Thermoelectric Conductivities at Finite Magnetic Field and the Nernst Effect

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.


Introduction
Strongly coupled electron systems show many interesting phases such as non-Fermi liquid, high T c superconductor and pseudo gap phase. One of the most important and basic experimental observables in investigating those systems is conductivity: electric(σ), thermoelectric(α,ᾱ), and thermal(κ) conductivity. Therefore, it is essential to develop a theoretical method to compute conductivity to explain and guide experiments. However, because of strong coupling, most perturbative analysis of quantum field theory do not work easily and we still don't have a reliable systematic theoretical tool for conductivity.
Gauge/gravity duality is one of the promising approaches for strong coupling problems and it has been developing the methods to compute conductivity [1,2]. The early works dealt with the systems which have translation invariance. However, at finite charge density the DC conductivity in those systems is doomed to be infinite. To solve this infinite conductivity problem, it is essential to introduce the momentum relaxation. For this, several ideas have been proposed.
However, all models with momentum relaxation dealt with the case at zero magnetic field, except [23]. Because the transport properties at finite magnetic field, such as quantum Hall effect, the Nernst effect, and the Hall angle, are also important and basic probes for strongly correlated electron system, it is timely and essential to develop the methods to compute conductivity at finite magnetic field together with momentum relaxation. Indeed, without momentum relaxation, the holographic analysis on conductivity at finite magnetic field was one of the pioneering themes opening the AdS/CMT(condensed matter theory) [28][29][30]. The purpose of our paper is to extend and generalise those works, namely, to study conductivities at finite magnetic field with momenturm relaxation 1 . This paper is a companion of [19,32,33] in the sense that all DC/AC electric(σ), thermoelectric(α,ᾱ), and thermal(κ) conductivities are analysed thoroughly.
We consider a general class of Einstein-Maxwell-dilaton theory with axion fields imposing momentum relaxation. First, we derive the analytic formulas for DC electric(σ), thermoelectric(α,ᾱ), and thermal(κ) conductivity in terms of black hole horizon data following 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 QCP(quantum critical point). As we approach to the QCP or the superconducting domain, the strength of the Nernst signal becomes stronger and shows non linear dependence on the magnetic field, which is very 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 can consider the case with a finite impurity effect that can not be covered in [29,31].
After discussions on a general class of model, as an explicit example, we study in detail the Dyonic black hole background [28][29][30], modified by the specific axion fields introduced in [16]. We numerically compute AC electric(σ), thermoelectric(α,ᾱ), and thermal(κ) conductivity and confirms their zero frequency limits agree to the DC formula that we have derived analytically. It recovers the result in [28] if momentum relaxation vanishes. We discuss the momentum relaxation effect on 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 Einstein-Maxwell-dilaton theory with axion fields and derive general formulas of DC electric, thermoelectric, and thermal conductivities at finite magnetic field as well as the Nernst 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 AC electric, thermoelectric, and thermal conductivities 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 nearing completion we learned of [37][38][39] which have some overlap with ours. [37] deals with a massive gravity model at finite magnetic field. [38] considers the same class of model as ours. [39] obtains general expressions for conductivities at finite magnetic field using the memory matrix formalism.

General analytic DC conductivities at finite magnetic field
In this section, we derive analytic formulas for DC conductivities(σ, α,ᾱ,κ) in the presence of the magnetic field, from a general class of Einstein-Maxwell-Dilaton theory with axion fields (χ 1 , χ 2 ) where F = dA and all Φ 1 (φ), Φ 2 (φ), and Z(φ) are not negative for positive energy condition. Without magnetic field, analytic formulas for conductivities(σ, α,ᾱ,κ) were computed in [34] while, with magnetic field, only σ were presented in [23]. Here we compute all the other conductivities in the presence of magnetic field. We employ the method developed in [8,34], but finite magnetic field gives some technical subtlety, which we will explain how to deal with. The action (2.1) yields equations of motion:

4)
To study the system with finite charge and background magnetic field(B) we take the gauge potential as To impose the momentum relaxation by breaking translational invariance in x, y space, we choose the axion fields [14,15] The metric anasatz consistent with the choice (2.6) and (2.7) is with φ = φ(r). We consider the case in which the black hole solution exists and its horizon is located at r = r h , i.e. U (r h ) = 0. However, within our ansatz, it turns out that to satisfy the equations of motion (2.2)-(2.5). First, the Maxwell equation (2.3) yields a conserved charge ρ in r-direction by which we can obtain the background solutions for given Z(φ) and V (φ). For example, if we choose with the AdS radius L, the background becomes the AdS-dynonic black hole geometry without momentum relaxation (k = 0) or modified AdS-dyonic black hole geometry with momentum relaxation (k = 0), which will be discussed in section 3 explicitly.
