Coherent/incoherent metal transition in a holographic model

Journal of High Energy Physics, Dec 2014

We study AC electric (σ), thermoelectric (α), and thermal \( \left(\overline{\kappa}\right) \) conductivities in a holographic model, which is based on 3+1 dimensional Einstein-Maxwell-scalar action. There is momentum relaxation due to massless scalar fields linear to spatial coordinate. The model has three field theory parameters: temperature (T), chemical potential (μ), and effective impurity (β). At low frequencies, if β < μ, all three AC conductivities (σ, α, \( \overline{\kappa} \)) exhibit a Drude peak modified by pair creation contribution (coherent metal). The parameters of this modified Drude peak are obtained analytically. In particular, if β ≪ μ the relaxation time of electric conductivity approaches to \( 2\sqrt{3\mu }/{\beta}^2 \) and the modified Drude peak becomes a standard Drude peak. If β > μ the shape of peak deviates from the Drude form (incoherent metal). At intermediate frequencies (T < ω < μ), we have analysed numerical data of three conductivities (σ, α, \( \overline{\kappa} \)) for a wide variety of parameters, searching for scaling laws, which are expected from either experimental results on cuprates superconductors or some holographic models. In the model we study, we find no clear signs of scaling behaviour.

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Coherent/incoherent metal transition in a holographic model

Keun-Young Kim 0 3 Kyung Kiu Kim 0 3 Yunseok Seo 0 1 Sang-Jin Sin 0 2 0 Open Access , c The Authors 1 Research Institute for Natural Science, Hanyang University 2 Department of Physics, Hanyang University 3 School of Physics and Chemistry, Gwangju Institute of Science and Technology We study AC electric (), thermoelectric (), and thermal () conductivities in a holographic model, which is based on 3+1 dimensional Einstein-Maxwell-scalar action. There is momentum relaxation due to massless scalar fields linear to spatial coordinate. The model has three field theory parameters: temperature (T ), chemical potential (), and effective impurity (). At low frequencies, if < , all three AC conductivities (, , ) exhibit a Drude peak modified by pair creation contribution (coherent metal). The parameters of this modified Drude peak are obtained analytically. In particular, if the relaxation time of electric conductivity approaches to 23/2 and the modified Drude peak becomes a standard Drude peak. If > the shape of peak deviates from the Drude form (incoherent metal). At intermediate frequencies (T < < ), we have analysed numerical data of three conductivities (, , ) for a wide variety of parameters, searching for scaling laws, which are expected from either experimental results on cuprates superconductors or some holographic models. In the model we study, we find no clear signs of scaling behaviour. 1 Introduction 2 3 4 Introduction AdS-RN black branes with scalar sources General action AdS-RN black brane General numerical methods with constraint Electric/thermal/thermoelectric AC conductivities Green functions and Transport coefficients Optical conductivity and coherent/incoherent metal Thermoelectric and thermal conductivity Holographic methods (gauge/gravity duality) provide novel tools to study many properties of strongly correlated systems by analysing the corresponding higher dimensional gravity theories [14]. In particular it gives a new way of computing transport coefficients such as viscosity, relaxation time, and electric/thermal conductivities as well as various equilibrium thermodynamic quantities. In this paper we focus on electric, thermoelectric, and thermal conductivities of strongly coupled systems by holographic methods. The early works on this subject, holographic conductivity, have dealt with the systems with translation invariance [2]. However, any system with finite charge density and translation invariance will exhibit an infinite electric DC conductivity. The reason is straightforward: a constant electric field will accelerate charges indefinitely because there is no momentum dissipation, which is implied by translation invariance. Real condensed matter systems will not have translation symmetry. It is broken by a background lattice or impurities. To remedy this infinite conductivity problem, there have been a number of proposals to introduce the momentum dissipation effect in the framework of holography. They fall into two classes: models with inhomogeneous boundary conditions (IBC) and homogeneous boundary conditions (HBC).1 In IBC models, one gives some bulk fields inhomogeneous boundary conditions breaking translation invariance explicitly [914]. One may introduce a spatially modulated scalar 1There is an earlier conceptually different