Thermoelectric conductivities at finite magnetic field and the Nernst effect

Journal of High Energy Physics, Jul 2015

We study the thermoelectric conductivities of a strongly correlated system in the presence of a magnetic field by the gauge/gravity duality. We consider a class of Einstein-Maxwell-Dilaton theories with axion fields imposing momentum relaxation. General analytic formulas for the direct current (DC) conductivities and the Nernst signal are derived in terms of the black hole horizon data. For an explicit model study, we analyse in detail the dyonic black hole modified by momentum relaxation. In this model, for small momentum relaxation, the Nernst signal shows a bell-shaped dependence on the magnetic field, which is a feature of the normal phase of cuprates. We compute all alternating current (AC) electric, thermoelectric, and thermal conductivities by numerical analysis and confirm that their zero frequency limits precisely reproduce our analytic DC formulas, which is a non-trivial consistency check of our methods. We discuss the momentum relaxation effects on the conductivities including cyclotron resonance poles.

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Thermoelectric conductivities at finite magnetic field and the Nernst effect

Received: March Thermoelectric conductivities at finite magnetic field and the Nernst effect Keun-Young Kim 0 1 2 4 5 Kyung Kiu Kim 0 1 2 4 5 Yunseok Seo 0 1 2 3 5 Sang-Jin Sin 0 1 2 3 5 0 We consider a class 1 Seoul 133-791 , Korea 2 Gwangju 500-712 , Korea 3 Department of Physics, Hanyang University 4 School of Physics and Chemistry, Gwangju Institute of Science and Technology 5 Open Access , c The Authors We study the thermoelectric conductivities of a strongly correlated system in the presence of a magnetic field by the gauge/gravity duality. of Einstein-Maxwell-Dilaton theories with axion fields imposing momentum relaxation. General analytic formulas for the direct current (DC) conductivities and the Nernst signal are derived in terms of the black hole horizon data. For an explicit model study, we analyse in detail the dyonic black hole modified by momentum relaxation. In this model, for small momentum relaxation, the Nernst signal shows a bell-shaped dependence on the magnetic field, which is a feature of the normal phase of cuprates. We compute all alternating current (AC) electric, thermoelectric, and thermal conductivities by numerical analysis and confirm that their zero frequency limits precisely reproduce our analytic DC formulas, which is a non-trivial consistency check of our methods. We discuss the momentum relaxation effects on the conductivities including cyclotron resonance poles. respondence and; the; Nernst; effect; Holography and condensed matter physics (AdS/CMT); Gauge-gravity cor- 1 Introduction 2 General analytic DC conductivities at finite magnetic field 2.1 Nernst effect 3 Example: dyonic black branes with momentum relaxation 3.1 3.2 4.1 5 Conclusions Introduction Model with massless axions 3.3 DC conductivities: Hall angle and Nernst effect 4 Numerical AC conductivities Equations of motion and on-shell action Numerical method 4.3 AC conductivities and the cyclotron poles Strongly coupled electron systems show many interesting phases such as non-Fermi liquid, high Tc superconductor and pseudo gap phase. Some of the most important and basic theoretical method to compute conductivities to explain and guide experiments. However, due to strong coupling, the perturbative analysis of quantum field theory does not work and we don’t have a reliable systematic tool to compute them. Gauge/gravity duality is an approach for such strong coupling problems, and it has been developed as a method for conductivity [1, 2]. Some early works treated systems which have translation invariance. However, at finite charge density the direct current (DC) conductivities in such systems are infinite. To solve this problem, it is essential to introduce momentum relaxation. For this, several ideas have been proposed. The most straightforward way is to impose inhomogeneous boundary conditions of the bulk fields to break translation invariance [3–8]. Massive gravity models studied in [9– 13] give mass terms to gravitons, which break the spatial diffeomorphisms (not the radial and temporal ones), and consequently break momentum conservation of the boundary field theory with translational invariance