To compute the conductivities, we consider small perturbations around the background obtained by (2.11)-(2.13) δG ti = t δf (2) i (r) + δg ti (r) , (2.16) δG ri = e v(r) δg ri (r) , (2.17) where i = 1, 2 and t denotes the time coordinate. δf (1) i (r) and δf (2) i (r) are chosen where E i is the electric field and −ζ i is identified with the temperature gradient [34]. In 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 a main motivation to introduce specific forms of (2.19) and (2.20). Furthermore, the electric current and heat current can be computed as the boundary(r → ∞) value of J i and Q i given by which were identified in [34] at B = 0 and they are still valid at finite B.
Our task is to plug the solutions of fluctuation equations into (2.21) and (2.23) and read off the coefficients of E i and ζ i . First, the Maxwell equation yields so, after integration from the horizon(r h ) to infinity, we have the electric current Notice that without magnetic field, since the current J i is independent of the radial coordinate we can calculate it simply from the horizon data. Next, let us turn to the heat current, Q i (2.23). It is convenient to start with the the derivative of Q i After using the Einstein equations for fluctuations with the ansatz (2.17)-(2.18), we have (2.28) In summary, we have two boundary currents: where Q i (r h ) and J i (r h ), functions at horizon, which can be further simplified from the regularity condition at the black hole horizon [8,34] (2.31) Thus the boundary currents yield ti also can be replaced by the horizon data by using the equations of motion (2.27). Expansion of the last two equations in (2.27) give us two algebraic equations for δg Finally, the conductivities are obtained by differentiating the boundary currents (J i (∞), Q i (∞)) with respect to external electric field(E i ) or thermal gradient (ζ i ): where we put hat on the conductivity notations to distinguish them from the ones without magnetization current (2.50)-(2.52) [29]. More explicitly, with (2.36), the general DC conductivity formulas are given as follows.
(i) The electric conductivity and Hall conductivity: (2.42) (ii) The thermoelectric coefficients: (iii) Thermal conductivities: Σ i terms in the off-diagonal thermoelectric and thermal conductivity (2.44), (2.48) comes from the contribution of magnetization current by external magnetic field, which should be subtracted [29]. Therefore, the DC conductivities(σ ij , α ij ,κ ij ) read which are expressed in terms of the black hole horizon data.

Nernst effect
The thermoelectric conductivites play an important role in understanding high T C superconductors. In the presence of magnetic field, a transverse electric field can be generated by the external transverse or longitudinal thermal gradient. The former is called 'Seebek' effect and the latter is called 'Nernst' effect. The charge current, J, can be written in terms of the external electric field and thermal gradient as follows; where σ and α are 2 × 2 matrices. In the absence of charge current, Based on the definition of the Nernst effect, the Nernst signal (e N ) is defined as The Nernst signal is one of the important observables in understanding high T C superconductor. In conventional metal, the Nernst signal is linear in B so the Nernst coefficient ν ≡ e N /B can be used. However, in the normal state of cuprates, e N can be highly nonlinear in B. For example, see Figure 3(b), which is a plot of our numerical analysis similar to experimental results in [35].