idea. It considers a model of a small number (Nf ) of charged degrees of freedom in a bath of a large number (Nc) of neutral degrees of freedom. If Nf is parametrically much smaller than Nc, the momentum of charged degrees of freedom can be absorbed into a bath. For example, see [58]. black hole of Einstein-Maxwell-scalar system, we may understand the translation symmetry breaking by the Ward identity (2.13) hTi = F hJ i + hOi , In HBC models, one does not impose explicit inhomogeneous boundary conditions, but find a way to break translation invariance effectively. A few models have been studied. Massive gravity approach [1518] introduces mass terms for some gravitons. It breaks bulk diffeomorphism invariance and consequently violates the conservation of the stress-energy tensor in the boundary field theory. Some models exploit a continuous global symmetry of the bulk theory [14, 19, 20], where, for example, the global phase of a complex scalar field breaks translational invariance.2 In [23], a simple model with massless scalar fields linear in spatial coordinate, breaking translation symmetry, was introduced.3 It was extended further in [24, 25]. On a technical level, IBC models require to solve complicated coupled partial differential equations (PDE) because of explicit inhomogeneous boundary condition. An advantage of HBC models is that they allow to deal with coupled ordinary differential equations (ODE) because the stress tensor still remains independent of field theory directions and all bulk fields can be treated as functions of the holographic direction. This technical advantage enables us to analyse a model more easily and extensively. Thus it will make possible more analytic and universal understanding on momentum dissipation mechanism at strong coupling, even though its microscopic field theory interpretation is unclear yet. In this paper, we study AC electric, thermoelectric, and thermal conductivities of a HBC model proposed in [23], focusing on a Drude nature at low frequencies and scaling laws at intermediate frequencies. The model we study is based on the Einstein-Maxwell-scalar coordinate are considered so that translation symmetry is broken. Because they enter the and gauge field still can be homogeneous in field theory direction. Furthermore, to have isotropic bulk fields the identical scalar field is introduced for every field theory spatial direction. In this model, the DC electric conductivity [23], thermoelectric and thermal conductivity [26] were computed analytically and our focus is on AC conductivities. AC electric conductivity was also studied in [25] and here we analyse it in greater detail as well as thermoelectric and thermal conductivities. For AC conductivities in other HBC models including massive gravity models we refer to [14, 2729]. At low frequencies, the Drude peak of electric conductivity has been observed in many holographic models with momentum dissipation. For example see [912]. 2Some of these models may be related to IBC models [19]. In a similar spirit, there are models utilising a Bianchi VII0 symmetry to construct black holes dual to helical lattices [21, 22]. 3This model may be understood also based on [19]. A single massless complex scalar with constant in (2.6) of [19] gives rise to a massless axion linear in the x1 direction. a quasi-particle picture. However, it was shown that this Drude-like peak can be realised even when there is no quasi-particle picture at strong coupling if the translation symmetry is broken weakly [30]. In this context, metal without quasi-particle can be divided into two classes: coherent metal with a Drude peak and incoherent metal without a Drude peak [31]. However, since our model is based on AdS-RN black brane solution, there will be a term containing the contribution from pair production affected by net charge density, which we controlling the strength of the translation symmetry breaking, we may investigate how coherent/incoherent metal phase is realised.4 Indeed, In our model, we find that when and a modified Drude form is reduced to a standard Drude from. Also we confirm the sum rule is satisfied for both cases, Drude and non-Drude. For thermoelectric and thermal conductivities, qualitatively the same results are obtained. potential, it was shown experimentally that certain high temperature superconductors in the normal phase exhibit scaling law models in a following modified form. behaviours have been produced while in [14, 19, 25] no scaling law has been observed. In our model we have analysed electric, thermoelectric, and thermal conductivities in a wide range of parameters for both scaling laws (1.4) and (1.5). However it seems that there is no robust scaling law, which agrees to the conclusion in [25]. From holographic