Holographic DC conductivity for a powerlaw Maxwell field
Abstract
We consider a neutral and static black brane background with a probe powerlaw 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 chargeconjugation symmetric contribution, negative magnetoresistance and Mottlike behavior.
1 Introduction
The idea of the membrane paradigm was started by Damour [1] and then developed further by Thorne et al. [2, 3]. Later, the membrane paradigm has been applied to various field theories by following a more systematic actionbased derivation proposed by Parikh and Wilczek in [4]. 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. In the presence of momentum dissipation, the DC conductivity can also be calculated in a similar way [12, 13, 14, 15] since there appeared to exist the radially independent zero mode of some current in bulk. Recently, Donos and Gauntlett obtained the DC thermoelectric conductivity in Einstein–Maxwell theory [16] by solving a system of Stokes equations on the black hole horizon for a charged fluid.
This paper is a followup paper of our previous paper [30]. In [30], 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 MaxwellChernSimons Lagrangian for small fields. Clearly, the powerlaw Maxwell field with \(p\ne 1\) 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 \(p\ne 1\) in this paper while they are not in [30].
In this paper, we will consider a neutral and static black brane background with a probe powerlaw 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 in the probe limit, the properties of magnetotransport have been investigated in holographic Dirac–Born–Infeld models [31]. Later in [32], the backreaction effects of matter fields on the Dirac–Born–Infeld models was considered.
The remainder of our paper is organized as follows: In Sect. 2, we briefly review the membrane paradigm for a powerlaw Maxwell field. The holographic DC conductivity of the dual conserved current is studied in Sect. 3. In Sect. 4, we conclude with a brief discussion of our results. We use convention that the Minkowski metric has signature of the metric \(\left( +++\right) \,\)in this paper.
2 Membrane paradigm
3 DC conductivity from gauge/gravity duality
3.1 Even positive integer p
We plot \(y\left( x\right) =x\left( x^{2}1\right) ^{p1}\) for \(p=2\) in Fig. 1a. In fact, \(y\left( x\right) \) and hence \(\sigma _{D}\) 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 \(\sigma _{D}\) is nonnegative and real, Eq. (27) shows that the green segment of \(y\left( x\right) \) in Fig. 1a is unphysical. Therefore, we only need to consider the blue and red segments to find the inverse function of \(y\left( x\right) \), which is plotted in Fig. 1b. As shown in Fig. 1b, there is a discontinuity at \(y=0\) for \(x\left( y\right) \), which, as will be shown later, indicates possible phase transitions at \(y=0\). Using \(x\left( y\right) \) in Fig. 1b, we plot \({\tilde{\sigma }}_{D}\) versus \({\tilde{\rho }}\) and \({\tilde{B}}\) in Fig. 2. It shows in Fig. 2 that \({\tilde{\sigma }}_{D}\) is continuos everywhere but the derivative \(\partial _{{\tilde{\rho }}}{\tilde{\sigma }}_{D}\) changes the sign at \({\tilde{\rho }}=0\). These observations imply that there might exist two phases for \({\tilde{\rho }}>0\) and \({\tilde{\rho }}<0\), respectively, and a continuous phase transition could occur at \({\tilde{\rho }}=0\).
Since \(x\left( 0\right) =1\), Eq. (27) shows that the DC conductivity \(\sigma _{D}\) vanishes at zero charge density, which implies that the main contribution to \(\sigma _{D}\) is from momentum relaxation for the charge carriers in the system. As shown in Fig. 2, \(\sigma _{D}\) increases with increasing \(\left \rho \right \) at constant B, which is a feature similar to the Drude metal. For the Drude metal, a larger charge density provides more available mobile charge carriers to efficiently transport charge. At constant \(\rho \), \(\sigma _{D}\) decreases with increasing B, which means a positive magnetoresistance.
3.2 Odd positive integer p
 Charge conjugation symmetric contribution. At zero charge density, \(\sigma _{D}\) has a nonzero value, if \({\tilde{B}}\ne 0\),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 [33, 34].$$\begin{aligned} \sigma _{D}\left( 0,{\tilde{B}}\right) =\frac{g_{zz}^{\frac{d3}{2}}\left( r_{h}\right) p}{2^{p1}}{\tilde{B}}^{2p2}, \end{aligned}$$(34)

Negative magnetoresistance. We plot \({\tilde{\sigma }}_{D}\) versus \({\tilde{B}}\) for \({\tilde{\rho }}=0\), 0.1, 0.3, 0.5, and 0.7 in Fig. 5a. Figures 4 and 5a show that \(\partial {\tilde{\sigma }}_{D}/\partial {\tilde{B}}>0\), which gives a negative magnetoresistance at given temperature and charge density.

Mottlike behavior. We plot \({\tilde{\sigma }}_{D}\) versus \({\tilde{\rho }}\) for \({\tilde{B}}=0.7\), 0.8, 0.9, 0.95, and 1 in Fig. 5b. Therefore we can see from Figs. 4 and 5b that \(\partial {\tilde{\sigma }}_{D}/\partial \rho <0\) for the green phase. This can be explained by the electronic traffic jam: strong enough ee interactions prevent the available mobile charge carriers to efficiently transport charges [35]. Note that a class of holographic models for Mott insulators, whose gravity dual contained NLED, was studied in [35].
3.3 Temperature dependence of DC conductivity
Sign of \(\partial \sigma _{D}/\partial T\) in all cases
Even p  Purple, green and blue phases for odd p  Orange and red phases for odd p  

\(d<4p1\)  Metal \(\left( \partial \sigma _{D}/\partial T<0\right) \)  Metal \(\left( \partial \sigma _{D}/\partial T<0\right) \)  Insulator \(\left( \partial \sigma _{D}/\partial T>0\right) \) 
\(d>4p1\)  Insulator \(\left( \partial \sigma _{D}/\partial T>0\right) \)  Insulator \(\left( \partial \sigma _{D}/\partial T>0\right) \)  Metal \(\left( \partial \sigma _{D}/\partial T<0\right) \) 
\(d=4p1\)  \(\partial \sigma _{D}/\partial T=0\)  \(\partial \sigma _{D}/\partial T=0\)  \(\partial \sigma _{D}/\partial T=0\) 
4 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 powerlaw 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 powerlaw 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 blackhole mechanics required that the DC conductivities of the stretched horizon and in the boundary theory are real and nonnegative. Imposing \(\sigma _{D}\ge 0\), 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 at \({\tilde{\rho }}=0\). 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 magnetoresistance and Mottlike behavior. We also discussed the temperature dependence of the DC conductivity. We found that the DC conductivity \(\sigma _{D}\) was independent of the temperature of the boundary theory when \(d=4p1\). Note that the powerlaw Maxwell field action is conformally invariant for \(d=4p1\).
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 powerlaw Maxwell field black hole is certainly interesting. Second, we assumed that the powerlaw 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 \(\rho \) and B were time independent and homogeneous in the boundary theory.
Notes
Acknowledgements
We are grateful to Houwen Wu and Zheng Sun for useful discussions. This work is supported in part by NSFC (Grant nos. 11005016, 11175039, 11375121, and 11747171) and Natural Science Foundation of Chengdu University of TCM (Grants nos. ZRYY1729 and ZRQN1656). Discipline Talent Promotion Program of “Xinglin Scholars” (Grant no. QNXZ2018050) and the key fund project for Education Department of Sichuan (Grant no. 18ZA0173)
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