Ward Identity and Homes' Law in a Holographic Superconductor with Momentum Relaxation

We study three properties of a holographic superconductor related to conductivities, where momentum relaxation plays an important role. First, we find that there are constraints between electric, thermoelectric and thermal conductivities. The constraints are analytically derived by the Ward identities regarding diffeomorphism from field theory perspective. We confirm them by numerically computing all two-point functions from holographic perspective. Second, we investigate Homes' law and Uemura's law for various high-temperature and conventional superconductors. They are empirical and (material independent) universal relations between the superfluid density at zero temperature, the transition temperature, and the electric DC conductivity right above the transition temperature. In our model, it turns out that the Homes' law does not hold but the Uemura's law holds at small momentum relaxation related to coherent metal regime. Third, we explicitly show that the DC electric conductivity is finite for a neutral scalar instability while it is infinite for a complex scalar instability. This shows that the neutral scalar instability has nothing to do with superconductivity as expected.


Introduction
Holographic methods have provided novel and effective tools to study strongly correlated systems [1][2][3][4] and they have been applied to many condensed matter problems. In particular, holographic understanding of high T c superconductor is one of the important issues. After the first holographic superconductor model proposed by Hartnoll, Herzog, and Horowitz (HHH) 1 [5,6], there have been extensive development and extension of the model. For reviews and references, we refer to [2,3,7,8].
In this paper, we study the HHH holographic superconductor model with massless scalar fields linear in spatial coordinate [44], where the strength of momentum relaxation is identified with the proportionality constant to spatial coordinate. The property of the normal phase of this model such as thermodynamics and transport coefficients were studied in [26,[28][29][30][50][51][52]. The superconducting phase was analysed in [43,44]. In particular, optical electric, thermoelectric and thermal conductivities of the model have been extensively studied in [29,44,51,52]. Building on them, we further investigate interesting properties related to conductivities and momentum relaxation. There are three issues that we want to address in this paper: (1) conductivities with a neutral scalar hair instability, (2) Ward identities: constraints between conductivities, (3) Homes' law and Uemura's law. We explain each issue in the following.
(1) In a holographic superconductor model of a Einstein-Maxwell-scalar action [6,7], a superconducting state is characterized by the formation of a complex scalar hair below some critical temperature. In essence, the complex scalar is turned on by coupling between the maxwell field and complex scalar through the covariant derivative. Interestingly, it was also observed [6,7] that a different mechanism for the instability forming neutral scalar hair 2 is possible. This instability was not associated with superconductivity because it does not break a U (1) symmetry, but at most breaks a Z 2 symmetry Φ → −Φ. Therefore, in this system with a neutral scalar hair, it is natural to expect that DC electric conductivity will be finite contrary to the case with a complex scalar hair (superconductor). However, to our knowledge, it has not been checked yet. In the early models without momentum relaxation, this question is not well posed since electric DC conductivity is always infinite due to translation invariance and finite density. In this paper, in a model with momentum relaxation, we show that DC electric conductivity is indeed finite with a neutral scalar hair.
(2) It was shown [3,55], in normal phase without momentum relaxation, there are two constraints relating three transport coefficients: electric conductivity(σ), thermoelectric conductivity(α) and thermal conductivity(κ). The constraints can be derived by the Ward identity regarding diffeomorphism. Thanks to these two constraints, α andκ can be obtained algebraically once σ is computed numerically. This is why only σ is presented in the literature [2]. In our model, there is an extra field, a massless scalar for momentum relaxation, and it turns out there are three Ward identities of six two-point functions: σ, α andκ and three more two-point functions related to the operator dual to a scalar field. Therefore, the information of σ alone cannot determine α andκ. If we know three twopoint functions then the Ward identities enable us to compute the other three two-point functions. In this paper, following the method in [3], we first derive the Ward identities for two-point functions analytically from field theory perspective. Next, we confirm them numerically from holographic perspective. This confirmation of the Ward identities also demonstrates the faithfulness of our numerical method.