unbroken [11]. Holographic Q-lattice models and those with massless linear dilaton/axion fields [14–23] take the advantage of a continuous global symmetry of the bulk theory. Some other models utilise a Bianchi VII0 symmetry to construct black holes dual to helical lattices [24–27]. All these models yield finite DC conductivities as desired. However, all models with momentum relaxation, except [23], did not include the magnetic field. Since the transport properties at finite magnetic field such as the quantum Hall effect, the Nernst effect, and the Hall angle, are also important basic probes for strongly correlated electron system, it is timely and essential to develop the methods for them in the presence of momentum relaxation. Indeed, the holographic analysis on conductivities at finite magnetic field was one of the pioneering themes opening up the AdS/CMT (condensed matter theory) [28–30]. The purpose of our paper is to extend them by implementing momenturm relaxation holographically.1 This paper is also a companion of [19, 32, 33], where thoroughly in the absence of magnetic field. We consider a general class of Einstein-Maxwell-dilaton theories with axion fields imfollowing the method developed in [34]. Based on these formulas we discuss the model independent features of the Nernst signal. Notice that the Nernst signal (2.55) is zero in the holographic model without momentum relaxation, since the electric conductivity is infinite. Thus, momentum relaxation is essential for the Nernst effect. The Nernst signal has interesting properties which could support the existence of the quantum critical point (QCP). As we approach to the QCP or the superconducting domain, the strength of the Nernst signal becomes stronger and shows a non-linear dependence on the magnetic field, which is different from the expectation based on the Fermi liquid theory [35].2 See [29, 31], for pioneering works on the Nernst effect by the holographic approach and the magnetohydrodynamics with a small impurity effect. We deal with similar topics by means of a general class of holographic models encoding momentum relaxation, where we assume that momentum relaxation is related to finite impurity density, which could be large. Note, however, that this relation is not proven yet. After discussions on a class of models, we study in detail the dyonic black hole background [28–30], modified by the specific axion fields introduced in [16]. We numerically their zero frequency limits agree to the DC formulas that we have derived analytically. It recovers the results in [28] if momentum relaxation vanishes. We discuss the momentum relaxation effect on the conductivities including the cyclotron resonance poles, which was first observed in [30]. This paper is organized as follows. In section 2, we consider a general class of EinsteinMaxwell-dilaton theories with axion fields and derive general formulas for the DC electric, thermoelectric, and thermal conductivity at finite magnetic field as well as the Nernst 1In [31], momentum relaxation is introduced perturbatively in the hydrodynamic limit. 2These anomalous behavior may be described by a liquid of quantized vortices and anti-vortices in non-superconducting phase. See [37] for a speculative point of view. signal. In section 3, as an explicit example, we analyse the dyonic black brane with the axion hair and discuss the Hall angle and the Nernst effect. In section 4, we continue our analysis on the model introduced in section 3. We compute the AC electric, thermoelectric, and thermal conductivity numerically. The momentum relaxation effect on AC conductivities and the cyclotron resonance poles are discussed. We compare the zero frequency limit of our numerical AC conductivities with the DC analytic formulas derived in section 3. In section 5 we conclude. Note added. While this work was near completion, we noticed the appearance of [38–40] which have some overlap with ours. [38] deals with a massive gravity model at finite magnetic field. [39] considers the same class of models as ours. [40] obtains general expressions for conductivities at finite magnetic field using the memory matrix formalism. General analytic DC conductivities at finite magnetic field the presence of a magnetic field, from a general class of Einstein-Maxwell-Dilaton theories S = sented in [23]. Here we compute all the other conductivities as well. We employ the