A simple holographic superconductor with momentum relaxation

Journal of High Energy Physics, Apr 2015

We study a holographic superconductor model with momentum relaxation due to massless scalar fields linear to spatial coordinates(ψ I = βδ Ii x i ), where β is the strength of momentum relaxation. In addition to the original superconductor induced by the chemical potential(μ) at β = 0, there exists a new type of superconductor induced by β even at μ = 0. It may imply a new ‘pairing’ mechanism of particles and antiparticles interacting with β, which may be interpreted as ‘impurity’. Two parameters μ and β compete in forming superconducting phase. As a result, the critical temperature behaves differently depending on β/μ. It decreases when β/μ is small and increases when β/μ is large, which is a novel feature compared to other models. After analysing ground states and phase diagrams for various β/μ, we study optical electric(σ), thermoelectric(α), and thermal(\( \overline{\kappa} \)) conductivities. When the system undergoes a phase transition from normal to a superconducting phase, 1/ω pole appears in the imaginary part of the electric conductivity, implying infinite DC conductivity. If β/μ < 1, at small ω, a two-fluid model with an imaginary 1/ω pole and the Drude peak works for σ, α, and \( \overline{\kappa} \), but If β/μ > 1 a non-Drude peak replaces the Drude peak. It is consistent with the coherent/incoherent metal transition in its metal phase. The Ferrell-Glover-Tinkham (FGT) sum rule is satisfied for all cases even when μ = 0.

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A simple holographic superconductor with momentum relaxation

Received: January A simple holographic superconductor with momentum relaxation Keun-Young Kim 0 2 Kyung Kiu Kim 0 2 Miok Park 0 1 Open Access 0 c The Authors. 0 0 Gwangju 500-712 , Korea 1 School of Physics, Korea Institute for Advanced Study 2 School of Physics and Chemistry, Gwangju Institute of Science and Technology We study a holographic superconductor model with momentum relaxation due to massless scalar fields linear to spatial coordinates(I = Iixi), where is the strength of momentum relaxation. In addition to the original superconductor induced by the chemical potential() at = 0, there exists a new type of superconductor induced by even at = 0. It may imply a new 'pairing' mechanism of particles and antiparticles interacting with , which may be interpreted as 'impurity'. Two parameters and compete in forming superconducting phase. As a result, the critical temperature behaves differently depending on /. It decreases when / is small and increases when / is large, which is a novel feature compared to other models. After analysing ground states and phase diagrams for various /, we study optical electric(), thermoelectric(), and thermal() conductivities. When the system undergoes a phase transition from normal to a superconducting phase, 1/ pole appears in the imaginary part of the electric conductivity, implying infinite DC conductivity. If / < 1, at small , a two-fluid model with an imaginary 1/ pole and the Drude peak works for , , and , but If / > 1 a non-Drude peak replaces the Drude peak. It is consistent with the coherent/incoherent metal transition in its metal phase. The Ferrell-Glover-Tinkham (FGT) sum rule is satisfied for all cases even when = 0. momentum; relaxation; Holography and condensed matter physics (AdS/CMT); Gauge-gravity cor- 1 Introduction 2 Metal/superconductor phase transition 2.1 Optical conductivity A One point functions Introduction Normal (metallic) phase Superconducting phase Criterion for instability and quantum phase transitions Fluctuations for optical conductivity: equations and on-shell action Numerical method Electric/thermal/thermoelectric conductivites Holographic methods (gauge/gravity duality) has provided novel tools to study diverse strongly correlated systems [14]. In particular, after a pioneering model of superconductor in holographic methods by Hartnoll, Herzog, and Horowitz(HHH) [5, 6], there have been extensive development of the model. We refer to [2, 3, 7] for reviews and references. The HHH model is translationally invariant. Because a translationally invariant system with finite charge density cannot relax momentum, the HHH model will exhibit an infinite electric DC conductivity even in the normal metal phase. Therefore, to construct more realistic superconductor models, it is important to incorporate momentum relaxation in the framework of holography. One way to include momentum relaxation is to break translational invariance by imposing explicit inhomogeneous boundary conditions such as a spatially modulated scalar field or temporal U(1) gauge