(3) Homes' law and Uemura's law are empirical and material independent universal laws for high-temperature and some conventional superconductors [56,57]. The law states that, for various superconductors, there is a universal material independent relation between the superfluid density (ρ s ) at near zero temperature and the transition temperature (T c ) multiplied by the electric DC conductivity (σ DC ) in the normal state right above the transition temperatureT c .ρ whereρ s ,T c and σ DC are scaled to be dimensionless, and C is a dimensionless universal constant: C ≈ 4.4 or 8.1. They are computed in [45] from the experimental data in [56,57]. For in-plane high T c superconductors and clean BCS superconductors C ≈ 4.4. For c-axis high T c superconductors and BCS superconductors in the dirty limit C ≈ 8.1. Notice that momentum relaxation is essential here because without momentum relaxation σ DC is infinite. There is another similar universal relation, Uemura's law, which holds only for underdoped cuprates [56,57]:ρ where B is another universal constant. In the context of holography Homes' law was studied in [45,58]. It was motivated [58] by holographic bound of the ratio of shear viscosity to entropy density (η/s) in strongly correlated plasma [1] and its understanding in terms of quantum criticality [59] or Planckian dissipation [60],where the time scale of dissipation is shortest possible. Since Homes' law also may arise in systems of Planckian dissipation [60] there is a good chance to find universal physics in condensed matter system as well as in quark-gluon plasma. In [45] Homes' law was observed in a holographic superconductor model in a helical lattice for some restricted parameter regime of momentum relaxation, while Uemura's law did not hold in that model. However, physic behind Homes' law in this model has not been clearly understood yet. For further understanding on Homes' law, in this paper, we have checked Homes' law and Uemuras' law in our holographic superconductor model. We find that Homes' law does not hold but Uemura's law holds at small momentum relaxation region, related to coherent metal regime. This paper is organised as follows. In section 2, we introduce our holographic superconductor model incorporating momentum relaxation by massless real scalar fields. The equilibrium state solutions and the method to compute AC conductivities are briefly reviewed. In section 3, the conductivities with a neutral scalar instability are computed and compared with the ones with a complex hair instability. In section 4, we first derived Ward identities giving constraints between conductivities analytically from field theory perspective. These identities are confirmed numerically by holographic method. In section 5, after analysing conductivities at small frequency, we discuss the Home's law and Uemura's law in our model. In section 6 we conclude.

Equilibrium state
In this section we briefly review the holographic superconductor model we study, referring to [26,29,51,52,61] for more complete and detailed analysis. We consider the action 3 where x M = {t, x, y, r} and r is the holographic direction. R is the Ricci scalar and Λ = −3/L 2 is the cosmological constant with the AdS radius L = 1. We have included the field strength F = dA for a U (1) gauge field A, the complex scalar field Φ with mass m, two massless scalar fields, ψ I (I = 1, 2). The covariant derivative is defined by for which we make the following ansatz: In the gauge field, A t (r) encodes a finite chemical potential or charge density and B plays a role of an external magnetic field. Φ(r) is dual to a superconducting phase order parameter, . . with two undetermined coefficients J Φ and O Φ , which are identified with the source and condensate respectively. The dimension ∆ of the condensate is related to the bulk mass of the complex scalar by m 2 = ∆(∆ − 3). In this paper, we take m 2 = −2 and ∆ = 2 to perform numerical analysis. ψ I is introduced to give momentum relaxation effect where β is the parameter for the strength of momentum relaxation. For β = 0, the model becomes the original holographic superconductor proposed by Hartnoll, Herzog, and Horowitz (HHH) [5,6].
First, if Φ(r) = 0 (no condensate), the solution corresponds to a normal state and its   analytic formula is given by where r h is the location of the black brane horizon defined by U (r h ) = 0, m 0 ≡ r 3 h − β 2 r h 2 + n 2 +B 2 4r h , and n is interpreted as charge density. It is the dyonic black brane [62] modified by β due to ψ I [52]. The thermodynamics and transport coefficients(electric, thermoelectric, and thermal conductivity) of this system was analysed in detail in [52]. In the case without magnetic field, see [29]. Next, if Φ(r) = 0, the solution corresponds to a superconducting state with finite condensate and its analytic formula is not available 4 . For B = 0, the solutions are numerically obtained in [6] for β = 0 and in [44] for β = 0. For example we display numerical solutions for some cases in Figure 1, where we set r h = 1 and plot dimensionless quantities scaled by µ: U (r)/µ 2 , A t /µ, and χ. For B = 0, due to the generation of vortex our ansatz (2.5) should be modified. In this paper we will not consider this case and refer to [2,63,64].