method developed in [8, 34], but a finite magnetic field poses some technical subtlety. We will explain how to treat it. The action (2.1) yields equations of motion: RMN − 2 gMN L− 2 (∂M φ)(∂N φ)− 2 i=1,2 1 X Φi(φ)(∂M χi)(∂N χi)− 1 X ∂Φi (∂χi)2 − − 4 1 F 2 ∂Z(φ) = 0 , = 0 , where, L is the Lagrangian density of the action (2.1). we take the gauge potential as To study the system at finite chemical potential with a background magnetic field (B), We choose the axion fields A = AM dxM = a(r)dt + (xdy − ydx) . which break translational invariance and can give rise to momentum relaxation [14, 15]. A metric anasatz consistent with the choice (2.6) and (2.7) is ds2 = GMN dxM dxN = −U (r)dt2 + dr2 + ev1(r)dx2 + ev2(r)dy2, v1(r) = v2(r) =: v(r) , −gZ(φ)F rt = Z(φ)ev(r)a0(r) , which is identified with the number density in the boundary field theory. The axion equation (2.4) is trivially satisfied. The Einstein equation (2.2) and scalar equation (2.5) become v00(r) = − 2 e−2v(r) U 00(r) = + ev(r)Z(φ) 2β2Φ(φ) + ev(r)U (r) v0(r)2 − φ0(r)2 if we choose the background becomes the AdS-dynonic black hole geometry with the momentum relaxation. We will discuss it in section 3 in detail. To compute the conductivities for the general background, we consider small perturbations around the background obtained by (2.11)–(2.13) δAi = t δfi(1)(r) + δai(r) , δGti = t δfi(2)(r) + δgti(r) , δfi(1)(r) = −Ei + ζia(r) , spite of the explicit t dependence in (2.15) and (2.16) all equations of motion of fluctuations turn out to be time-independent, which is the reason to introduce the specific forms of (2.19) and (2.20). Furthermore, the electric current and heat current can be computed as the boundary (r → ∞) values of J i(r) and Qi(r), J i(r) = Z(φ)√−g F ir(r) − a(r)J i(r) , Our task is to plug the solutions of the fluctuation equations into (2.21) and (2.23) Therefore, the current at the boundary is given by 0 = ∂M Z(φ)√−g F iM = ∂r Z(φ)√−g F ir + ∂t Z(φ)√−g F it = ∂rJ i − B ij e−v(r)ζj Z(φ) . J i(∞) = J i(rh) + B ij ζj Z ∞ Next, let us turn to the heat current, Qi, (2.23). It is convenient to start with the derivative − a0(r)J i − a(r)J i0 = B ij Ej e−v(r) − 2B ij ζj a(r)e−v(r)Z(φ) . After using the Einstein equations for fluctuations with the ansatz (2.17)–(2.18), 2B2Z(φ)e−2v(r) + 2β2Φ(φ)e−v(r) + U (r) v02(r) − φ02(r) 2B2Z(φ)e−2v(r) + 2β2Φ(φ)e−v(r) + U (r) v02(r) − φ02(r) −Bρe−v(r)δgtx −BZ(φ)U (r)δa0x(r) where Qi(rh) and J i(rh) are functions at horizon, which can be further simplified by the regularity condition at the black hole horizon [8, 34] J i(∞) = J i(rh) + B ij ζj Σ1 , Q (∞) = Qi(rh) + B ij Ej Σ1 + B ij ζj Σ2 , i Ei ln(r − rh) + · · · , δgti(r) ∼ δgt(ih) + O (r − rh) + · · · , δgri(r) ∼ e−v(rh) δgt(ih) δχi(r) ∼ χi(h) + O (r − rh) + · · · . Thus, the boundary currents yield Q (∞) = −4πT δij δgt(jh) + B ij Ej Σ1 + B ij ζj Σ2 , i using the equations of motion (2.27). The near horizon expansion of the last two equations in (2.27) gives Bρe−vh δgt(yh) − BZhEy − ρEx − 4πevh T ζx , − Bρe−vh δgt(xh) − BZhEx − ρEy − 4πevh T ζy , we end up with a relatively simple expression for the heat current; dr0e−v(r0)Z φ(r0) − 2B ij ζj Z ∞ Z ∞ ≡ Qi(rh) + B ij Ej Σ1 + B ij ζj Σ2 . In summary, we have two boundary currents: − β2ρevh Φh Ei − B(ρ2 + B2Zh2 + β2evh ZhΦh) ij Ej + 4πT evh (B2Zh + β2evh Φh) ζi − 4πT evh Bρ ij ζj , (ii) The thermoelectric conductivity: (iii) The thermal conductivity: ∂J i(∞) ∂Ej 1 ∂J i(∞) 1 ∂Qi(∞) 1 ∂Qi(∞) ∂Ej = −(ρ δike−vh + B ike−vh Zh) ∂Ej = −(ρ δike−vh + B ike−vh Zh) T = −4πδik δgt(kh) + ij B = −4πδik δgt(kh) + ij B ∂Ej where we put hats on the conductivities to distinguish them from the ones where magnetization current are taken out [29]. More explicitly, with (2.36), the general DC conductivity formulas are given as follows. (i) The electric conductivity and Hall conductivity: Z ∞ Z ∞ current and the energy magnetization current, which should be subtracted [29]. In particular, in the case of the dyonic black hole in section 3.2,3 Σ1 = MB and Σ2 = 2(ME−μM) , B where M is the magnetization and ME is the energy magnetization. The relation between − T − T which are expressed in terms of the black hole horizon data. Nernst effect The thermoelectric conductivites play an important role in understanding high Tc superconductors. In the presence of a magnetic field, a transverse electric field can be generated by a transverse or longitudinal thermal gradient. The