Thermoelectric DC conductivities with momentum dissipation from higher derivative gravity

We present a mechanism of momentum relaxation in higher derivative gravity by adding linear scalar fields to the Gauss-Bonnet theory. We analytically computed all of the DC thermoelectric conductivities in this theory by adopting the method given by Donos and Gauntlett in [arXiv:1406.4742]. The results show that the DC electric conductivity is not a monotonic function of the effective impurity parameter $\beta$: in the small $\beta$ limit, the DC conductivity is dominated by the coherent phase, while for larger $\beta$, pair creation contribution to the conductivity becomes dominant, signaling an incoherent phase. In addition, the DC heat conductivity is found independent of the Gauss-Bonnet coupling constant.


Introduction
The AdS/CFT correspondence provides a powerful tool in probing many important phenomena of strongly correlated systems in condensed matter physics [1][2][3]. In the context of AdS/CFT, many charge transport coefficients such as DC conductivity, optical conductivity have been computed by considering the near-equilibrium filed theories on the boundary with gravity dual in the bulk. One can perturb the boundary by a time-dependent field with frequency ω to obtain the optical conductivity [1,3]. However, under this approach, when getting the DC conductivity with the limit ω → 0, one will confront the divergence due to the spatial translation invariance of the homogeneous gravitational backgrounds involved.
Unfortunately, it is well-known that in the real materials, the spatial translation invariance is not preserved i.e. the momentum are not conserved because of the presence of impurities and lattices.
Recently, a new approach to calculation of the DC conductivity have been developed in [27,28]. This approach does not rely on the zero frequency limit, but rather than a time-independent electric field as perturbation on the boundary. The DC conductivities can be obtained in terms of the horizon data by analysing regularity conditions to the holographic model where the momentum dissipation is due to linear spatial scalar fields.
Further discussions on the holographic massive gravity theory and Einstein-Maxwell theory with inhomogeneous, periodic lattices have been studied in [29] and [30] respectively.
In this paper, we generalize the strategy presented in [27,28] to calculate the DC conductivities in five-dimensional Einstein-Gauss-Bonnet-Maxwell-linear scalar field theory with momentum dissipation. The Gauss-Bonnet (GB) term in string effective action appears as the first curvature stringy correction to Einstein-Hilbert action when considering the semi-classical effect, then the higher order terms is dual to the finite corrections to the 1/N expansion of field theory on the boundary [31,32]. So in the framework of AdS/CMT, it is interesting to investigate the holographic conductivity of the quantum field theories with the higher derivative gravity dual before the string theory is fully understood [33][34][35][36]. Furthermore, to obtain the finite DC conductivities, we will also introduce spatially dependent massless field leads to the momentum dispassion [21]. Since we will focus on the isotropic bulk metric, we shall include three scalar fields that are linear in all the spatial directions.
The anisotropic solutions with one linear axion have been studied in [37][38][39][40], and the discussions for condensed matter with the anisotropic black brane dual can be found in [24,28,41]. This paper is organised as follows. In section 2, we present the exact solution for Einstein-Gauss-Bonnet-Maxwell gravity with linear scalar fields. Then, following [28], we calculate the DC electrical conductivity σ, thermal conductivityκ and thermoelectric α in terms of horizon data in section 3. The conclusions are presented in section 4.