AC conductivities
The purpose of this subsection is to briefly describe the essential points of a method to compute the AC thermo-electric conductivities. For more details and clarification regarding our model at B = 0, see [51,52] for normal phase and [44] for superconducting phase. At B = 0 see [29] for normal phase.
In order to study transport phenomena holographically, we introduce small bulk fluctuations around the background obtained in the previous subsection. For example, to compute electric, thermoelectric, and thermal conductivities it is enough to consider where i = x, y for B = 0 and i = x is enough for B = 0 thanks to a rotational symmetry in x − y space. For the sake of illustration of our method, we consider the case for B = 0 [44] and refer to [52] for B = 0. In momentum space, the linearized equations of motion around the background are 5 (2.9) Near boundary (r → ∞) the asymptotic solutions are The on-shell quadratic action in momentum space reads Here U (1) is the coefficient of 1/r when U (r) is expanded near boundary and n is charge density. The index ω in J a and R a are suppressed. The remaining task for reading off the retarded Green's function is to express R b in terms of J a . It can be done by the following procedure. First let us denote small fluctuations in momentum space by Φ a collectively. i.e. Φ a = (δa i , δh ti , δξ i ) . (2.13) Near black brane horizon (r = 1), solutions may be expanded as which corresponds to incoming boundary conditions for the retarded Green's function [65] and n a is some integer depending on specific fields, Φ a . The leading terms ϕ a are only free parameters and the higher order coefficients such asφ a are determined by the equations of motion. A general choice of ϕ a can be written as a linear combination of independent basis ϕ a i , Every ϕ a i yields a solution Φ a i (r), which is expanded near boundary as where δ a ≥ 1 and the leading terms S a i are the sources of i-th solutions and O a i are the corresponding operator expectation values. S and O can be regarded as regular matrices of order N , where a is for row index and i is for column index. A general solution may be constructed from a basis solution set {Φ a i }: with arbitrary constants c i 's. For a given J a , we always can find c i6 so the corresponding response R a may be expressed in terms of the sources J b With (2.20), the action (2.11) becomes where the range of ω is chosen to be positive following the prescription in [65] and the retarded Green's functions are explicitly denoted as Finally, the thermo-electric conductivities are related to the retarded Green's functions as (2.23)

Conductivities with a neutral scalar hair instability
By the numerical method reviewed in the previous subsection, the electric, thermoelectric and thermal conductivities of the model (2.1) have been computed in various cases [29,44,52]. As an example, in Figure 2, we show the results for B = 0, q = 3 [44], which is reproduced here for easy comparison with new results in this paper. Figure 2 shows AC electric conductivity (σ(ω)), thermoelectric conductivity (α(ω)), and thermal conductivity (κ(ω)) for β/µ = 1 and q = 3 at different temperatures. The colors of curves represent the temperature ratio, T /T c , where T c is the critical temperature of metal/superconductor phase transition. T /T c = 3.2, 1, 0.89, 0.66, 0.27 for dotted, red, orange,green, and blue curves respectively. In particular, the dotted curve is the case above T c and the red curve corresponds to the critical temperature. The first row is the real part and the second row is the imaginary part of conductivities. One feature we want to focus on in Figure 2 is 1/ω pole in Im[σ] below the critical temperature. There is no 1/ω pole above the critical temperature. By the Kramers-Kronig relation, the 1/ω pole in Im[σ] implies the existence of the delta function at ω = 0 in Re[σ]. It means that in superconducting phase the DC conductivity is infinite while in normal phase the DC conductivity is finite due to momentum relaxation.