It is also known that the Nernst signal in curates is very large compared to a usual ferromagnetic conductor and shows interesting features depending on the type of dopping or materials [35]. One explanation of the large value of the Nernst signal relies on the existence of vortex-liquid in superconductor. In the vortex-liquid state, a temperature gradient produces vortex motion. Because of 2π phase singularity at the vortex core, there should be a phase slippage. The time variation of the phase can produce an electrochemical potential difference through the Josephson equation. This Josephson voltage can be expressed as a transverse electric field which is detected as the Nernst signal. We refer to [35] for more details.
Now we have the general formulas of the DC transport coefficients, (2.50) and (2.51), we can compute a general Nernst signal (2.55) which is expressed in terms of the black hole horizon data. This formula may guide in constructing more realistic models for strongly correlated systems and in understanding the physics behind it. There are two comments on the general feature of the Nernst signal (2.56). First, the Nernst signal is proportional to k 2 for small value of k and it goes as 1/k 2 for large value of k. It has the maximum at k = k max of which dependence on ρ, B is universal regardless of the detail of background geometry. Second, in the limit, ρ → 0, the Nernst signal becomes where τ L is the lattice time scale [23], In this regime, relevant to quantum critical point, the Nernst signal is proportional to the lattice time scale. Furthermore, The Nernst signal increase as temperature decreases if the temperature dependence of τ L is weaker than T , which is controlled by Φ h in (2.59). This will be a restriction for model building.
3 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 obtained from (2.1) with the following conditions where L is the AdS radius which will be set to be 1 from here. Adding the Gibbons-Hawking term, we start with where γ is the determinant of the induced metric on the boundary. K is the trace of the extrinsic curvature tensor K M N 3 which is defined by −γ P M γ Q N ∇ (P n Q) and n is the outward pointing unit vector normal to the boundary 4 . From here we will take a unit such that 16πG = 1. For the holographic renormalisation, we have to add a counter action and the finite renormalized action is Since the boundary terms do not change the equations of motion, the equations (2.2)-(2.5) are valid and yield, with (3.1), We want to find a solution of the equations of motion, describing a system at finite chemical potential(µ) and temperature(T ) in an external magnetic field(B) with momentum dissipation. It turns out the dyonic black brane solution modified by the axion hair (2.7) do the job. i.e.
where r h is the location of the the brane horizon and (3.8)

Thermodynamics
To obtain a thermodynamic potential for this black hole solution we compute the on-shell Euclidean action(S E ) by analytically continuing to Euclidean time(τ ) of which period is the inverse temperature where S E is the Euclidean action. By a regularity condition at the black hole horizon the temperature of the system is given by the Hawking temperature, 10) 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 thermodynamic potential(Ω) and its density(W): where V 2 = dxdy. The potential density W can be expresses in terms of thermodynamic variables as where ε is energy density, ρ is charge density and we used the relations which will be derived at the end of this subsection, (3.22). Since pressure P = −W, we have a Smarr-like relation Notice that the pressure is not equal to T xx since The magnetization(M) is We find the first law of thermodynamics by combining the variation of (3.13) with respect to T, µ, β 2 and B: 19) and the variation of (3.15). We close this subsection by summarizing the procedure to compute one-point functions of the boundary energy-momentum tensor(T µν ), current(J µ ) and scalar operators(O I ) dual to χ I . Our metric is of the following form 20) where N and V µ are a lapse function and a shift vector. The extrinsic curvature tensor has non-vanishing components, where D µ is a covariant derivative of the boundary metric γ µν . In terms of aforementioned variables, we define 'conjugate momentum' of fields as Thus the one point functions yield