former is called the ‘Seebeck’ effect and the latter is called the ‘Nernst’ effect. The electric current, J~, can be written in terms of the external electric field and the thermal gradient as follows; Based on the definition of the Nernst effect, the Nernst signal (eN ) is defined as The Nernst signal in cuprates shows different features from conventional metals, so it is one of the important observables in understanding high Tc superconductors. For example, in conventional metals the Nernst signal is linear in B, while in the normal state of a cuprate it is bell-shaped as a function of B. See, for example, figure 12 in [35]. At a fixed B, the Nernst signal increases as temperature decreases in the normal state of a cuprate and, in turn, near the superconducting phase transition the Nernst signal becomes much stronger than conventional metals as shown in figure 20 in [35]. Now we have the general formulas for the DC transport coefficients, (2.50) and (2.51), we can compute a general Nernst signal (2.55) eN = ρ4 + 2evh β2ρ2ZhΦh + Zh2(B2ρ2 + e2vh β4Φ2 ) , h 3The magnetization and energy magnetization current in a general setup were computed in [39]. which is expressed in terms of the black hole horizon data. By playing with the parameters, more realistic models showing aforementioned cuprate-like properties and furthermore in understanding the physics of strongly correlated systems. There are two comments on general features of the Nernst signal (2.56). First, it is In this regime, relevant to the quantum critical point, the Nernst signal is proportional to Example: dyonic black branes with momentum relaxation Model with massless axions As an explicit model, we consider the Einstein-Maxwell system with massless axions. The action is given by (2.1) with the following choices where L is the AdS radius which will be set to be 1 from now on. Adding the GibbonsHawking term, we start with S0 = 4 √ −g R + 6 − 4 − 2 1 F 2 1 X2 (∂χI )2 3 √ the extrinsic curvature tensor KMN N ∇(P nQ), where n is the outward Sc = dx3√ 1 X2 γμν ∂μχI ∂ν χI , 5In our case, nM = 0, 0, 0, 1/pU (r) . See (3.7). and the finite renormalized on-shell action is Sren = lim (S0 + Sc)on-shell . Since the boundary terms do not change the equations of motion, the equations (2.2)–(2.5) are valid and yield, with (3.1), RMN = 2 gMN R+6− 41 F 2 − 12 XI=21(∂χI )2 + 1 2 I 1 X ∂M χI ∂N χI + 2 FM P FNP , ∇M F MN = 0 , We want to find a solution of the equations of motion, describing a system at finite mentum relaxation. It turns out the dyonic black brane solution modified by the axion hair (2.7) does the job. I.e. ds2 = −U (r)dt2 + dr2 + r2(dx2 + dy2) , U (r) = r2 − 2 − r B = qmrh , where rh is the location of the horizon and m0 = rh3 1 + Thermodynamics To obtain a thermodynamic potential for this black brane solution, we compute the onperiod is the inverse temperature SE = −iSren , where SE is the Euclidean action. By a regularity condition at the black brane horizon the temperature of the system is given by the Hawking temperature, T = U 0(rh) = 3rh − and the entropy density is given by the area of the horizon, Plugging the solution (3.7) into the Euclidean renormalised action (3.4), we have the SE = (−m0 − β2rh + qm2rh) ≡ T namic variables as W = is obtained by using the relation which is derived as follows. We want to compute one-point functions of the boundary metric is of the following form ds2 = N2dr2 + γμν (dxμ + V μdr)(dxν + V ν dr) , define ‘conjugate momenta’ of the fields as −γ Kμν − Kγμν − 2γμν + Gμν [γ] − 2 ∂μχI ∂ν χI + Thus, the one point functions are hOI i = rl→im∞ √ which yield (3.14). Notice that the pressure is not equal to hTxxi since P = hTxxi + rhβ2 − rhqm2 . The magnetization (M) and the energy magnetization (ME ) are M = − = −qm , ME = − 1 − r2 1 − r δAy = x δB + μδBE r δE = T δs + μδρ − rhδ(β2) − MδB , δW = −MδB − sδT − rhδ(β2) − ρδμ , We find the first law of thermodynamics and the variation of (3.18). DC conductivities: Hall angle and Nernst effect In this section we study the DC conductivities of the dyonic black brane with momentum relaxation. Because we have derived the general formulas in section 2, we only need to plug model-dependent information (3.1) and (3.7) into (2.50)–(2.52). The electric conductivities yield where temperature dependence drops out and we recover the results expected on general grounds from Lorentz invariance, agreeing with [28]. In the limit B → 0, the expressions 6For details we refer to appendix C of [29]. which reproduces the result in [16, 19]. The thermoelectric and thermal conductivities read Therefore, we make plots of conductivities in figure 1, where we scaled the variables by T increases, the electric and thermoelectric conductivity generally increases while its overall 2-dimensional shape does not change qualitatively. However, the thermal conductivities strange metal phase, we are interested in the temperature dependence of the Hall angle, which is proportional to 1/T 2. In our case, we numerically found that the Hall angle ranges between 1/T 0 and 1/T 1 (figure 2(b)). In the large T regime, the Hall angle always scales as 1/T . It can be seen also from the formula (3.29), where if T is large compared to the other scales, rH ∼ T so tan θH ∼ 1/T . The Nernst signal (2.56) yields eN = As discussed in section 2.1, the Nernst signal is linear in B in conventional metals while it becomes bell-like in the normal state of cuprates [35]. To see whether our model can capture this feature, we make a three dimensional plot of the Nernst signal as a function of The blue line is almost straight while the green and red ones are bell-like. Therefore, we find that our system shows the transition from the normal metal (blue line) to cuprate-like red curves are similar to figure 12 in [35] and it was proposed that the bell shape can be explained by the dynamics of the vortex liquid in non-superconducting phase [36, 37]. It was also interpreted as an evidence of the pseudo-gap phase [35]. If we take this point of view, our model might be relevant to the pseudo-gap phase. One may wonder if the blue line is qualitatively similar to the green and red ones for higher B/T 2. As B/T 2 increases the blue line reaches the maximum and has almost plateaued out, which is not bell-shaped. Furthermore, the numerical value of B/T 2 at the (red, green, blue) Numerical AC conductivities So far we have discussed the DC conductivities. In this section, we will consider the AC conductivities. To be concrete, we continue to investigate the model in the previous section, namely, the dyonic black brane with the momentum relaxation by the axion fields. Equations of motion and on-shell action should not be changed by this gauge choice.7 Indeed, we will show the zero frequency limit section 2. It serves as a good consistency check of our numerical method. To compute conductivities holographically it is consistent to turn on linear fluctuations where r2 in the metric fluctuation (4.2) is introduced to make the asymptotic solution of the linearized equations for the Fourier components are given as follows. 7There is a subtle issue on the gauge choice and holographic renormalization. For more details we refer − Einstein equations: − Maxwell equations: − Scalar equations: − r4 − − h0t0i = 0 , Among these eight equations, only six are independent. Near the black hole horizon (r → 1)8 the solutions are expanded as hti = (r − 1)ν±+1 ht(iI) + ht(iII)(r − 1) + · · · , ai = (r − 1)ν± a(I) + ai(II)(r − 1) + · · · , i ψi = (r − 1)ν± ψ(I) + ψi(II)(r − 1) + · · · , i out that the 4 parameters ai(I) and ψ(I) may be chosen to be independent since ht(iI) and i all higher power coefficients can be determined by them. hti = ht(i0) + ai = ai(0) + where the leading terms ht(i0), a i(0), and ψi(0) are independent constants, which fix ht(i2), ψ(2) i leading terms, hti , ai(0), and ψi(0). The leading terms play the role of sources for the (0) Expanding the renormalized action (3.4) around the dyonic black brane background and using the equations of motion, we obtain a quadratic on-shell action: Sr(e2n) = lim 4r3 − pU (r) contributions from the horizon, which is the prescription for the retarded Green’s function [41]. In particular, with the spatially homogeneous ansatz (4.1)–(4.3), the quadratic action in momentum space yields Sr(e2n) = hti − 2m0h¯t(i0)ht(i0) + a¯ where V2 is the two dimensional spatial volume R dxdy. The argument of the variables with to obtain complex two point functions [41]. The on-shell action (4.11) is nothing but the generating functional for the two-point point functions from the first two terms in (4.11). The other three terms are nontrivial linearity of the equations (4.4)–(4.7) makes it easy to find out a linear relation between { i , ht(i3), ψ(3)} and {ai(0), ht(i0), ψ(0)}. In the following subsection we will