Unlike the studies in [44], here we set q = 0. Between finite q and zero q, there is a qualitative difference in the instability of a Reissner-Nordstrom AdS black hole [6]. The origin of the superconductor (or superfluidity) instability responsible for the complex scalar hair Φ may be understood as the coupling of the charged scalar to the charge of the black hole through the covariant derivative In other words, the effective mass of Φ defined by m 2 eff ≡ m 2 − q 2 |g tt |A 2 t can be compared with the Breitenlohner-Freedman (BF) bound. The BF bound for AdS d+1 is − d 2 4 ≡ m 2 BF . The effective mass m 2 eff may be sufficiently negative near the horizon to destabilize the scalar field since |g tt | becomes bigger at low temperature 7 . Based on this argument one may expect that when q = 0 the instability would turn off. However, it turns out that a Reissner-Nordstrom AdS black hole may still be unstable to forming neutral scalar hair, if m 2 is a little bit bigger than the BF bound for AdS 4 . It can be understood by the near horizon geometry of an extremal Reissner-Nordstrom AdS black hole. It is AdS 2 × R 2 so scalars above the BF bound for AdS 4 may be below the bound for AdS 2 . These two instability conditions can be summrized by one ineqaulity [44] As the temperature of a charged black hole is decreased, gtt develops a double zero at the horizon.
Here, we see m 2 eff can be below the BF bound when q = 0. However, it was discussed in [6,7] that the instability to forming neutral scalar hair for q = 0 is not associated with superconductivity because it does not break a U (1) symmetry, but at most breaks a Z 2 symmetry Φ → −Φ. Therefore, it would be interesting to see if the DC conductivity is infinite or not in the background with a neutral scalar hair. 8 Without momentum relaxation (β = 0) this question is not well posed since the DC conductivity is always infinite with or without a neutral scalar hair due to translation invariance and finite density. Now we have a model with momentum relaxation (β = 0), we can address this issue properly.
To have an instability at q = 0 we choose the same parameters as Figure 2: m 2 = −2 and β/µ = 1. For q = 0, m 2 eff = −1/2, which is below the BF bound (3.1).  1, 0.84, 0.62, 0.45, 0.32 for red, orange,green, and blue curves respectively. A main difference of Figure 3 from Figure 2 is the disappearance of 1/ω pole in Im[σ] below T c . It confirms that the neutral scalar hair has nothing to do with superconductivity as expected.
In Figure 3 it is not easy to see the conductivities in small ω regime, so we zoom in there in Figure 4. Contrary to the conductivity of normal component in superconducting phase, the DC electric conductivity is not so sensitive to temperature and increases as temperature decreases, which is the property of metal. The thermoelectric and thermal conductivities decrease as temperature increases except a small increase of thermoelectric conductivity near the critical temperature. As a cross check, we have also computed these DC conductivities analytically by using the black hole horizon data according to the method developed in [50]. Since there is no singular behavior in the conductivities as ω → 0 we may regard the real scalar field here as the dilaton in [50] and the conductivities read where s, Q and T are the entropy density, charge density and temperature in the dual field theory. They are given by s = 4πr 2 h , Q = lim r→∞ r 2 e χ/2 A t (r) and T = U (r h ) 4π e (χ(∞)−χ(r h ))/2 . The analytic values are designated by the red dots in Figure 4 and they agree to the numerical values very well. For a special case with µ = 0, in Figure 5, we see that σ(ω) = 1, different from superconducting case (q = 0 shown in [44]), but α(ω) = 0, same as superconducting case.

Superfluid density with a complex scalar hair
We have found that for q = 0 there is no 1/ω pole in Im[σ], of which strength corresponds to superfluid density. To understand it better, we derive an expression for superfluid density for q = 0. Let us start with the Maxwell equation, Once we assume that all fields depend on r and t and the fluctuations are allowed only for the x-direction, the x-component of the Maxwell equation reads The integration of (3.5) from horizon to boundary gives the boundary current (3.6) By hydrodynamic expansion for small ω, it turns out that the first term and the second term goes to zero as while the last term goes to constant. Here we used the expansions of the fileds near horizon where ζ is a constant residual gauge parameter fixing δg rx = 0 [51], andâ x ,ĥ x andχ x can be expanded asâ (3.9) With the following source-vanishing-boundary conditions except δA x , the current (3.6) can be interpreted as Because only the last term of (3.6) contribute to J x for ω = 0, as discussed in (3.7), the superfluid density K s (the strength of the 1/ω pole of Im[σ]) is given by This shows how the hairy configuration Φ contributes to K s . If q = 0, K s vanishes, which confirms our numerical analysis.