perspective, the computation of electric, thermoelectric, and thermal conductivities are related to the Dynamics of three bulk fields fluctuations (metric, gauge, scalar fields). Their dynamics are determined by equations of motion, a system of second order coupled ODEs. From the on-shell quadratic action for these fluctuations we can read off the retarded Greens functions relevant to three conductivities. In the case that many bulk fields are coupled, the computation of the holographic retarded Greens functions is not very straightforward. To facilitate solving this important problems we introduce a 4The same question was addressed based on analytic DC conductivities in [22, 24]. systematic numerical method following [33, 34] adapted to our purpose. This method, used to compute conductivities in this paper, can be applied to other models and problems. It will be useful especially when many fields are coupled and the system has constraint coming from the residual gauge symmetry. This paper is organised as follows. In section 2, after reviewing Einstein-Maxwell theory with massless scalar fields in general, we focus on a specific ground state solution to introduce momentum relaxation. To set up the stage for AC conductivities, we summarise equations for small fluctuations of relevant metric, gauge and scalar fields around the ground state. In section 3, we present a general numerical method to compute retarded Greens functions when many fields are coupled. By using this method, in section 4, we compute AC electric, thermoelectric, and thermal conductivities. At low frequencies we focus on the shape of the peak, Drude or non-Drude, and at intermediate frequencies we search for possible scaling laws. In section 5 we conclude. AdS-RN black branes with scalar sources In this section we briefly review the holographic model of momentum relaxation studied in [23]. We summarize essential minimum to set up stage for our study, AC conductivities, and refer to [23, 25] for more details and extensions. General action boundary M Let us start with the Einstein-Maxwell action on a four dimensional manifold M with SEM = 4 1 F 2 2 3 and the cosmological constant l are equal to 1 . The second term is the Gibbons-Hawking term required for a well defined variational problem with Dirichlet boundary conditions. extrinsic curvature. In order to have a momentum relaxation effect, we include two free massless scalars 4 g 2 1 X2 (I )2 . RMN = 1 F 2 2 1 X2 (I )2 + 1 X M I N I + M F MN = 0 , Given the solutions of these equations of motion, the holographically renormalised action (Sren) [35] is obtained by the on-shell action of where Sc is the counter term Sc = dx3 1 X2 I I coordinate system For a general understanding of Sren, it is useful to employ the Fefferman-Graham Near the boundary the solutions are expanded as ds2 = g = g(0) + 2g(2) + 3g(3) + , I = (0) + 2(2) + 3(3) + , I I I where leading terms g(0), A(0), (0) are chosen to be functions of the boundary coordinates I (incoming) boundary condition at the horizon. With small fluctuations, the renormalisation on shell action up to linear order in fluctuations reads Sr(e1n) = g(0) OI respectively. Their expectation values are The constraint (2.9) in terms of the one point function (2.11) yields the Ward identities hT i = hOI iI(0) + F(0)hJ i , which correspond to the invariance of the renormalised action under a U(1) transformation AdS-RN black brane We want to study the field theory at finite charge density and finite temperature with momentum dissipation. A gravity dual will be a charged black brane solution with broken translation symmetry. Indeed the equations (2.3)(2.4) admit the following solutions [36] ds2 = GMN dxM dxN = f (r)dt2 + m0 = r03 1 + 4r02 2r02 These analytic solutions have been reported in [36] and explored further in the context of functions, metric and gauge field are not, thanks to equal contributions from two scalars for two spatial coordinates. However, with only one scalar field, the solutions are anisotopic and this case has been studied in [37, 38]. dual field theory: from which, r0 yields T = r0 = 3r0 the parameter which controls momentum relaxation. The parameter m0 obtained by the density. In summary, for solutions (2.14)(2.17), one point function (2.11) is T tti = 2m0 , T xxi = hT yyi = m0 , hO1i = 0 , Now we want to study the responses of this system for small perturbations. In particular we are interested in the electric conductivity, which is related to the boundary current operators J~. Because of rotational symmetry in x-y space, it is enough to consider Jx. Since this operator is dual to the bulk gauge fields Ax, we consider a following linear fluctuation around the background The fluctuation is chosen to be independent of x and