field At, which mimicks an ionic lattice [8, 9]. These models successfully yield a finite DC conductivity as well as interesting features in optical lattice) was applied to the HHH model in [13] and, interestingly, many properties of the bismuth-based cuprates were observed. In this method, however, because of inhomogeneity of dynamic fields, the equations of motion become a complicated coupled partial differential equations (PDE). It is technically involved and less flexible than ordinary differential equations (ODE), though conceptually clear. Therefore, it will be efficient and complementary if we can analyse the system with ODEs. In this line a few ideas have been proposed and developed. Massive gravity models [1417] introduces mass terms for some gravitons. It breaks bulk diffeomorphism invariance and consequently breaks translation invariance in the boundary field theory. Holographic Q-lattice models [18, 19] exploit a continuous global symmetry of the bulk theory, where, for example, the global phase of a complex scalar field breaks translational invariance. Models with massless scalar fields linear in spatial coordinate [2023] take advantage of the shift symmetry.1 This model is related to Q-lattice models. For example, a massless complex scalar with constant in (2.6) of [18] gives a massless axion linear in a spatial direction. Also there are models utilising a Bianchi VII0 symmetry to construct black holes dual to helical lattices [2729]. All these models give us ODEs and yields a finite DC conductivity as expected. Furthermore, for a large class of models, the analytic DC conductivity formulas are available in terms of the data of the black hole horizon [30]. Building on this development based on ODE, it is natural to revisit the holographic superconductor models. The superconductor model combined to the massive gravity models and Q-lattice models have been studied in [31] and [32, 33] respectively. For massless scalar models, an anisotropic background case with one scalar field was addressed in [34, 35]. As in non-superconducting cases, the properties of theses ODE-based superconductor models qualitatively agree to the PDE-based model with ionic lattice [13]. In this paper, we study a holographic superconductor model based on a massless scalar model for isotropic background. The model consists of two parts: the HHH action [13] and two massless real scalar fields [20]. The HHH action is a class of Einstein-Maxwell-complex have isotropic bulk fields, the identical scalar field is introduced for every spatial direction The HHH model without massless real scalar sector has been studied extensively. See [2, 3] for review. It has two phases, normal metallic phase and superconducting phase. Without the complex scalar field, the black hole is Reissner-Nordstrom type and the system is in normal metal phase. With a finite complex scalar hair, the system is in superconductor phase. With massless real scalar fields, a normal metal phase still exist [20, 24] and its thermodynamic and transport coefficients were studied: the DC electric conductivity [20], DC thermoelectric and thermal conductivity [30], optical electric conductivity [22], optical electric, thermoelectric and thermal conductivities [23]. Having studied the metal phase of the model, we want to investigate the superconducting phase. First, we will examine the condition in which a superconductor phase may and the properties of superconductor. 1This model with analytic solutions have been reported in [24] without specific applications to gauge/gravity duality. In holographic context a model with only one scalar field was studied for an anisotopic background in [25, 26] Interestingly, we find that there exists a new type of superconducting phase even when other models. Our main tool to analyse the superconducting phase is optical conductivities: the computation of these conductivities are related to the classical dynamics of three coupled bulk field fluctuations(metric, gauge, scalar fields). By computing the on-shell quadratic action for these fluctuations we can read off the retarded Greens functions relevant to three conductivities. Most papers deal with only electric optical conductivity. However, for better understanding, it will be good to have a complete set of three conare available [30, 36], but not for optical conductivities. In [23], a systematic numerical method to compute all three conductivities in a system with a constraint were developed based on [37, 38]. The method was applied to the normal metallic phase of our model, producing numerical conductivities reliably [23], and we will use the same method for the superconducting phase in this paper.2 One of the main results of [23] is numerical demonstrations of coherent/incoherent metal transition.3 In [42] metal without quasi-particle at strong coupling was classified by two classes: coherent metal with a Drude peak and incoherent metal without a Drude peak. fit well to the Drude form modified by K0: where Ks is supposed to be proportional to superfluid density. In superconducting phase, 2For another numerical analysis on three optical conductivities we refer to [39]. 