explain how to a(1) find such a relationship numerically in a more general setup and apply it to our case. Numerical method A systematic numerical method with multi fields and constraints was developed in [19] based on [42, 43]. We summarize it briefly for the present case and refer to [19, 33] for around a background. Suppose that they satisfy a set of coupled N second order differential equations and the fluctuation fields depend on only t and r: (r → ∞). For example, p = 2 in (4.2). Φa(r) = (r − 1)νa± ϕa + ϕ˜a(r − 1) + · · · , (outgoing) boundary condition. In order to compute the retarded Green’s function we choose the incoming boundary condition [41]. This choice reduces the number of independent parameter from 2N to N . There may be further reductions by Nc if there are Nc ϕa1 ϕa2 ϕa3 . . . ϕaN−Nc Every column vector ϕaˆi yields a solution with the incoming boundary condition, denoted (near boundary) , Φian(r) ≡ Φˆia(r)cˆi → Sˆacˆi + · · · + i (near boundary) , with real constants cˆi. We want to identify Sˆacˆi with the independent sources J a but if i there are constraints, it is not possible since a > ˆi. However, in this case, there may be Nc other solutions corresponding to some residual gauge transformations [19, 33] (near boundary) , solution together with the ingoing solutions. are two sets of additional constant solutions of the equations of motion (4.4)–(4.7) hti = ht0i , ai = − Sa¯i = y = Therefore, the most general solution reads Φian(r) + Φca(r) ≡ J a + · · · + where we defined J a and Ra. For arbitrary sources J a we can always find cI =  10 cI = (S−1)I aJ a, Ra = OaI cI = OaI (S−1)I bJ b. Sr(e2n) = J¯aAabJ b + J¯aBabRb , The general on-shell quadratic action in terms of the sources and the responses can be the action (4.11) is the case with: J a = hhtt(yx0)  , Ra = hhtt(yx3)  , into the action (4.22) we have which yields the retarded Green’s function Sr(e2n) = V2 Z ∞ dω J¯a Aab + BacOcI (S−1)I b J b, Gab = Aab + BacOcI (S−1)I b . In summary, to compute the retarded Green’s function, we need four square N × N matrices, A, B, S and O. The matrices A and B can be read off from the action (4.22), which is given by the on-shell expansion near the boundary. The matrices S and O are obtained by solving a set of the differential equations. Part of them comes from the solutions with incoming boundary conditions and the others may be related to the constraints. Notice that the Green’s functions do not depend on the choice of initial conditions (4.14). In the case of the dyonic black branes in section 4.1, we may construct a 6 × 6 matrix of the retarded Green’s function. We will focus on the 4 × 4 submatrix corresponding to the electric current J i and the ht(i0) is dual to the energy-momentum tensor T ti. From the linear response theory, we have the following relation between the response functions and the i direction. Notice that the electric and heat current here contain the contribution of magnetization, so we use the conductivities with hat ((2.37)–(2.40)). By taking into account diffeomorphism invariance [1, 2, 19], (4.28) can be expressed as From (4.27) and (4.29) with the magnetization subtraction (2.50)–(2.52), the conductivities are expressed in terms of the retarded Green’s functions as follows i(GiTjT −GiTjT (ω=0)−μ(GiJjT +GiTjJ −μGiJjJ )) − T −(∇j T )/T AC conductivities and the cyclotron poles In this section we present our numerical results. Some examples of the AC electric conductivity are shown in figure 5 and 8; the thermoelectric conductivity is in figure 6; and the thermal conductivity is in figure 7. Before discussing the AC nature of the conductivities, we first examine the DC limit in section 3.3. The comparisons are shown in figure 4, where all conductivities are plotted lines were drawn by the analytic expressions, (3.23), (3.24), (3.25)–(3.28) and the red dots serves as a supporting evidence for the validity of our analytic and numerical methods. Indeed this agreement is not so trivial in technical perspective. In the DC computation we turned on hri and read off the physics from the horizon data while in the AC computation Let us turn to the AC properties of the conductivities. Figure 5, 6, and 7 show given by the analytic formulas (3.23), (3.24), (3.25)–(3.28). They agree to the numerical curves become flatter, which is expected from stronger momentum relaxation. Notice that because the gauge field for B (2.6) breaks