DC conductivities: Hall angle and Nernst effect
In this section we study DC conductivities of the dyonic black brane with momentum relaxation. Because we have derived 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 (3.24) In the clean limit, β → 0, these boil down to where temperature dependence drops out and we recover the results expected on general grounds from Lorentz invariance, agreeing to [28]. In the limit, B → 0, the expressions become which reproduces the result in [16,19]. The thermoelectric and thermal conductivities read (3.28) To see the effect of β and B on the electric conductivities, the formulas (3.23)-(3.28) are not so convenient since r h is a complicated function of T, B, µ, and β, as expressed in (3.10). Therefore, we make plots of conductivities in Figure 1, where we scaled the variables by T and fixed µ/T = 4. σ xx , α xx , andκ xx are qualitatively similar; the B dependence at fixed β is monotonic, while the β dependence at fixed B is not. σ xy , α xy , andκ xy are similar; the β dependence at fixed B is monotonic, while the B dependence at fixed β is not. See Figure 4 for the cross sections at B/T 2 = 1. As the chemical potential increases, the electric and thermoelectric conductivities generally increase but the overall shape does not change qualitatively. Thermal conductivity is different;κ xx becomes broaden whilē κ xy becomes sharpen. The Hall angle(θ H ) is defined as . (3.29) which agrees to the result reported in [23]. As shown in Figure 5 the Hall angle θ H ranges from π/2 to 0, since tan θ H goes to µB r h β 2 2µB r h β 2 as β → 0(β → ∞). The angle increases as B increases or β decreases( Figure 5(a)). In the 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 find that the Hall angle ranges between 1/T 0 and 1/T 1 (Figure 5(b)). In large T , the Hall angle always scales as 1/T , which can be seen also in the formula (3.29). If T is large compared to other scales, r H ∼ T so tan θ H ∼ 1/T . The Nernst signal (2.56) yields (3.30) As discussed in Section 2.1, the Nernst signal is linear in B in conventional metal while it becomes nontrivial in cuprates. To see our model can capture this feature we make a plot of the Nernst signal as a function of β/T and B/T 2 at fixed µ/T = 1 in Figure 3, where Figure (b) is the cross section of (a) at fixed β. We find that our system shows the transition from the normal metal(blue line) to curates-like state(green and red) as we decrease β, from the Nernst signal perspective. Interestingly, our curve is similar to Figure  12 in [35].

Numerical AC conductivities
So far we have discussed the DC conductivities and in this section we consider AC conductivities. To be concrete, we continue to investigate the model in the previous section, namely, the dyonic black brane with moment relaxation by axion fields.

Equations of motion and on-shell action
To compute conductivities holographically it will suffice to turn on linear fluctuations of bulk fields, δg ti , δA i and δχ i , where i = 1, 2 or i = x, y, at zero momentum in the x and y directions. The fields δg ti and δA i are related to the heat current and the electric current. They were introduced in [28] and here we add δχ i for momentum relaxation. This set of fields are consistent and do not source any other fields. Since the fields which have spatial vector indices, i, can be mixed at the linear level we have to take into account δg ri , δg ti , δA i and δχ i . However, we may choose a gauge δg ri = 0, since it has no dynamics along the holographic direction r. To see this explicitly we may turn on δg ri and examine the fluctuation equations. Non-zero δg ri do not increase the number of the equations of motion and δg ri can be gauged away. We refer to [33] for more details in the absence of magnetic field, since similar computations will work in our case. In section 2 and 3, we have derived the DC conductivities in a different gauge, g ri = 0. We will show that this gauge choice does not change physics by showing the results in this section with g ri = 0 agree to the result in section 3.
The fields, δg ti , δA i and δχ i , can be expressed in momentum space as where r 2 in the metric fluctuation (4.2) is introduced to make an asymptotic solution of h ti constant at the boundary(r → ∞), for the sake of convenience. From (3.5) and (3.6), the linearised equations for the Fourier components are given as follows.