3It was shown in [40] for anisotropic background. See also [41]. 4See [43] for discussions on universal bounds for thermoelectric diffusion constants in incoherent metal. where K0 is the contribution from pair production affected by net charge density. For all three optical conductivities fit to We have confirmed numerically the Ferrell-Glover-Tinkham (FGT) sum rule is satisfied in all cases we considered. Z This paper is organised as follows. In section 2, we introduce our holographic superconductor model (Einstein-Maxwell-complex scalar action with negative cosmological constant) incorporating momentum relaxation by massless real scalar fields. Background bulk solutions corresponding to superconducting phase and normal phase are obtained. By comparing on-shell actions of both solutions, we identify the phase transition temperature as a function of chemical potential and momentum relaxation parameter, which yields 3dimensional phase diagrams. In superconducting phase, we also compute condensates as a function of temperature for given chemical potential and momentum relaxation parameter. In section 3, we compute optical electric, thermoelectric, and thermal conductivities in superconducting phase and normal phase. In particular, in superconducting phase, we discuss the effect of momentum relaxation on conductivity in several aspects such as the appearance of infinite DC conductivity, Drude-nature of optical conductivity in small frequency range, two-fluid model, and Ferrell-Glover-Tinkham(FGT) sum rule. We also present a general numerical method to compute retarded Greens functions when many fields are coupled. In section 4 we conclude. Note added. After this paper was completed, we became aware of [33] which has overlap Metal/superconductor phase transition We start with the original holographic superconductor model proposed by Hartnoll, Herzog, and Horowitz(HHH) [5] dd+1x g 2 1 Xd1(I )2 . SHHH = SGH = 2 dd+1x g R + d d(d 1) 4 SGH, is the Gibbons-Hawking term, which is required for a well defined variational problem boundary, and K denotes the trace of the extrinsic curvature. To impose a momentum relaxation effect, we add the action of free massless scalars proposed in [20] M F MN + iq(DN DN ) = 0 , RMN 2 gMN d(d 1) 4 1 F 2 |D|2 m2||2 2 1 Xd1(I )2 now on. In order to construct a plane(x, y)-symmetric superconducting background, we take the following ansatz, A = At(r)dt , where non-zero At(r) is introduced for a finite chemical potential or charge density and a strength of impurity. Plugging the ansatz (2.8) into the equations of motion (2.4)(2.7), we have four equations (2.9)(2.12). Maxwells equation (2.4) yields r2 = 0 , 2 A0t At = 0 , Massless real scalar equations (2.6) is satisfied by the ansatz (2.8). The tt and rr components of Einsteins equations (2.7) give We will numerically solve a set of coupled equations of two second order differential equations (2.9)(2.10) and two first order differential equations (2.11)(2.12) by integrating the regularity condition at the horizon, six initial conditions are determined by three initial action the Hawking temperature(TH ) is given by 12 2m2L2(rh)2 2L2 2 2rh2 L2e(rh)(A0t(rh))2 e 21 (rh) . G(r) L2 + At(r) A(0) t 2L2 + G The coefficients of (2.16)(2.19) are identified with the field theory quantities as follows. hO /2 , J hO operator and its source, respectively. So J (i) should vanish for condensation of O of the black hole with the temperature of boundary field theory. Equivalently, we may rescale the time is computed as much easier since we have to shoot out from horizon. Then the field theory temperature where TH is defined in (2.15). However, analytic formulas from here are presented by There are two scaling symmetries of the equations of motion. The first symmetry is r a1r , (t, x, y) a1(t, x, y) , and the second one is r a2r , (t, x, y) (t, x, y)/a2 , , (t, x, y) = G = While performing numerics we work in terms of these tilde-variables. In practice, it is results, we need to scale them back carefully. Normal (metallic) phase Here we minimally summarize the properties of the metal phase to set up the stage for our paper, referring to [20] for more details. The normal phase of the system corresponds to the solution without condensate and the analytic