Holographic DC Conductivity for a Power-law Maxwell Field

We consider a neutral and static black brane background with a probe power-law Maxwell field. Via the membrane paradigm, an expression for the holographic DC conductivity of the dual conserved current is obtained. We also discuss the dependence of the DC conductivity on the temperature, charge density and spatial components of the external field strength in the boundary theory. Our results show that there might be more than one phase in the boundary theory. Phase transitions could occur where the DC conductivity or its derivatives are not continuous. Specifically, we find that one phase possesses a charge-conjugation symmetric contribution, negative magneto-resistance and Mott-like behavior.


I. INTRODUCTION
The idea of the membrane paradigm was started by Thibault Damour [1] and then developed further by Kip Thorne et al. [2,3]. Later, a more systematic action-based derivation was proposed by Parikh and Wilczek in [4], which could apply to various field theories. In the membrane paradigm, the observer at infinity sees that the black hole is equivalent with a thin fluid membrane living just outside the black hole's event horizon, and hence the black hole can be replaced by the fluid membrane. The membrane paradigm was originally proposed to study astrophysical black holes [5][6][7]. Realizing the membrane fluid could provide the long wavelength description of the strongly coupled quantum field theory at a finite temperature, researchers take a new interest in the membrane paradigm in the context of gauge/gravity duality [8][9][10][11]. In [10], the low frequency limit of the boundary theory transport coefficients could be expressed in terms of geometric quantities evaluated at the horizon by identifying the currents in the boundary theory with radially independent quantities in bulk. The method of [10] was later extended to calculate the DC conductivity in the presence of momentum dissipation [12][13][14][15], where the zero mode of the current, not the current itself, did not evolve in the radial direction. Specifically, the DC thermoelectric conductivity has recently been obtained by solving a system of Stokes equations on the black hole horizon for a charged fluid in Einstein-Maxwell theory [16].
Nonlinear electrodynamics (NLED) is an effective model incorporating quantum corrections to Maxwell electromagnetic theory. NLED is interesting per se, for example some models give finite self-energy of charged particles and can remove singularity at the classical level. Two famous NLED are Heisenberg-Euler effective Lagrangian [17] and Born-Infeld electrodynamics [18]. On the other hand, it is well-known that the Maxwell action enjoys the conformal invariance in four dimensions. A natural extension of the Maxwell action in (d + 1)-dimensional spacetime that is the conformally invariant is the action of a power-law Maxwell field [19]: where we define a nontrivial scalar A a is the electromagnetic field tensor, and A a is the electromagnetic potential.
The action (1) is conformally invariant provided p = (d + 1) /4. When d = 3, the action (1) recovers the standard Maxwell action. However, we don't confine ourselves to p = (d + 1) /4 in our paper. Instead, we shall consider a more general case from now on, in which p is an arbitrary positive integer. Coupling the power-law Maxwell field to gravity, various charged black holes were derived in a number of papers [19][20][21][22][23][24]. In the framework of gauge/gravity duality, holographic superconductors [25,26], action/complexity conjecture [27], and the DC conductivity in the massive gravity [28] were studied in presence of a power-law Maxwell field.
This paper is a follow-up paper of our previous paper [29]. In [29], we used the method of [10] to compute the DC conductivities of an conserved current dual to a probe nonlinear electrodynamics field in a general neutral and static black brane background. However, our previous paper dealt, primarily, with a NLED Lagrangian that would reduce to the Maxwell-Chern-Simons Lagrangian for small fields. Clearly, the power-law Maxwell field with p = 1 does not belong to this class of NLED models and would have some different predictions for the DC conductivities in the boundary theory. For example, when the charge density and magnetic field in the boundary theory vanish, the DC conductivities are zero for p = 1 in this paper while they are not in [29].
In this paper, we will consider a neutral and static black brane background with a probe power-law Maxwell field and the dual theory. The aim of this paper is to find an expression for the holographic DC conductivity of the dual conserved current and investigate the properties of the boundary theory, e.g the possible phases and the magnetotransport. Note that the properties of magnetotransport in holographic Dirac-Born-Infeld models have been discussed in a probe case [30] and taking into account the effects of backreaction on the geometry [31].
The remainder of our paper is organized as follows: In section II, we briefly review the membrane paradigm for a power-law Maxwell field. The holographic DC conductivity of the dual conserved current is studied in section III. In section IV, we conclude with a brief discussion of our results. We use convention that the Minkowski metric has signature of the metric (− + ++) in this paper.