-Einstein equations: -Maxwell equations: -Scalar equations: Among these eight equations, only six are independent. Near the black hole horizon (r → 1) 5 the solutions are expanded as where ν ± = ±i4ω/(−12 + q 2 m + 2β 2 + µ 2 ). In order to impose the incoming boundary condition relevant to the retarded Green's function [40], we have to take ν = ν + . It turns out that the 4 parameters a Near the boundary(r → ∞), the asymptotic solutions read where the leading terms h Expanding the renormalized action (3.4) around the dyonic background and using the equations of motion, we obtain a quadratic on-shell action : where δh ti ≡ r −2 δg ti (t, r), 'dot' denotes ∂ t , 'prime' denotes ∂ r . We have dropped the contributions from the horizon, which is the prescription for the retarded Green's function [40]. In particular, with the spatially homogeneous ansatz (4.1)-(4.3), the quadratic action in momentum space yields where V is the two dimensional spatial volume d 2 x. The argument of the variables with the bar is −ω. We dropped the range (−∞, 0), which is the complex conjugate of (4.11), to obtain complex two point functions [40]. The on shell action (4.11) is nothing but the generating functional for two-point Green's functions sourced by a ti , and ψ (0) . We may simply read off part of the two point functions from the first two terms in (4.11). The other three terms are nontrivial and we need to know the dependence of {a ti , ψ (0) }. However, the linearity of the equations (4.4)-(4.7) make it easy to find out the linear relation between {a ti , ψ (0) }. In the following subsection we will explain how to 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 [41,42]. We summarize it briefly for the present case and refer to [19,33] for more details. Let us start with N fields Φ a (x, r), (a = 1, 2, · · · , N ), which are fluctuation 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 : where r p is multiplied such that the solution Φ a ω (r) goes to constant at boundary. For example, p = 2 in (4.2).
Near horizon(r = 1), solutions can be expanded as Φ a (r) = (r − 1) ν a± (ϕ a +φ a (r − 1) + · · · ) , (4.13) where we omitted the subscript ω for simplicity and ν a+ (ν a− ) corresponds to the incoming(outgoing) boundary condition. In order to compute the retarded Green's function we choose the incoming boundary condition [40]. This choice reduces the number of independent parameter from 2N to N . There may be further reductions by N c if there are N c constraint equations. In general, the number of independent parameter is N − N c so we may choose N − N c initial conditions, denoted by ϕ aî (î = 1, 2, · · · , N − N c ): (4.14) Every column vector ϕ aî yields a solution with the incoming boundary condition, denoted by Φ aî (r), which is expanded as where S aî are related to the sources, which are the leading terms ofî-th solution, and O aî are related to the expectation values of the operators corresponding to the sources(δ a ≥ 1). A general solution constructed by Φ aî (r) is with real constants cî. We want to identify S â i cî with the independent sources J a but if there is a constraint it is not possible since a >î. However, in this case, there may be N c other solutions corresponding to some residual gauge transformations [19,33] Φ aī (r) → S ā i + · · · + O ā i r δa + · · · (near boundary) , (4.17) whereī runs from N − N c + 1 to N . This extra basis set generates a general solution Φ a c (r) = Φ ā i cī. In our case, N = 6 and N c = 2, which corresponds to the constraints g ri = 0. There are two sets of additional constant solutions of the equations of motion (4.4)-(4.7) where h 0 ti is arbitrary constant and i, j = 1, 2. These can be understood as residual gauge transformations keeping g ri = 0, which are generated by the vector fields, of which nonvanishing component is ξ x = x e −iωt and ξ y = y e −iωt respectively ( i are constants). i.e. L ξ A i = −q m ij ξ j , L ξ g ti = −iωr 2 ξ i and L ξ χ i = βξ i . Therefore, the most general solution reads where we defined J a and R a . For arbitrary sources J a we always can find c I where I = 1, . . . , N . The corresponding response R a is expressed as A general on-shell quadratic action in terms of the sources and the responses can be written as where A and B are regular matrices of order N and the argument ofJ a is −ω. For example, the action (4.11) is the case with: where the index ω is suppressed and 1 2 is the 2 × 2 unit matrix. Plugging the relation (4.21) into the action (4.22) we have which yields the retarded Green's function 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 differential equations. Part of them comes from the solutions with incoming boundary conditions and the others may come form the constraints. Notice that the Green's function do not depend on the choice of initial conditions (4.14).