solution is given by ds2 = G(r)dt2 + G(r) = A = 1 r m0 = rH3 4rh2 2rh2 The normal phase of the system is described by a charged black brane solution with nonis given by the Hawking temperature (2.14): TH = G0(rh) 3rh There is an important property of the geometry which we will rely on when discussing the instability of the metal phase. In the zero temperature limit, the near horizon geometry L22 = L2d+1 (d 1)2 + (d 2)22 d(d 1) Criterion for instability and quantum phase transitions Before performing a full numerical analysis to search superconducting phase, we first want to address a simpler question: when can the normal phase ((2.26)(2.30)) be unstable by small scalar field perturbations, which may develop into a hairy black hole with nonzero ground (2.26)(2.30): = G The normal phase is unstable if there is a normalisable solution with incoming boundary normalisable mode. Our numerical procedure is as follows. The equation (2.35) depends only on four In figure 1(a) there is a special curve(solid one) representing quantum phase tranBreitenlohner-Freedman(BF) bound. First, the effective mass of scalar field, which is read from (2.35), near horizon at zero temperature is me2ff = lim m2 + q2gttAt2 = m2 (b) Quantum phase transitions phase. The orange solid curve in (a) and (b) are the same. Second, the near horizon geometry of an extremal black brane is given by AdS2 R2 with the effective radius of AdS2 given by (2.33) real scalar field in AdS2 spacetime with the curvature radius of unity. Thus, we reach the same conclusion as (2.39) by (2.38). L22 = M 2L2d+1 = 4 Third, recall that, real scalar field in the AdSd+1 space with the radius Ld+1 is unstable with mass(M ) below the BF bound From these three data (2.36)(2.38), we conclude that the scalar field is unstable near m e2ff me2ff L22 = m2 Alternatively, this result can be obtained from (2.35). For the scalar field near horizon of the extremal black hole, (2.35) becomes above (below) the curve is stable or normal phase (unstable or superconducting) phase. It be always in the superconducting phase at zero temperature. Notice that, in this case, the Superconducting phase In the previous subsection we investigated the possibility of superconducting phase at finite temperature and zero temperature. Based on our result on quantum phase transition, transition. In this subsection we want to confirm our anticipation by constructing explicit superconducting background at finite temperature. Our numerical analysis is performed as follows. By shooting out from horizon we consider the condensate of the operator of dimension two, hO (2) . See (2.20). We may this paper. At high temperature we obtain only one solution, which agrees to an analytic solution of normal state (2.26)(2.30). At low temperature we find another solution with case it turns out that the superconducting solution always has a lower grand potential and becomes a ground state. The phase transition is continuous at a critical temperature(Tc). Figure 2 shows typical examples of phase diagrams for three points (a), (b), and (c) in figure 1(b). The three dimensional information in figure 2 may be summarized in a two practice, we have obtained such two dimensional plots first and rescaled them to make plots would be more convenient to represent overall features, even though all information can be compressed in two dimensional plots. Let us start with the point (a) and (b) in figure 1(b). They are always in superconthat the system undergoes a phase transition from superconducting phase to normal phase. Our numerical analysis confirms it and the phase diagram is shown in figure 2(a)(b), where (d) plot (a) and (b) together boundary at the critical temperature. Dark region below the surface is superconducting phase while region above the surface is normal phase. the meshed surface is the phase boundary at the critical temperature. Dark region below the surface is superconducting phase while region above the surface is normal phase. Figfigure 2(a)(b) to larger values and combine them for comparison, where figure 2(a) is red. The red mesh is above the black mesh, which means that a large q enhances superconductivity, as at the zero temperature in figure 1. However, the phase transition line coincides This competition is reflected on the phase boundary surfaces in figure 2. In words, the from the previous studies. In Q-lattice model [32, 33] and single scalar model [34] the critical temperature decreases as momentum relaxation effect increases while in ionic lattice model [13] the critical temperature increases monotonically as lattice effect increase. In figure 1(b), the point (c) is different from (a) and (b), in that the system at (c) at The color here