II. MEMBRANE PARADIGM
In [29], the electromagnetic membrane properties have been examined for a general NLED model via the method of [4]. In this section, we first give a quick review of the membrane paradigm in the framework of a power-law Maxwell field. In membrane paradigm, a timelike hypersurface, namely the stretched horizon, is put just outside the black hole horizon.
The stretched horizon is composed of a family of fiducial observers with world lines U a and possesses a spacelike outward pointing normal vector n a . The stretched horizon is denoted by S. To derive the Euler-Lagrange equations from the action restricted to the spacetime outside the stretched horizon S out , it is necessary to add a surface term S surf to S out to exactly cancel all the boundary terms. Consequently, the total action can be rewritten as where δS out + δS surf = 0 will give the correct equations of motion outside S.
For a power-law Maxwell field A a , the external action in a (d + 1)-dimensional spacetime is given by the action (1). To cancel the boundary contribution on the stretched horizon from the action (1), we add a surface term S surf where h ab = g ab − n a n b is the induced metric on S; we define and Note that j a s can be interpreted as the membrane current on the stretched horizon since n a j a s = 0. This current corresponds to the surface electric charge density ρ = −j a s U a and current density j a s = j a s − σU a . In this paper, we consider a general black brane background, the metric of which takes the form where indices {a, b} run over the (d + 1)-dimensional bulk space, {µ, ν} over d-dimensional constant-r slice, and {A, B} over spatial coordinates. We assume that there is an event horizon at r = r h , where g tt (r) has a first order zero, g rr (r) has a first order pole, and g zz (r) is nonzero and finite. The Hawking temperature of this black brane is Now put the stretched horizon at r = r 0 with r 0 − r h r h . This stretched horizon has n a = g rr (r 0 )δ ar and U a = − g tt (r 0 )δ at .
Thus, the membrane current (5) reduces to It showed in [29] that, on the stretched horizon, the NLED field strength has We then use eqns. (6), (10) and (11) to rewrite j A s as where the electric field measured by the fiducial observers on the stretched horizon is and s on the stretched horizon becomes From eqn. (12), we can read that the diagonal components of the conductivities of the stretched horizon are and the Hall components are zero. These fields also increase the black hole's entropy S in accord with the Joule-heating relation [3]: where α = g tt (r 0 ) is the renormalized factor [3].

III. DC CONDUCTIVITY FROM GAUGE/GRAVITY DUALITY
We now consider a probe power-law Maxwell field in the background of a (d + 1)dimensional black brane with the metric (7). For simplicity, we assume that this black brane is uncharged with trivial background configuration of the power-law Maxwell field.
This power-law Maxwell field is a U(1) gauge field and dual to a conserved current J µ in the boundary theory. The corresponding AC conductivities are given by where the boundary theory lives at r → ∞. The DC conductivities are obtained in the long wavelength and low frequency limit: We can compute the expectation value of the current J µ for the boundary theory by [29] where Π µ is the conjugate momentum of the field A µ with respect to r-foliation. When where ρ can be interpreted as the charge density in the dual field theory.
Identifying the currents in the boundary theory with radially independent quantities in the bulk, authors of [10] showed that the membrane paradigm fluid on the stretched horizon determined the low frequency limit of conductivities of a conserved current in the boundary theory, which was dual to a Maxwell field in bulk. Later, the method of [10] was extended to the NLED case in [29]. In particular, it showed there that, in the long wavelength and low frequency limit, i.e. ω → 0 and k → 0 with F ρσ and Π η fixed, the following quantities did not evolve in the radial direction: On the horizon, we have where we take the limit r 0 → r h . Here, s on the horizon becomes where we define For d = 2 and 3, the magnetic field is a scalar and a vector, respectively, and B can be treated as the magnitude of the magnetic field in the boundary theory. To express F rt (r h ) in terms of quantities in the boundary theory, we can use the following formula On the boundary, we have that, in the zero momentum limit, From eqn. (26), we can read that the diagonal components of the DC conductivities in the boundary theory: where, for later convenience, we definẽ When d = 3 and p = 1, eqn. (27) reproduces the well-known result in the Maxwell case [3] σ D = 1.
For p = 1, the DC conductivity σ D is zero in the absence of the magnetic field and charge density in the boundary theory, which is consistent with eqn. (44) with q = 0 in [28].
In the long wavelength and low frequency limit with ω → 0 and k → 0, we keep F µν and Π µ fixed and neglect higher µ-derivatives. This means that F µν and ρ are constant and homogeneous on the boundary. In this limit, one can relate the DC conductivity σ D in the boundary theory to σ s of the stretched horizon as which is also constant and homogeneous on the stretched horizon. Therefore in the long wavelength and low frequency limit, the rate of the black hole's entropy S becomes The second law of black hole mechanics implies that the DC conductivity σ D in the boundary theory is non-negative and real.
It is interesting to note that the function x (y) is usually a multivalued function, which indicates that there might exist more than one phase and possible phase transitions. For later convenience, we defineσ Here we require that σ D is non-negative and real on x (y).