y. It is allowed since all the background fields entering the equations of motion are independent of x and y. The gauge field fluctuaand all the other fluctuations can be decoupled. Since we will work in momentum space, that it goes to constant as r goes to infinity. In momentum space, the linearised equations around the background are h0t0x =0 , which are obtained from (2.3)(2.4). Among these four equations, only three are independent.8 We need to solve these equations satisfying two boundary conditions: incoming boundary conditions at the black hole horizon and the Dirichlet boundary conditions at ax = a(x0) + 8The equations (2.24)(2.27) may be decoupled in terms of gauge invariant combinations [23]. The equation governing electric conductivity turns out to be the same as the one in the massive gravity model [16], while the equations for thermal/thermoelectric conductivities are different. and the DC electric conductivity [23] is Sr(e2n) = lim r 2 which is derived from which was computed at the horizon (not at the boundary) by rewriting the DC conductivity technique using r-independent quantity is in line with [17, 39], but does not work for finite the following section. General numerical methods with constraint The analytic method used in [23] is efficient to obtain the DC electric conductivity. However, to compute AC electric conductivity together with AC thermal/thermoelectric conductivity we have to resort to a numerical method. Since the conductivities are related to the retarded Greens functions through the Kubo formula, we need to obtain an action (generating functional) including two sources. A natural holographic starting point is the on-shell renormalised action to quadratic order in fluctuation fields [2, 40, 41]. In momentum space the on-shell action with the fluctuations (2.21)(2.23) reads htx where V2 is the two dimensional spatial volume R dxdy. Notice that the boundary term at the horizon is deleted according to the prescription to the retarded green function [40]. The boundary values of the fields are interpreted as the sources of some dual field theory operators, so we may readily read off the two point functions from the first two terms in (3.1), while the other three terms look not straightforward. However, thanks to linearity of equations (2.24)(2.27), we can always linear relation the action is reduced to the schematic form as follows. Sr(e2n) = a little bit to be more succinct and economical.9 a = 1, 2, , N , Sr(e2n) = lim r 2 ddk hak(r)Aab(r, k)bk(r) + ak(r)Bab(r, k)rbk(r)i , A and B are regular matrices of order N . The renormalized action (3.5) is assumed to contain all the counter terms. For example, see (3.1) for an action and (2.24)(2.27) for a system of equations. When the differential equations are second order we need to give 2N boundary conthem J a) are at the boundary. For numerical integration, we have to convert the boundary value problem to the initial value problem, by considering 2N canonical initial data at the event horizon. We solve the initial value problem for N independent initial value set and judiciously combine N set of solutions such that final value of solution is identical to the boundary value we have chosen. The procedure will be independent of the chose of canonical initial data as we will show below. where the index a includes components of higher spin fields. rq is multiplied such that the general on-shell quadratic action in momentum space has the form of 9In some cases the equations may be separable in terms of master fields. However, our method applies to any number of coupled fields straightforwardly and we dont need to try to figure out master fields. a(r) = (r 1)a (a + a(r 1) + ) where we omitted the subscript k for simplicity and correspond to incoming/outgoing boundary conditions. To compute the retarded Greens function we choose the incoming boundary condition [40], fixing N initial conditions. The other N initial conditions correincoming boundary condition, a determines a through horizon-regularity condition so that we can determine the solution completely. For example, ia may be chosen as 1 1 1 . . . 1 1 . . . 