Ward identities: constraints between conductivities
In this section, we first analytically derive the Ward identities regarding diffeomorphism from field theory perspective. It gives constraints between conductivities(σ, α,κ) and twopoint functions related to the operator dual to the real scalar field. Next, these identities are confirmed by computing all two-point functions numerically from holographic perspective.

Analytic derivation: field theory
To derive the Ward identities, we closely follow the procedure in [3]   two-point functions: . (4.12) We consider the generating functional W [h αβ ,Ā µ ,ψ I ,Φ,Φ * ] invariant under diffeomorphism, x µ → x µ + ζ µ , and the variation of the fields can be expressed in terms of a Lie derivative with respect to the vector field ζ µ δh µν = (L ζh ) µν = ∇ µ ζ ν + ∇ ν ζ µ , (4.13) 14) For diffeomorphism invariance, the variation of W should vanish: which, after integration by parts, yields the Ward identity for one-point functions regarding diffeomorphism.
where D µ T µν (x) ≡ ∂ µ T µν (x) + Γ ν αβ T αβ (x) . By taking a derivative of (4.18) with respect to eitherh αβ (y),Ā α (y),ψ J (y) orΦ * (y), we obtain the Ward identities for the two-point functions: where the covariant derivatives act only on the operators of x. From here we consider a flat space,h µν = δ µν , and assume external fields such as F µν , ∂ µψI andΦ, are constant in space-time. We further assume translation invariance is not spontaneously broken in the equilibrium state, so all the one point functions, T µν , J µ , O I and O Φ , should be constant in space-time. In momentum space, the Ward identities (4.19)-(4.22) read Since we want to study transport coefficients, we analytically continue to Minkowski space, so that the Euclidian Green's functions can be continued to the Retarded Green's functions. Thus, the Ward identities (4.23)-(4.26) become In particular, we consider 2+1 dimensional system in an equilibrium state with the constant expectation values for the energy-momentum and current with finite or zero condensate O Φ and O I = 0. To this system we apply a constant external magnetic field with a background scalarψ I : We take k µ = (ω, 0, 0) to focus on the spatially homogeneous AC conductivity induced by the small external electric field and temperature gradient along i(= x, y) direction. Under these conditions the Ward identities (4.27)-(4.29) becomes where i, j, k run over x and y. It is convenient to introduce the complexified combinations defined as (4.36) With this notation, (4.33)-(4.35) can be rewritten as ± ω JT ± − B JJ ± + ω n ± iβ JS ± = 0 , (4.37) ± ω T T ± − B T J ± + ω ± iβ T S ± = 0 , (4.38) ∓ ω ST ± − B SJ ± ∓ iβ SS ± = 0 , (4.39) or, in terms of the heat current Q = T − µJ, ± ω JQ ± + (±µω − B) JJ ± + ω n ± iβ JS ± = 0 , (4.40) Finally, using the Kubo formulas for conductivities 10 we obtain the relations between the conductivities: where (4.51) 10 The complexified conductivities are denoted by X± ≡ Xxy ± iXxx, where X = σ, α,ᾱ,κ.