A. Even Positive Integer p
We plot y (x) = −x (x 2 − 1) p−1 for p = 2 in FIG. 1(a). In fact, y (x) and hence σ D in all the cases of p being even positive integer show very similar behavior as in that of p = 2.
So for concreteness, we shall focus on the case of p = 2. Bearing in mind that σ D is nonnegative and real, eqn. (27) shows that the green segment of y (x) in FIG. 1(a) is unphysical.
Therefore, we only need to consider the blue and red segments to find the inverse function of y (x), which is plotted in FIG. 1(b). As shown in FIG. 1(b), there is a discontinuity at y = 0 for x (y), which, as will be shown later, indicates possible phase transitions at y = 0.   . 3(a). Unlike the p = 2 case, x/y ≤ 0 and henceσ D is non-negative for all points on y (x) in the p = 3 case. So the physical inverse function of y (x) is plotted in FIG. 3(b). As To determine the stable phases and the transition points, one needs to find the thermodynamic potential in a specific boundary theory, which, however, is beyond the scope of this paper.
For the blue, orange, red, and purple phases, the behavior of the DC conductivity σ D is similar to that in the p = 2 case, i.e. σ D = 0 at zero charge density, σ D increases with increasing |ρ| at constant B, and σ D decreases with increasing B at constant ρ. However, the green phase has some interesting features: • Charge conjugation symmetric contribution. At zero charge density, σ D has a non-zero value, ifB = 0, which can be interpreted by a incoherent contribution due to intrinsic current relaxation and independent of the charge density. This contribution is also known as the charge conjugation symmetric contribution [32,33].
Therefore we can see from FIGs. 4 and 5(b) that ∂σ D /∂ρ < 0 for the green phase.
This can be explained by the electronic traffic jam: strong enough e-e interactions prevent the available mobile charge carriers to efficiently transport charges [34]. Note that a class of holographic models for Mott insulators, whose gravity dual contained NLED, was studied in [34].

C. Temperature Dependence of DC Conductivity
To discuss the temperature dependence of the DC conductivity, we can express σ D in terms of ρ and B: When d = 4p−1, the power-law Maxwell field action (1) is conformally invariant. In this case, the DC conductivity σ D is independent of the geometric quantities evaluated at the horizon, especially the Hawking temperature T of the black brane. So the DC conductivity σ D does not depend on the temperature of the boundary theory when the power-law Maxwell field in bulk is conformally invariant. In fact, the dual conserved current is also scale invariant.
For this scale invariant current at finite temperature, all nonzero temperatures should be equivalent since there is no other scale with which to compare the temperature.
For d = 4p − 1, we can now discuss the temperature dependence of the DC conductivity by relating r h to the Hawking temperature T . For simplicity and concreteness, we consider the Schwarzschild AdS black brane where we take the AdS radius L = 1, and r h determines the Hawking temperature of the black hole: Since x (y) behaves differently for small y depending on whether p is an even or odd integer, it will be convenient to consider these two cases separately.
When p is even, one has that x 2 (y) ∼ 1 for y 1. In the case with ρ = 0 and nonzero B, one has σ D = 0. In the case with nonzero B and ρ, one has that, when d > 4p − 1,  in the blue and orange phases; x (y) ∼ −y in the green phase. In the case with ρ = 0 and nonzero B, we find that in the green phase, and σ D = 0 otherwise.
From the monotonicity of x (y), we can also determine whether each phase is a metal or an

IV. DISCUSSION AND CONCLUSION
In this paper, we extended the method of [10] to study the electrical transport behavior of some boundary field theory in the presence of a power-law Maxwell gauge field. In particular, we first calculated the conductivities of the stretched horizon of some general static and neutral black brane in the framework of the membrane paradigm. Since the conjugate momentum of the power-law Maxwell field encoded the information about the conductivities both on the stretched horizon and in the boundary theory and, in the zero momentum limit, did not evolve in the radial direction, we obtained the DC conductivity of the dual conserved current in the boundary theory. We also found that the DC conductivity could be expressed in terms of the electromagnetic quantities and the temperature of the boundary theory.
In the context of the membrane paradigm, we found that the second law of black-hole mechanics required that the DC conductivities of the stretched horizon and in the boundary theory are real and non-negative. Imposing σ D ≥ 0, we showed that, when p was an even integer, there might be two phases in the boundary theory, and a continuous phase transition could occur atρ = 0. When p was an odd integer, there might be five phases in the boundary theory, and the transitions among them could be considered as first order phase transitions. Specifically, it showed that the green phase possessed a charge conjugation symmetric contribution, a negative magneto-resistance and Mott-like behavior. We also discussed the temperature dependence of the DC conductivity. We found that the DC conductivity σ D was independent of the temperature of the boundary theory when d = 4p−1.
Note that the power-law Maxwell field action is conformally invariant for d = 4p − 1.
Finally, we discuss the assumption and limitation of our calculations. First, we assumed that the black brane background was neutral, and hence there was no background charge density in the boundary theory. Since the low frequency behavior of the conductivities depends crucially on whether there is a background charge density [12], investigating the behavior of the DC conductivity in a boundary theory dual to a charged power-law Maxwell field black hole is certainly interesting. Second, we assumed that the power-law Maxwell field was a probe field and neglected the backreaction on the bulk spacetime metric. One would like to study the effects of backreaction on the bulk spacetime metric and DC conductivity in the boundary theory. Third, we carried out our calculations in the zero momentum limit, in which the conjugate momentum did not evolve along the radial direction in bulk, and the electromagnetic quantities ρ and B were time independent and homogeneous in the boundary theory.