matches the color of the lines in figure 2(a). In other words, we compute condensate along the vertical line(temperature) standing on the colored-lines in figure 2(a) looks similar to the superconducting dome in cuprate superconductor phase diagram when superconductivity by comparing figure 2(a) and (c). For example, in figure 3, we show the condensate as a function of temperature. At the critical temperature condensate starts forming and increases continuously as temperature goes down. At very small temperature our numerics becomes not reliable and we did not matches the color of the lines in figure 2(a). In other words, we compute condensate along the vertical line(temperature) standing any point on the colored-lines in figure 2(a). The the lower bound. We finish this subsection by discussing on the on-shell action of the ground state. To calculate a thermodynamic potential for the black hole solutions we calculate the on-shell SE = iSren , 7We have not yet obtained the solution at zero T . The data for the plot is numerically computed up to example, following [45]. Let us first consider the Euclidean bulk action SbEulk = 4 g Lbulk , Thus the bulk Lagrangian is In superconducting phase which defines Lbulk. It can be computed following [6]. The xx-component of the Einstein equation gives a useful relation: Gxx = r (Lbulk R) + r2 R + Lbulk = R . where Sren consists of four-dimensional Sbulk and three dimensional Sbdy: Sren SHHH + S + SGH + Sct . Sbulk Sbdy The first three terms are defined in (2.1), (2.3) and (2.2) respectively, and the last term is the counter term for holographic renormalisation [6]: Sct = 3 L 2 I I + 1(nM M + nM M ) + 2||2/L outgoing normal vector. See appendix A for more details. SbEulk = After adding SbEdy the total on-shell action is finite and reads SE = SbEulk + SbEdy = dx1dx2W = where we consider a homogeneous and equilibrium system. Thus V2 is the volume of the system and the length of the Euclidean time circle is identified with inverse temperature(T ). W is a thermodynamic potential per unit volume: W = G 1 Z 1 . Z derived from (A.10) and the definition of the Bekenstein-Hawking entropy, the equa W = which agrees to [20]. In superconducting phase, = hTtti = 2G(1) , = J t = A(1) , s = 4rh2 , where G(1) and A(1) are numerical values. Lack of the analytic relation such as (2.55) = hTtti = 2m0 , = J Optical conductivity sidering small fluctuations of relevant gauge, metric, scalar fields around the normal and superconducting background we obtained in the previous section. From here on, we set Fluctuations for optical conductivity: equations and on-shell action Z d eitax(, r) , of which boundary dual operator is electric current. The fluctuation is chosen to be independent of x and y, which is allowed since all the background fields affecting the equations of motion are independent of x and y. Because of rotational symmetry in x-y plane, it is Z d eitr2htx(, r) , constant as r goes to infinity. are derived from (2.4)(2.7): In momentum space, the linearized equations of motion around the background (2.8) 2 2 we have the analytic solutions ((2.26)(2.30)) but for superconducting phase we have it We solve these equations with two boundary conditions: incoming boundary conditions at the black hole horizon and the Dirichlet boundary conditions at the boundary. First, near the black hole horizon (r 1) the solutions are expanded as htx = (r 1)+1 ht(xI) + ht(xII)(r 1) + , ax = (r 1) a(xI) + a(xII)(r 1) + , = (r 1) (I) + (II)(r 1) + , htx = ht(x0) + ax = a(x0) + and we fix the values of the leading terms as boundary conditions. Ge 2 axax0 2 Plugging the solutions into the renormalized action (2.42), we have a quadratic order where we discarded the contribution from the horizon as the prescription for the retarded Greens function [46]. In particular, with the spatially homogeneous ansatz (3.1)(3.3), the quadratic action in momentum space yields Sr(e2n) = a(x0)ht(x0) + 2G(1)ht(x0)ht(x0) + a(x0)a(1) x 3ht(x0)ht(x3) + 3(0)(3) , (3.14) where V2 is the two dimensional spatial volume R dxdy and we omit the term proportional to be positive following the prescription in [46]. The on shell action (3.14) plays a role of the generating functional for two-point Greens functions from the first two terms in (3.14). The other three terms are nontrivial and to linearity of equations (3.4)(3.6), we can always find out the linear relation between such a relationship in a more general setup in the following subsection and continue the computation in that setup. Numerical method on-shell action S(2) = 2 A systematic numerical method for a system with multi fields and constraints were developed in [23] based on [37, 38]. We summarise it briefly and refer to [23] for more details. where the index a may include components of higher spin fields. For convenience, rp is a(r) = (r 1)a (a + a(r 1) + ) , boundary conditions. To compute the retarded Greens function we choose the incoming boundary condition [46], fixing N initial conditions. The other N initial conditions, denoted 1 1 1 . . . 