Numerical confirmation: holography
In the previous subsection we have derived the Ward identities for two-point functions from field theory perspective. Here we show that those Ward identities indeed hold in our holographic model studied in [29,44,52]. More concretely, our goal is to compute σ, α,κ, JS , QS , SS numerically and plug them into the Ward identities (4.44)-(4.45) and (4.56)-(4.58) to check if they add up to zero or not. For B = 0, the conductivities σ, α,κ were reported in [52] and reproduced in Figure  2. Here in Figure 6 we display the other two-point correlation functions related to the real scalar operator, JS , QS , and SS . Contrary to Figure 2 there is no divergence at ω = 0, which is also shown in their small ω expressions (5.10)-(5.12). By using the data in Figure 2 and 6 we numerically compute the left hand side of three Ward identities (4.56)-(4.58). The numerical sums for all considered temperatures are shown together in Figure 7. All of them vanish(< 10 −15 ), confirming analytic formulas. We have also checked three other cases: 1) B = 0 and β/µ = 0.1, 2) B = 0 and µ = 0, 3) B = 0. It turned out that all numerical sums vanish too. For completeness, we show the numerical data for these three cases in the appendix A.

Homes' law and Uemura's law
We first analyse small ω behaviours of the two-point correlation functions based on our numerical results and Ward Identities both in superconducting and normal phase. After identifying superfluid density and normal component density in the two fluid model of superconductor we check Homes' law and Uemura's law.

Conductivities at small ω
For β = 0, the Ward identities (4.56)-(4.58) become simplified , This relation was reported in [2] for normal phase and here we have shown it still holds for superconducting phase. By these relations, once σ is obtained, α andκ are completely determined. In both normal and superconducting phase, Im[σ] turns out to have 1/ω pole by numerical computation so Re[σ] is infinite by the Kramers-Kronig relation [2,6,29]. Therefore, by (5.1), Re[α] and Re[κ] are also infinite and Im[α] and Im[κ] have 1/ω poles. In normal phase it is due to the absence of momentum relaxation and in superconducting phase there is another contribution due to condensate. For β = 0, the Ward identities (4.56)-(4.58) may be rewritten as In normal phase (see, for example, the dotted curve in Figure 2), the real and imaginary part of σ, α, andκ are all finite at ω = 0 due to the momentum relaxation (β = 0). At small ω, it is inferred that Re[ JS ] ∼ ω 2 from (5.2) and Im[ JS ] ∼ ω from (5.3). Also Re[ QS ] ∼ ω 2 from (5.4) and Im[ QS ] ∼ ω from (5.5) 12 . Finally, the small ω behaviour of SS is determined by JS and QS via (5.6). In superconducting phase (see for example the solid curves in Figure 2), unlike normal phase, Im[σ] and Im[κ] have 1/ω poles, which implies the existence of delta functions at ω = 0 in the corresponding real parts. In summary, the small ω behaviours can be written as

JS
where σ DC , α DC ,κ DC are real value of conductivities at ω = 0, while σ I , α I ,κ I are imaginary values linear to ω. K s is introduced as a strength of the pole of Im[σ], which can be identified with the superfluid density [40]. In (5.7)-(5.12), K s is the only parameter characterising the superconducting phase and if we set K s = 0 the expressions works for the normal phase. The two-point functions related to the scalar operator S ( JS , QS , SS ) diverge when β goes to zero at small ω. We have confirmed that (5.7)-(5.12) agree to the numerical results in Figure 6,12 and 13. If we define a normal fluid density (K n ) as the Ward identity (5.3) yields the charge conservation n = µK s +µK n . In Figure 8, we plot n/µ 2 (dotted curves) and K s /µ(solid curves) versus T /T c at several µ/β s. The difference between the dotted and solid curve at a given µ/β is K n /µ. As temperature approaches to zero 13 , K n vanishes for µ/β 2 (Figure 8(b)) while K n is nonzero for µ/β 2 (Figure 8(a)). Interestingly, it seems that this transition conicide with the coherent/incoherent metal transition studied in [29], where the metal state of this model was classified as coherent state with a well defined Drude peak in AC conductivity for µ/β 2 and incoherent state without a Drude peak for µ/β 2 14 . In coherent state, the normal fluid density K n can be used as an input parameter to fit the Drude formula in the two fluid model of holographic superconductors [40]. The non-zero K n at zero temperature for large momentum relaxation has been also observed in a holographic superconductor dual to a helical lattice [45].

