the width of the condensation gap is not proportional to the size of the condensate and at low temperatures the condensation gap tends to zero for the strong coupling. Analysing the frequency dependence of conductivity we found that at strong coupling the relation ω g ≃ 2Ω g ≃ 4T c holds, where Ω g is the energy gap for charged excitation, which is very near the prediction ω g ≃ 3.54T c in BCS theory.
We argued that these results suggest that the derivative coupling in the gravity bulk can have a dual interpretation on the boundary corresponding to impurities concentrations in a real material. This correspondence can be understood from the fact that the coupling of a scalar field to Einstein tensor alters the kinematical state of the scalar field a behaviour which it is also exhibited by the quasiparticles moving in a material with impurities.
We believe that the probe limit captures all the essential features of the problem under study. Nevertheless, it would be interesting to extend this study beyond the probe limit. Assuming a spherically symmetric ansatz for the metric the field equations (17), (21) and (22) have to be solved. However, even their numerical solution is a formidable task mainly because of the presence of the energy-momentum tensor (20) resulting from the coupling of the scalar field to Einstein tensor. A more realistic approach would be to follow the perturbative methods employed in [22,24].