1 1 . . . 1 (near boundary) , where Sia are the sources(leading terms) of i-th solution and Oia are the operator expectation matrices of order N , where the superscript a runs for row index and the subscript i runs for column index. J a + + a(r) = ia(r)ci Siaci + + Oi c (near boundary) 1 Z Sr(e2n) = 2 0 (2)d Jk Aab(k) + BacOic(S1)ib(k) J 1 Z d with real constants cis. For any given J a we can always find ci and the corresponding response Ra is expressed as Ra = Oiaci = Oia(S1)i J b . b A general on-shell quadratic action in momentum space has the form of Sr(e2n) = as a row matrix. For example, the action (3.14) can be written in the form (3.23) with A = 0 2G(1) 0 , B = 0 3 0 , for a careful derivation of the gauge invariant Greens function matrix. Notice that for one field case without mass term, this is the well known structure of the retarded Greens function: A = 0 and GR O/S. In summary, to compute the retarded Greens function. We need four square matrices of order N (the number of fields): A, B, S, O. A and B can be read off from the action (3.23), after taking care of all divergences by counter terms. To construct regular matrices, S and O, we solve a set of differential equations N times with independent initial conditions. The For our equations there is one subtlety caused by a symmetry of the system. Solving the equations near horizon with the expansions (3.7)(3.9) we find that only two of a(xI), residual gauge orbit. tion (2.4)(2.7), since the residual gauge transformation leaves the linearised equation of motion invariant. Therefore, our procedure is equivalent to formally adding a constant solutions of the equations to the solution set {Sia}. With the matrices S and O, which is numerically computed, we may construct a 3 3 matrix of the retarded Greens function. We will focus on the 22 submatrix corresponding to a(x0) and ht(x0) in (3.10). Siince a(x0) is dual to U(1) current Jx and ht(x0) is dual to energymomentum tensor Ttx From the linear response theory, we have the following relation between the response functions and the sources: (xT )/T As shown in [2, 3, 23], by taking into account a diffeomorphism invariance, (3.29) can be expressed as 0.66, 0.27 (dotted, red, orange, orange, green, blue) 13.2, 3.5, 1, 0.95, 0.7, 0.4, 0.25 (dashed, dotted, red, orange, green, blue, purple) The purple line fits well too. data in figure 4 and solid lines are Drude-like fits. The comparison of (3.28) and (3.30) yields Electric/thermal/thermoelectric conductivites color of curves represents temperature ratio, T /Tc, where Tc is the critical temperature. The numerical values of temperature ratio are shown in the caption. In particular the dotted black curve8 is for the temperature above Tc, which is in metal phase and the red curve corresponds to the critical temperature (in practice, it is slightly higher than the black curves in figure 4 and 7, but distinguishable in figure 8. Therefore, in normal phase the DC conductivity is finite due to momentum relaxation and in superconducting phase the DC conductivity is infinite, which is one of the hallmarks of superconductor. from normal components in superconducting phase, implying a two-fluid model. For small at higher temperature, the peak becomes sharper. For the sake of comparison we used a Therefore we zoom in figure 4(a) in figure 5(a). The data points well fit to the formula orange, green) better fit to which is shown in figure 5(b). K0 is related to pair creation and it was necessary also in Indeed (3.32) is understood as an approximation of (3.33) when K0 is negligible compared becomes zero at low temperature (green and blue line in figure 4(c)), but it is possible that Kn is finite even at zero T [13, 31, 32]. Drude peak becomes a non-Drude peak [23]. The figure 5 in [23] suggests that the critical the fit of figure 5 (b) is not as good as (a) and starts deviating from (3.33).9 Ferrell-Glover-Tinkham (FGT) sum rule: FGT Z dRe[n() s()] 2 Ks = 0 , standing of the range of applicability of the Drude model, it is important to analyse Kn more extensively. FGT -0.002 13.2, 3.5, 1, 0.95, 0.7, 0.4, 0.25 (dashed, dotted, red, orange, green, blue, purple) temperatures with the same colors in figure 4. 