1 is expanded as10 where Sia denote the leading terms of i-th solution and Oia are the sub-leading term dewritten as regular matrices of order N , where the superscript a runs for row index and the subscript i runs for column index. The general solution is a linear combination of them: let (near boundary) , In this case, the second term of (3.5) may be written as J a = Siaci . with real constants cis. We need to choose ci such that the combined source term matches the boundary value J a: no constraint related to the diffeomorphic invariance, it can be done simply by noting that ci is expressed in terms of J a Bac(r, k)(crc1Ock)(S1)ibi J b + := [Cab(r, k)] J b + . the independence of the choice of the initial condition (3.7), because the different choice right multiplications in the solutions: S SR, O OR. With (3.10) and (3.12) the final boundary action yields Sr(e2n) = 1 Z where we reinserted the subscript k. Since matrices A, C are independent of J , the retarded Greens function is GaRb = Aab(, k) + Cab(, k) . example, see g(2) and (2) in (2.8). Those terms can be taken care of by counter terms and we dont write I them here to focus on essential ideas. Notice that for one field case without mass term, this is the well known structure of the of the coefficients of the subleading term and the leading term. 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 boundary action (3.5). S and O are given from the solution of a set of differential equations. We have to solve N times with 2N independent initial conditions to construct regular matrices of A + C. The precise form of C is shown in (3.12). Our story so far is for the system without constraint. In actual calculation, Einstein the space spanned by {S~1, , S~N1}. Then, equation always contains constraint equations (CE) due to the residual diffeomophism invariance of the linearised equation of motion. We describe how to fix this complication for the case of one CE for notational convenience. Generalization to two or more CE is straightforward. At the horizon, the CE relates the (initial) field values. For example, the last component of ~j can be determined by other components. So the space of initial value vectors (IVV) is N-1 dimensional subspace. Solving differential equation using such N-1 IVV gives, of course, only N-1 solutions. However, when we give boundary conditions, we formally assign N boundary values Ji. As a consequence, eq. (3.10) is not invertible! To N1 X S~j cj + S~0 c0 = J~. Now this equation is invertible. We can find proper c0 and cis for any choice of initial condition vectors to fit the given J~. Notice that S~0 does not generate true degree of freedom since it is along the diffeomorphism orbit direction. It is not hard to see why S~0 actually satisfies the on-shell condition: this happens since the residual gauge transformation leaves is not a true solution satisfying in falling conditions, it can serve as an element of a basis of the on-shell J space and on shell condition. that in this proof of basis independence, we used the fact that the differential equations define a linear map U (r) : ia Uba(r)ib, so that the evolution operator U (r) is a left multiplication (acting on upper index a), while basis change is a right multiplication acting on index i, so that initial data ia at the horizon and the final solution Sia at the boundary is multiplied by the same matrix R. In order to check the validity of our numerical method and code, we computed AC the figure 6 of [2]. It is a nontrivial consistency check of our method since the plot in [2] corresponding to delta functions has been obtained by solving a single equation of the gauge field ax, while we have solved coupled equations of ax and gtx. Of course if the coupled equations can be decoupled as shown in [2] there is no point of solving coupled equations. However, because this decoupling is not always possible it is important to develop a systematic and efficient method for coupled fields cases. In addition to the agreement of figure 1 to figure 6 of [2] 3 4r02 K = r0 figure 1(c) from figure 1(b). Electric/thermal/thermoelectric AC conductivities we want to attack our main problem, AC conductivity with moment dissipation generated can read off the conductivities from the action (3.1). To closely follow the general methods presented in section 3 we rewrite the action as s(re2n) Sr(e2n) = lim r Z d ha(r)Aab(r, )b(r) + a(r)Bab(r, )rb(r)i , B = f (r) 0 0 r2f (r) To compute the matrix C in (3.12) we have to solve the equations (2.24)(2.27), which we rewrite here setting r0 = 1: Since only three equations are independent we may solve any three of them. Near the black hole horizon (r 1) the solutions are expanded as r4 h0t0x =0 , 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) + With incoming boundary condition and initial values (3.7) at horizon we numerically integrate the equations from the horizon. For our equations there is one subtlety caused by a symmetry of the system. Analysing a complete basis to reconstruct a general boundary value vector J~. However, there is a S~0 = (a0x, ht0x, 0)T = (0, 1, i/)T . Notice that S~0 satisfies the equations of motion, as we mentioned in earlier section. Since S~0 happened to be r-independent, it is equivalent to formally add a constant solutions ax = 0 , htx = ht0x , introduced in [46]. It is interesting that (4.10) is similar to (3.8) of [19], which is