Homes' law and Uemura's law
Homes' law and Uemura's law are material independent universal scaling relations observed in high temperature superconductor as well as conventional superconductors [45,[56][57][58]. Uemura's law appearing in underdoped cuprates is and Homes' law satisfied in a broader class of materials is where B and C are material independent universal constants. Here, the superfluid density (ρ s ), temperature(T ), and conductivity(σ DC ) are all dimensionless [45]. In this subsection we use momentum relaxation strength parameter(β) as our scale so we chooseρ s = K s /β andT = T /β. In our model, there are two free parameters, µ/β and q. Thus universality of B and C means that B and C are independent of µ/β and q. To check this it is convenient to fix q first, and make plots of B and C vs µ/β for Uemura's law and Homes' law respectively. To compute B and C, the superfluid densityρ s (= µ/β · K s /µ) can be read off from the solid curves in Figure  8, where the curves do not reach to T = 0 because of instability of numerical analysis. Therefore, we extrapolated the curves up to zero temperature to readρ s at T = 0. The conductivity σ DC can be read in Figure 2 or analytically σ DC = 1+µ 2 /β 2 in our model [26]. The transition temperatureT c has been computed numerically in [44]. Our numerical results ofρ s ,T c and σ D C for q = 3 are shown in Figure 9. From Figure 9 we may expect that there is a linear relation betweenρ s andT c at least for large µ/β, which supports Uemura's law. To see if this is the case also for small µ/β we make a plot of B vs µ/β in Figure 10(a), where we find that Uemura's law holds only for µ/β 2, of which data are red dots. Interestingly, the parameter regime µ/β 2 (red dots) belongs to coherent metal regime, where the optical conductivity of normal component shows a Drude peak behaviour. Furthermore, this regime corresponds to Figure  8  dots belong to incoherent regime, where a optical conductivity loses a Drude behaviour. They correspond to Figure 8(a) and there is a gap between charge density and superfluid density at zero temperature. Also, for different values of q, we find that Uemura's law is satisfied for large µ/β but with a different constant B. For example, for q = 2, B ≈ 6.87 and for q = 6, B ≈ 4.64 in the regime of µ/β 2 ( Figure 10(c)). Since Uemura's law is observed in underdoped regimes, if β can be interpreted as a doping parameter our result will be consistent with phenomena.
Based on our results on Uemura's law (Figure 10(a)) and σ DC (Figure 9(c)), we may anticipate if Homes' law is satisfied. If σ DC is quickly decreasing function approaching to constant for µ/β 2 we may have a chance to obtain Homes' law. However, our σ DC does not show that behaviour. Therefore, as shown in Figure 11, Home's law does not hold in both coherent regime (red dots) and incoherent regime (blue dots). In Figure 11(a), for large µ/β, C(=ρ s /(σ DCT c )) approaches to a constant value, but it is zero. It simply means that σ DC goes to infinite as momentum relaxation goes to zero. Figure 11(b) is another representation, a plot ofρ s versus σ DCT c , where it is also clear that there is no linear relation between betweenρ s and σ DCT c . For different values of q, we considered q = 2 and q = 6 and obtained figures qualitatively similar to Figure 11, so Homes' law seems not satisfied for different values of q either.
Homes' law may be understood based on Planckian dissipation, for which the time scale of dissipation is shortest possible [60] . In summary, the left hand side of (5.16), superfluid density is proportional to density of mobile electrons in superconducting state (n S ). The right hand side of (5.16), σ DC is proportional to density of mobile electrons in normal state (n N ) times relaxation time (τ ), and the relaxation time is inversely proportional to the temperature (Planckian dissipation): where k B is Boltzmann's constant and proportionality constants of the relations are material independent. Notice that thanks to the Planckian dissipation T c is cancelled out in Homes' law, leaving universal constant /k B . Finally if we use another empirical law, Tanner's law, n S = n N /4, Homes' law is obtained. In our model, it turns out a kind of Tanner's law holds in coherent regime (µ/β ≥ 2). In Figure 8(b), all curves coincide and it means n s /n N does not depend on µ/β, which is the qualitative content of Tanner's law. Therefore, if our system were Planckian dissipator in coherent regime, we would have seen Homes' law. The relaxation time τ for our model can be written as where T in the denominator is extracted to mimic the form of Planckian dissipation [45].