0.66, 0.27 (dotted, red, orange, orange, green, blue) for T > Tc since the spectral weight is constant in metal phase [23]. We computed (3.34) numerically for all cases in figure 4 and showed that the FGT sum rule is satisfied up to 103 in figure 6. invariance. So the delta function must have a different origin, which may be a new type of superconductivity. Interestingly, even in this case, the FGT the sum rule works. The deficit of spectral function may be interpreted as a deficit of particle-anti-particle pairs, 0.66, 0.27 (dotted, red, orange, orange, green, blue) 13.2, 3.5, 1, 0.95, 0.7, 0.4, 0.25 (dashed, dotted, red, orange, green, blue, purple) represent the same temperature. which will condense. It may imply a new pairing mechanism of particles and anti particles decreases quickly as temperature goes down as shown in figure 10. The thermoelectric confirmed numerically. Conclusions which agrees to the Ward identity [47]. and Re[G12(0)] is finite if is finite. Then, Im[(0)] = ReG12(0) In this paper, we studied a simple holographic superconductor model incorporating momentum relaxation. The model consists of two parts: the HHH model [5, 6] and massless scalar fields sector for momentum relaxation [20], where the strength of momentum relaxation is One of the interesting features of the model is that the existence of a new type of impurity [23]. The electric optical conductivity of this new superconductor satisfies the FGT sum rule too. The deficit of the spectral weight may be interpreted as a deficit of particle-anti-particle pairs which are condensed. ent from the previous studies. As momentum relaxation effect increases, in a Q-lattice model [32, 33] and a single scalar model [34] the critical temperature decreases while in the ionic lattice model [13] the critical temperature increases. The condensate has the upper where Ks and Kn are supposed to be proportional to the superfluid density and normal 1, K0 becomes negligible. The However, the Ferrell-Glover-Tinkham (FGT) sum rule is satisfied for all cases regardless and relations. The temperature dependence of the parameters are of physical importance. superfluid density, proportional to Ks, will be essential to investigate Homes law [48] holographically [49]. It will be also useful to obtain analytic formula for DC conductivities from the horizon data in superconducting phase as in metallic phase [30]. While the model we considered shows many interesting features as metal and superconductor, it also has shortcomings. The electric DC conductivity in normal phase is temperature independent and the insulator phase is lacking. It would be interesting to consider superconducting phase without those shortcomings. Indeed, there is a simple generalization of the model that provides a temperature dependent DC conductivity and an insulating phase at small temperature [50], so it would be interesting to construct a superconductor model based on this background. It would be also interesting to extend our model to the d-wave superconductors [51, 52] and to consider dynamical gauge fields [53]. Acknowledgments We would like to thank Kiseok Kim, Dongsu Park, Yunseok Seo and Sang-Jin Sin for valuable discussions and correspondence. The work of KYK and KKK was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT & Future Planning(NRF-2014R1A1A1003220). M. Park was supported by by the National Research Foundation of Korea (NRF) funded by the Korea government with the grant No. 2013R1A6A3A01065975 and TJ Park Science Fellowship of POSCO TJ Park Foundation. We acknowledge the hospitality at APCTP(Aspects of Holography, Jul. 2014) and Orthodox Academy of Crete(Quantum field theory, string theory and condensed matter physics, Sep. 2014), where part of this work was done. One point functions We briefly summarize how to compute one point functions holographically. Let us consider an ADM decomposition as follows: is given as K = 21N 0 . d4xg R(3) KK + K2 + 1 X2 (I )2 |D|2 m2||2 , Sct = Let us consider a renormalised action(Sren) consisting of a bare action(S0) and a counter action(Sct) by taking into account the holographic renormalization: Sren = S0 + Sct , SreIn | = grI + where the variations from the counter action are 2 L2 + G[] L2 I I + 1 (21||nAA|| + 2||2/L) , The expectation values of the energy momentum tensor, the current and the scalar operators in the dual field theory can be computed as hTi = lim r , hJ i = lim r3 1 2 r r r =1 = lim r2 /L . r Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. (2009) 343001 [arXiv:0904.1975] [INSPIRE]. phase transitions, arXiv:1110.3814 [INSPIRE]. Duality, Hot QCD and Heavy Ion Collisions, arXiv:1101.0618 [INSPIRE]. 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Keun-Young Kim, Kyung Kiu Kim, Miok Park. A simple holographic superconductor with momentum relaxation, Journal of High Energy Physics, 2015, 152, DOI: 10.1007/JHEP04(2015)152