a condition imposed at infinity to extract the gauge-invariant conductivity.11 They first choose initial data such that the final solution lies on the gauge orbit and then using the gauge transfor i ht(x0) = 0 is nothing but the equation of the gauge orbit (in the space of leading component of solutions) passing (a0x, 0, 0). one can calculate electric conductivity by a(x1)/a(x0). In this approach, one can calculate only electric conductivity and shooting is cumbersome. simultaneously. Again, we emphasise that true physical degree of freedom is 2 dimension and S~0 does not generate a true physical degree of freedom. Green functions and Transport coefficients Having computed numerically the matrices S and O, we may construct a 3 3 matrix of retarded Greens function. We will focus on the 2 2 submatrix corresponding to a(x0) and ht(x0) in (4.8). Since a(x0) is dual to U(1) current Jx and ht(x0) is dual to energy-momentum tensor Ttx the Greens function matrix may be written as 11We thank Aristomenis Donos and Jerome Gauntlett for pointing out this similarity. where we introduced the second term for notational simplicity. From the linear response theory, we have the following relation between the response functions and the sources: We want to relate the Greens functions (4.11) to phenomenological transport coefficients. Our goal is to study the electric, thermal, thermoelectric conductivities defined as (xT )/T temperature gradient. By taking into account a diffeomorphism invariance [2, 3], (4.13) can be expressed as corresponding to a delta function in (a) transition to incoherent metal is manifest. are the analytic DC values (4.16). Comparing (4.14) and (4.12) we have response theory: one has to subtract static term whenever the latter is non-zero [47]. This is related to the fact that for the causal Green functions, we have to apply source fields only in the past not in future. Without such subtraction, we would have unphysical term [27]. In summary, we numerically compute G11, G12, G21, G22 by (3.14) and combine them as (4.15) for physical conductivities. electric conductivity respectively. In figure 4 and 5 we analyse the conductivity at small maximum value of the peak (DC conductivities) decreases. In this variation, we checked that the area of the real part of the conductivity does not change numerically. The area inferred from the imaginary part of the conductivity. This is an example of a sum rule and we have confirmed it for various parameters. Numerical DC conductivities agree to the analytic result (2.29) There are two issues on finite frequency regime: one is Drude-like peaks at small frequency and the other is possible scaling laws at intermediate frequency regime. Let us is broken weakly and we expect to have a Drude form according to [30]. For large values much these peaks can resemble the Drude model, Let us examine the Ward identity. At the level of fluctuation the Ward identity (2.13) is Comparing with the Drude model 12We thanks the referee to point this out. , it turns out that hOi hpxi, our model is based on AdS-RN black brane solution, there will be a contribution from pair K = r0 r0 = which is defined in (2.19). effect. For T time (4.24) yields The expression (4.24) is not very illuminating so we make a plot of the relaxation time We compare numerical data (blue dotted lines) with a Drude model (red solid To check the validity of our analytic expression of the Drude model (4.19) with parameters (4.20) and (4.24), we have made numerical plots for a wide range of parameters and compared with (4.19). Figure 5(a,d) and (c,f) are examples showing a good agreement of numerical data to (4.19) and deviation from (4.19) respectively.14 Blue dotted lines are 13This form of Drude conductivity implicitly appeared in [48] with shifted pole due to the magnetic field. 14Similar plots were obtained independently by Blaise Gouteraux and Richard Davison and presented at the workshop, Holographic methods and application, Iceland, August, 2014. numerical data and red solid curves are the analytic expression (4.19). In figure 5 (c,f) if fitting curve is slightly improved, but it is still deviated from (4.19). In these examples, peak is not a Drude form. It is a concrete realisation of coherent/incoherent transition Next, we want to investigate the scaling property in the intermediate frequency regime. superconductors in the normal phase exhibits scaling law models with momentum dissipation. In models studied in [9, 10, 12] modified scalings (4.28) have been reported while in [14, 19, 25] no scaling law have been observed. With our model we have analysed several cases for a wide range of parameters to search a scaling behaviour. However it seems that there is no robust scaling