translation invariance in the same way as the axion fields do. There is a peak in the curves in figures 5–7, which is related to the cyclotron resonance of as arising from interactions between the positively charged current and the negatively charged current of the fluid, which are counter-circulating. In the hydrodynamic regime increases. It is indirectly shown in the plots since all curves become flat, which may reflect frequencies as B increases, which is consistent with the hydrodynamic analysis (4.32). same symmetry is preserved. It is demonstrated in figure 9(d)(e)(f); the figure (d) and (f) position of the poles to the negative imaginary direction. This implies the width of the and also fits well to the hydrodynamic analysis (black dashed line) for small magnetic field. in figure 10 and additional similar numerical data for different parameters, we found the following relation at small B ω∗ = ωc − iγ = ωc0 + c1β2B − i(γ0 + c2β2) , 0, 2, 3, 4 (gray, red, green blue). The dotted lines are the results by the hydrodynamic analysis (4.32). figure 10(a) are linear to B. However, at large magnetic field B we numerically found the In the presence of dissipation, the cyclotron frequency was shown [29] to be changed as suspect that the analysis in [29], where B/T 2 imaginary part in (4.34) is consistent with the hydrodynamic calculation [29]. However, 1 is assumed, is valid in the limit c1B is small. We leave this issue and the analytic justification of the specific form (4.34) for a future project. Conclusions In this paper, we have computed the electric, thermoelectric, and thermal conductivity at finite magnetic field by means of the gauge/gravity duality. First, by considering a general class of Einstein-Maxwell-Dilaton theories with axion fields imposing momentum relaxation, we have derived the analytic DC conductivities, which are expressed in terms of the black hole horizon data. As an explicit model we have studied the dyonic black hole modified by a momentum relaxation effect. The background solution is analytically obtained and the AC electric, thermoelectric, and thermal conductivity were numerically computed. The zero frequency limit of the numerical AC conductivities agree to the DC formulas. This is a non-trivial consistency check of our analytic and numerical methods to compute conductivities. Our numerical method can be applied to other cases in which multiple transport coefficients need to be computed at the same time. The Nernst signal, the Hall angle, and the cyclotron resonance pole were discussed following [28–30]. Our general analytic formulas of the Nernst signal can be used to build a realistic model and to investigate the universal properties of the model. In particular, in the dyonic black hole case, the Nernst signal is a bell-shaped curve as a function of the magnetic field, if momentum relaxation is small. It is similar to an experimental result in the normal state of cuprates [35], which was speculatively explained by a vortex-liquid effect [36, 37]. For large momentum relaxation the Nernst signal is proportional to the magnetic field, which is a typical property of conventional metals. The Hall angle for the dyonic black hole was computed explicitly. The Hall angle ranges between 1/T 0 and 1/T 1 and scales as 1/T for large T . However, in the strange metal phase, it was known that the detail both numerically and analytically. It is important to compare our AC conductivities with the general expressions based on the memory matrix formalism [40, 44]. It would be also interesting to compare our AC conductivity results with [45], where the AC electric conductivities have been studied at finite magnetic field in the probe brane set up, focusing on the transport at quantum Hall linear-T resistivity, so it is worthwhile to start with the models having that property and then investigate the Hall angle and the Nernst effect in those models. Acknowledgments 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). The work of SS and YS was supported by Mid-career Researcher Program through the National Research Foundation of Korea (NRF) grant No. NRF-2013R1A2A2A05004846. 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Keun-Young Kim, Kyung Kiu Kim, Yunseok Seo, Sang-Jin Sin. Thermoelectric conductivities at finite magnetic field and the Nernst effect, Journal of High Energy Physics, 2015, 27, DOI: 10.1007/JHEP07(2015)027