With the matrices S and O, we may construct a 6 × 6 matrix of the retarded Green's function. We will focus on the 4 × 4 submatrix corresponding to a where every G ij αβ is a 2 × 2 retarded Green's function with i = x, y for given α and β. The sub-induces α, β denote the operators corresponding to the sources. i.e. a ti is dual to energy-momentum tensor T ti . From the linear response theory, we have the following relation between the response functions and the sources: where J i , T ti , a tj are understood as 2 × 1 column matrices, with i = x, y. We want to relate these Green's functions to the electric (σ), thermal (κ), and thermoelectric (α,α) conductivities defined as where Q i is the heat current, E i is an electric field and ∇ i T is a temperature gradient along x 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 (4.29)  Together with (4.27), (4.29) and the magnetazation subtraction (2.50-2.52), the conductivities are expressed in terms of the retarded Green's functions as follows (4.30)

AC conductivities and the cyclotron poles
In this section we present our numerical results. Examples of AC electric conductivity are shown in Figure 5 and 8; thermoelectric conductivity is in Figure 6; and thermal conductivity is in Figure 7. Before discussing AC nature of conductivities we first examine the DC limit(ω → 0) of AC conductivities and compare them with the analytic expressions derived in section 3.3. The comparisons are shown in Figure 4, where all conductivities are drawn as a function of β/T with other parameters fixed, µ/T = 4, B/T 2 = 1, for example. Solid lines are analytic expressions (3.23), (3.24), (3.25)-(3.28) and red dots are read off from AC numerical results in the limit ω → 0. Both results agree and 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 DC computation we turned on h ri and read off physics from the horizon data while in AC computation we worked in the gauge h ri = 0 and considered full evolution in r-direction.
Let us turn to AC properties of conductivities. Figure 5, 6, and 7 show β dependence of electric, thermoelectric, and thermal conductivities respectively. Dotted lines are the cases with β = 0 and red, green, blue curves are for β/T = 0.5, 1, 1.5. The red dots in the real part of the conductivities are the values of the analytic formula, which agree to the numerical AC conductivities in the limit ω → 0. As β increases the curves become flatter,   which is expected from stronger momentum relaxation. Notice that there is no 1/ω pole in the imaginary parts of the conductivities even in β = 0 case, contrary to B = 0 case. It is because the gauge field for B (2.6) breaks translation invariance in the same way as the axion fields.
There is a peak in the curves in Figures 5-7, which is related to the cyclotron resonance pole, the position of the pole of the conductivity in complex ω plane [29,30] ω * ≡ ω c − iγ , (4.31) where the cyclotron frequency ω c is the relativistic hydrodynamic analog of the free particle case, ω f = eB/mc. However, the resonance due to ω * here should be understood to be due to a collective fluid motion rather than to free particles. A damping γ could be thought of as arising from interactions between the positively charged currents and the negative charged current of the fluid, which are counter-circulating. Because β is related to momentum relaxation, it is expected that γ will increase as β increases. It is indirectly shown in the plots since all curves become flat, which may reflect the fact the pole goes away from the real ω axis. It turns out that ω c tend to increase as β increase, although it is not so clear in the plots. It will be discussed later in Figure 10 and the equation (4.34). In Figure 5-7, B is fixed and we focused on the effect of β. To show the effect of B at fixed β we make Figure 8, where red, orange, green, and blue curves are for B/T 2 = 0.5, , 1, 2, 4 respectively. A relatively big µ/β = 8 has been chosen since the peak is shaper when µ/β is bigger. We find the peaks of the curves shift towards higher frequencies as B increases, which is consistent with the hydrodynamic analysis (4.33).
To see the cyclotron pole directly in complex ω plane, we introduce the following combination σ ± = σ xy ± iσ xx , (4.32) for easy comparison with [29,30]. We make density plots of |σ + | in complex ω plane, which are shown in Figure 9. Figure 9(a),(b),(c) are the cases with β = 0 and 9(d),(e),(f) are the cases with β/T = 10. We choose µ and q m such that q 2 m + µ 2 to be constant. First, at β = 0 we recover the result of [30] 6 . White areas correspond to poles and dark areas are zeroes of σ + . There is a symmetry: µ → q m and q m → −µ at β = 0 inherited from the electromagnetic duality of the bulk theory [30]. Since this bulk duality holds at finite β we expect the same symmetry is preserved. It is demonstrated in the figure 9(d)-(f); the figure (d) and (f) are symmetric under the exchange of the white and dark region. The finite β shifts the position of the poles to the negative imaginary direction. This implies the width of the peak increases at real ω axis as discussed previously.