law. -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 robust scaling law under change of parameters. These plots are presented to show how the constant shift C in (4.28) can improve the scaling behaviour. do not see a scaling behaviour of the form (4.27). Now we want to investigate if there is a modified scaling law motivated by previous holographic models [9, 10, 12, 16]. which is shown in figure 7. Interestingly, in this case, the constants B and C in (4.28) determined. However, this approximate scaling behaviour is not precise15 and robust under change of parameters. We present figure 7 to show how the constant shift C in (4.28) can improve the scaling behaviour of figure 6(c) even though it is not an evidence of a scaling behaviour. After numerical analysis with a wide variety of parameters and cases we do not see a scaling behaviour of the form (4.28), which agrees to the conclusion in [25]. Thermoelectric and thermal conductivity analytically computed in [26] In order to discuss the Wiedemann-Franz law, we compute the ratio of the DC thermal conductivity to the DC electric conductivity as follows where we took low temperature limit in the last expression since the Wiedemann-Franz law L = but the numerical values are different from the Fermi-liquid case, as expected in a nonFermi liquid, see e.g. [50]. At small frequencies, like electric conductivity, thermoelectric and thermal conductivbe obtained analytically by using the hydrodynamics results in [4345]. Figure 9 shows an excellent agreement of numerical data to (4.34), where the blue dots are numerical values are fitting curves of the from (4.34). that three relaxation time are almost the same. This is because in all three cases the Drude frequencies, we do not see any scaling law unlike [10]. Conclusions based on the 3+1 dimensional Einstein-Maxwell-scalar action. Momentum is dissipated due to massless scalar fields linear to every spatial coordinate. purity. There are two more free parameters in the model: temperature (T ) and chemical given analytically. They depend on only holographic direction because the scalar field gauge, and scalar fields) relevant for three conductivities can be chosen to be functions of only the holographic direction, so the computations can be done by coupled ODEs rather 16We thanks the referee for pointing this out. independent constants, but the numerical values are different from the Fermi-liquid case. We presented a concrete realisation of coherent/incoherent transition induced by impurity conductivities show a modified Drude peak. For example, for electric conductivity, with different parameter values. For example, the relaxation times are different for three below (2.20). If we can use massive one the constant plateaux will disappear. some holographic models [9, 10]. but we find no robust scaling law, which agrees to the conclusion in [14, 19, 25]. In [51] a mechanism to engineer scaling laws was provided, where translation symmetry is not broken. It would be interesting to generalize it to our case. by Ward identities, and once electric conductivity is given the other two are algebraically determined [2, 3]. In our model the relationship between them are more complicated, involving the background scalar fields. It will be interesting to understand how their We introduced a general numerical method to compute the holographic retarded Greens functions when many fields are coupled. This method, used to compute three conductivities in this paper, can be applied also to other models and problems such as [19, 20, 22, 24, 25]. It would be interesting to extend our analysis to dyonic black holes and holographic superconductors [49]. It would be also interesting to study the models based on other free massless form fields introduced in [36], which may be used to engineer certain desired properties of condensed matter systems. Acknowledgments We would like to thank Yunkyu Bang, Richard Davison, Aristomenis Donos, Jerome Gauntlett, Xian-Hui Ge, Blaise Gouteraux, Takaaki Ishii, Yan Liu, Ya-Wen Sun, and Marika Taylor 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 (NRF2014R1A1A1003220). The work of SS and YS was supported by Mid-career Researcher Program through the National Research Foundation of Korea (NRF) grant No. NRF2013R1A2A2A05004846. YS was also supported in part by Basic Science Research Program through NRF grant No. NRF-2012R1A1A2040881. 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), and at CERN (Numerical holography, Dec. 2014), where part of this work was done. Open Access. 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Keun-Young Kim, Kyung Kiu Kim, Yunseok Seo, Sang-Jin Sin. Coherent/incoherent metal transition in a holographic model, Journal of High Energy Physics, 2014, 170, DOI: 10.1007/JHEP12(2014)170