Black brane solutions in Einstein-Maxwell-Gauss-Bonnet gravity with linear scalar fields
We begin with the following five-dimensional action of Einstein-Maxwell-Gauss-Bonnet gravity with three scalar fields.
where 2κ 2 = 16πG 5 is the five-dimensional gravitational coupling and Λ = −6 is cosmological constant.α is Gauss-Bonnet coupling constant with dimension (length) 2 and is Gauss-Bonnet term 1 . φ i (x µ )(i = 1, 2, 3) are 3 massless scalar fields and U(1) gauge field strength is defined as F µν = (dA) µν . 1 We follow the conventions of curvatures as [42] The equations of motion are easily obtained as We will consider homogeneous and isotropic charged black brane solutions, and then work with the following planar symmetric ansatz: where the UV boundary is defined as r → ∞ and N is a constant which can be fixed shortly by requiring that the geometry of the spacetime should asymptotically approach to the conformaly flat metric at the UV boundary.
To obtain the metric homogeneous, we also assume the scalar fields are linearly dependent on the three spatial coordinates: and gauge field as So the Maxwell equations and Einstein equations can be solved exactly where µ is the chemical potential of the dual field theory on the boundary, r H is the black The temperature can be evaluated directly from the Euclidean continuation of the metric (4), that is Since the entropy of GB black hole satisfies the area formula, from the Bekenstein-Hawking entropy formula, we obtain the entropy density of horizon Finally, we discuss the UV and IR behavior of the solution. First, near the UV boundary r → ∞, to guarantee that the spacetime is asymptotically conformally flat, we set N 2 = 1 2 1 + √ 1 − 8α and assumeα ≤ 1 8 . Note that the Einstein limit is obtained by taking the limitα → 0, in which the solution (4) reduces to the metric of [21]. To understand the geometry near horizon, we define a new coordinate u, then at T = 0, one can readily check that the extremal black brane geometry is topologically equivalent to AdS 2 × R 3 : where L is the curvature radius of AdS 2 : So we can see that in the absence of U (1) gauge field, the extremal black brane geometry can still be achieved, and the radius of AdS 2 will be bigger with the increase of Gauss-Bonnet coupling constantα .

DC conductivities
In this section, we will evaluate the DC electrical conductivity σ, thermal conductivityκ and thermoelectric conductivity α in terms of horizon data.

Electric conductivity
In order to compute the conductivities, we consider the perturbations as follows: Then linearizing the Maxwell equation, Einstein equations and Klein-Gordon equation, we can obtain four independent equations of perturbations: where the prime denotes derivatives with respect to r. Note that for the special case of a 2 1 = b 2 2 = c 2 3 = β 2 , Einstein equation (16) implies the equation of motion for δφ (18). From the (15), one can obtain a radially conserved current which is a constant. The Einstein equation (16) simply gives Then it is straightforward to see that the equation of motion for δφ can be simplified as To completely determine the solution of perturbation equations, we also need to impose the boundary condition for fluctuation. First, near the UV boundary r → ∞, the scaler field perturbation δφ should be regular, since (3f +rf ) rf ∼ r −1 as r → ∞ which is consistent with (21). δg tx behaves as r −1 which can be seen from (17).
Now we consider the asymptotic behavior near the horizon r = r H . Since we consider the boundary condition at the future horizon, we will use ingoing Eddiongton-Finklestein coordinates (v, r) defined as v = t + dr f (r)N here. First, the gauge field should be regular at the future horizon, which means that A x ∼ −Ev + .... So from the (14), we conclude that δA x should satisfies near horizon r = r H . On the other hand, it is easy to see that the singular part of the metric (4) can be expressed as in the ingoing Eddington-Finklestein coordinates. We can see from (20) that δg rx ∼ 1 r−r H is divergence as r → r H , so to get the metric non-singular at the horizon, we should require the metric perturbation behaves as Note that we have used the assumption of δφ is regular at the horizon.
Since electric current J is radial conserved, the DC electric conductivity can easily obtained by evaluation of the (19) at the horizon: Note that, in the Einstein limitα → 0, it will reproduce the result of [21].
As a demonstration, we plot the conductivity as a function of temperature in Fig 1, which reflects that the ground state of the model we are studying is not an insulator. It behaves more like semiconductors, since for semiconductors, there are insufficient mobile carriers at low temperatures and resistance is high; but as one heats the material, more and more of the lightly bound carriers escape and become free to conduct. However for normal metals there are plenty of mobile carriers and the motion of the lattice atoms due to thermal energy causes them to interfere with the transport of mobile carriers through the lattice. Thus, the conductivity of metals decreases as temperature goes up. We can see from Fig.1 that what we obtained does not correspond to normal metals.  The dependence of conductivity on β are shown in Fig.1 (b), in which we can see that in the small β limit, σ ∝ 1 β 2 means that it is dominated by the coherent phase. But as β becomes larger, σ ∝ β implies that the contribution of the pair production becomes stronger, leading to an incoherent phase [22]. This phenomena strongly signals that there is a competition effect between the Drude conductivity and conductivity due to pair creation at the horizon.
The dependence of conductivity on the Gauss-Bonnet coupling constantα is shown in Fig.1 (c). The upper bound of the Gauss-Bonnet coupling constant and its relation with the causality has been investigated in [43][44][45][46][47][48]. We can see that the DC conductivity increases as the Gauss-Bonnet constant becomes bigger. Now let us examine the behaviour of electric conductivity at low temperature. It is easy to see that, in the limit of T µ, the σ behaves as which means that the electric conductivity σ is finite as T → 0, indicating the metallic behaviour. On the other hand, for the case T µ, we have To obtain the transport coefficientᾱ, we will use a two-form [28]: where K µ is the Killing vector. Obviously, the metric (4) possesses the Killing vector K µ = 1 N ∂ t , then it is straightforward to check that √ −gH rx is conserved . We can deduce that As discussed in [28], the quantity Q should be identical to the heat current in the x-direction via calculation of holographic stress tensor [51][52][53]. Note that Q is independent of r, so after evaluation at the horizon, one can obtain Q = 8EπT r 2 H µ kN 2 , thenᾱ = ∂Q T ∂E is given bȳ