The magnetic field dependence of the cyclotron poles at different value of β is shown in Figure 10, where (a) and (b) shows the real part(ω c ) and imaginary part(−γ) respectively. The gray,red,green,blue dots are numerical results for β = 0, 2, 3, 4 respectively while the black dashed line is the analytic result at β = 0 for small B from hydrodynamic analysis [30] At β = 0, our numerical result(gray dots) agrees to [30] and also fits well to the hydrodynamic analysis(black dashed line) for small magnetic field. We investigate how ω * changes as β is introduced. Based on the red,green,blue dots in Figure 10 and additional similar Figure 9. A density plot of |σ + | in complex ω plane. White areas correspond to poles and dark areas are zeroes of σ + . β = 0 for (a)(b)(c) and β/T = 10 for (d)(e)(f). (b) Im ω * (= −γ) Figure 10. The magnetic field dependence of the position of the cyclotron resonance pole. β = 0, 2, 3, 4(gray, red, green blue). Dotted lines are the results from hydrodynamic limit analysis(4.33).
numerical data for different parameters, we find the following relation holds at small B where c 1 and c 2 are dimensionful constants independent of β and B. For this formula we focused on small B region where the dots in Figure 10(a) are linear to B. At large magnetic field B we find the tendency numerically that ω c = c 3 B, where c 3 is independent of β.
In the presence of dissipation, the cyclotron frequency is shown [29] to be changed by while ω c is intact. In our case τ imp is proportional to 1/β 2 for small β so the shift of the imaginary part in (4.34) is consistent with hydrodynamic calculation [29]. However, our result implies that the cyclotron frequency(ω c ) also shifted by c 1 β 2 B ∼ B/τ imp . We suspect that the analysis in [29], where B/T 2 1 is assumed, is valid in the limit c 1 B is small. We leave further investigation of this issue and an analytic justification of the specific form (4.34) as a future project.

Conclusions
In this paper, we have computed electric, thermoelectric, and thermal conductivities at finite magnetic field by means of the gauge/gravity duality. First, by considering a general class of Einstein-Maxwell-Dilaton theory with axion fields imposing momentum relaxation, we have derived 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 AC electric, thermoelectric, and thermal conductivities are numerically computed. The zero frequency limit of the numerical AC conductivities agree to the DC formula. This is a nontrivial consistency check of our analytic and numerical methods to compute conductivities. Our procedure can be applied to other cases in which multiple transport coefficients need to be computed at the same time.
In particular, the Nernst signal, the Hall angle, and the cyclotron resonance pole were discussed following [28][29][30]. Our general analytic formulas of the Nernst signal can be used to build a realistic model and to investigate universal properties of the model. The Nernst signal of the dyonic black hole shows a typical vortex-liquid effect when momentum relaxation is comparable to chemical potential. For small momentum relaxation the Nernst signal behaves as in a conventional metal. The Hall angle for the dyonic black hole has been computed explicitly. The Hall angle ranges between 1/T 0 and 1/T 1 and scales as 1/T for large T . The cyclotron pole we found are consistent with the hydrodynamic result at β = 0 [29,30]. They are shifted by a momentum relaxation effect(β = 0), which is proportional to the inverse of the relaxation time, 1/τ imp ∼ β 2 . Our preliminary numerical analysis suggest the specific modification of the cyclotron frequency (4.34). We plan to examine it in more detail both numerically and analytically. Furthermore, it is important to compare our AC conductivities with the general expressions based on the memory matrix formalism [39,43].
It would be interesting to compare our AC conductivity results with [44], 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 critical points [45]. It would also be of interest to apply our formalism and numerical method to other specific models with momentum relaxation. The metal phase of our model at B = 0 does not have the property of linear-T resistivity; it is worth while to start with the models respecting it and then investigate the Hall angle and the Nernst effect in those models.