Since our system does not show Homes' law it is not a Planckian dissipator, which means f is not universal near T c . Indeed we may induce that f ∼ µ 2 /β 2 because T c ∼ µ/β from Figure 9(b) and τ ∼ µ/β from the analysis in [29].
Our results on Uemura's law and Homes' law are different from the previous work [45], where a superconductor model in a helical lattice was studied. In the model, there are two parameters corresponding to the strength of momentum relaxation effect: the lattice strength λ and the helix pitch p, and it was found that Homes' law held for restricted parameter regime (not in small momentum relaxation, but for rather large values of λ and p) while Uemura's law did not hold. In particular, Homes' law was observed in insulating phase near phase transition. However, in our model there is no insulating phase and it may be a reason why two models show different results. There are other differences between two models. The model in helix lattice is anisotropic five dimensional model, while our model is isotropic four dimensional.

Conclusion and discussions
In this paper, we analysed a holographic superconductor model incorporating momentum relaxation. Building on previous works [29,44,51,52], we focused on three issues, where momentum relaxation plays an important role. (1) Ward identities: constraints between conductivities, (2) conductivities with a neutral scalar hair instability, (3) Homes' law and Uemura's law.
In holographic methods, we often need to solve complicated differential equations which do not allow analytic solutions, so it is important to develop reliable and systematic numerical methods. Computing AC conductivities is such an example, for which we have developed a numerical method. However, to make sure our numerics are reliable and robust, it will be good to have a cross-check. The Ward identity serves as a nice cross-check of our numerical method since we can compare our numerical results with the independently derived analytic formula. When there is a neutral scalar instability we explicitly showed that the DC electric conductivity is finite, while it is infinite for a complex scalar instability. This shows that the neutral scalar instability has nothing to do with superconductivity as expected.
Homes' law is very interesting and important not only because of its material independent universality but also a possible relation to quantum criticality and Planckian dissipation, which also underpins the universal bound of the viscosity to entropy in strongly correlated systems such as quark-gluon plasma. We have checked Homes' law and Uemura's law in our model. It turns out that Homes' law does not hold and Umeura's law holds for small momentum relaxation related to coherent metal regime. Our results are different from [45], where a holographic superconductor in a helical lattice was considered and it was shown that Homes' law is satisfied for some restricted parameter regime in insulating phase, while Uemuras' law is not satisfied at all. The difference may be due to the existence of insulating phase and/or anisotropy in a model with a helical lattice. To clarify it, it will be helpful to study Homes' law in different holographic superconductor models such as anisotropic massless scalar model, Q-lattice model, or massive gravity model [68].
Regarding Homes' law and Uemura's law, there may be an issue in the identification of the superfulid density. We have found that superfluid density and total charge density at zero temperature do not agree at large momentum relaxation, similarly to the case in a helical lattice [45]. Because the Ferrell-Glover-Tinkham (FGT) sum rule still holds in our model [44], it is possible that part of the low frequency spectral weight are transferred to intermediate frequencies instead of the superfluid pole. Therefore, as a cross check, it will be good to compute the superfluid density from the transverse response by the magnetic/London penetration depth, for which we need to solve for the transverse propagator at small non-zero momentum [45]. There is also another closely related quantity to superfluid density. By integrating a Maxwell's equation over the holographic coordinate r, we may define the charge density of hair outside the horizon, n hair , as where n is the charge density of the dual field theory and n h is interpreted as the charge density inside the horizon. In normal phase n hair = 0 while in superconducting phase n hair = 0. Therefore, n hair plays a role of order parameter of superconducting phase transition. For q = 0, n hair is zero so K s is zero, which is consistent with our results in section 3. However, it turns out that the numerical value of K s is different from n hair . We have checked Homes' law and Uemura's law by using n hair as the superfulid density, but it did not support Homes' law and Uemura's law. It will be interesting to find a physical meaning of n hair in the dual field theory and the precise relation to superfluid density.