Thermal and thermoelectric conductivities
To compute the thermoelectric and thermal conductivities, as in [28], we consider the fluctuations as follows: Then, similarly, the linearised Maxwell equation involves a conserved current and the linearised Einstein equation gives As the last section, the heat current is also obtained as To get the transport coefficients α andκ, we suppose δh(r) = −CN f (r) and δa(r) = −E + C N A t (r) which can cancel the time-dependent terms of the conserved current J and Q. To find the behaviours of the perturbations near the horizon, we switch to Kruskal Similar to the last section, for the purpose of the metric regularity at the horizon, the perturbation at the horizon should be required as Note that the positive sign in the first term of (36) is chosen to be satisfied the equation for δg tx .
Now the α andκ can be easily obtained. First, because J and Q are constants in r direction, then evaluating the two conserved currents (32) and (34) at the horizon, we get Consequently, the conductivities α andκ are given by At low temperature, these transport coefficients behave as while at high temperature, the behaviour is So we find that thermoelectric conductivity α is finite at T = 0, while thermal conductivity κ = 0, meaning that a heat gradient does not give rise to transport. The presence of Gauss-Bonnet term increases the thermoelectric and thermal conductivities. Our results implies that we can extend [28] to higher dimensions with higher derivative gravity terms.

Summary
In this paper, we studied holographic conductivities for the higher derivative gravity with momentum relaxation. We presented a exact solution for Gauss-Bonnet-Maxwell theory with scalar fields. Then we derived analytically the DC electric conductivity, thermal and thermoelectric conductivities of the dual conformal filed on the boundary in the Gauss-Bonnet-Maxwell theory with momentum dissipation. We found that when the Gauss-Bonnet coupling increases, all the conductivities become bigger. The exact form of the conductivities confirmed that the ansatz given in [28] is applicable even in Gauss-Bonnet gravity in AdS space.
Different from the conductivities discussed in [22], the DC electric conductivity derived in this paper is temperature dependent and basically it increases as the temperature goes up.
The DC electric conductivity does not vanish even at T → 0 limit. In our case, at T = 0 the black brane approaches AdS 2 × R 3 in the far IR with non-vanishing entropy density. This reflects that the ground states of our system are semiconductors or bad metals. The electric conductivity at zero temperature might be regarded as arising from charged particle-hole pairs evolution.
It is our interests for the future task to work on the viscosity bound and causality problem in this linear scalar fields modified Gauss-Bonnet theory. There are some very recent works on viscosity bound in anisotropic superfluid [49] and backreaction effects [50] in higher derivative gravity. We expect that the presence of the linear scalars may contribute some physics more interesting that would greatly change the causal structure of the boundary theory and the upper and lower bounds of the Gauss-Bonnet coupling constant.