Holographic transports from Born–Infeld electrodynamics with momentum dissipation
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Abstract
We study the Einsteinaxions AdS black hole from Born–Infeld electrodynamics. Various DC transport coefficients of the dual boundary theory are analytically computed. The DC electric conductivity depends on the temperature, which is a novel property comparing to that in RNAdS black hole. The effects of BornInfeld parameter on the transport coefficients are analyzed. Also, we study the AC electric conductivity from Born–Infeld electrodynamics with momentum dissipation. For weak momentum dissipation, the low frequency behavior satisfies the standard Drude formula and the electric transport is coherent for various correction parameter. While for stronger momentum dissipation, the modified Drude formula is applied and we observe a crossover from coherent to incoherent phase. Moreover, the Born–Infeld correction amplifies the incoherent behavior. Finally, we study the nonlinear conductivity in probe limit and compare our results with those observed in (i)DBI model.
1 Introduction
The gaugegravity duality [1, 2, 3] provides a new avenue to study strongly coupled systems, which is difficult to process in the traditional perturbation theory. As an implement of this holographic application, transport phenomenon attracts lots of concentration by studying the electricthermo linear response via gaugegravity duality. In the study, the introduction of momentum relaxation is required to describe more real condensed matter systems, such that finite DC transport coefficients can be realized. In order to include momentum relaxation in the dual theory, several ways are proposed in the bulk gravitational sectors.
A simple mechanism to introduce the momentum dissipation is in the massive gravity framework. In this mechanism, the momentum dissipation in the dual boundary field theory is implemented by breaking the diffeomorphism invariance in the bulk [4]. It inspired remarkable progress in holographic studies with momentum relaxation in massive gravity [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18].
Another mechanism is to introduce a spatialdependent source, which breaks the Ward identity and the momentum is not conserved in the dual boundary theory. An obvious way is the so called scalar lattice or ionic lattice structure, which is implemented by a periodic scalar source or chemical potential [19, 20, 21]. Also, we can obtain the boundary spontaneous modulation profiles in some particular gravitational models, which break the translational symmetry and induce the charge, spin or pair density waves [22, 23, 24, 25, 26]. These ways involve solving partial differential equations (PDEs) in the bulk. Another important way is to break the translation symmetry but hold the homogeneity of the background geometry. Comparing with the scalar lattice or ionic lattice structure, the advantage of this way is that we only need to solve ordinary differential equations (ODEs) in the bulk. Outstanding examples of this include holographic Qlattices [27, 28, 29, 30], helical lattices [31] and axion model [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. Holographic Qlattice model breaks the translational invariance via the global phase of the complex scalar field. Holographic helical lattice model possesses the nonAbelian Bianchi VII\(_0\) symmetry, where the translational symmetry is broken in one space direction but holds in the other two space directions. The translational symmetry is broken in holographic axion model by a pair of spatialdependent scalar fields, which are introduced to source the breaking of Ward identity. In addition, by turning on a higherderivative interaction term between the U(1) gauge field and the scalar field, we can also obtain a spatially dependent profile of the scalar field generated spontaneously, which leads to the breaking of the Ward identity and the momentum dissipation in the dual boundary field theory [43, 44].
In this paper, we will study the Einsteinaxions theory with Born–Infeld Maxwell field, i.e., the EinsteinBorn–Infeldaxions theory. We first construct the black brane solution by solving the equations of motion in the theory. Then we analytically compute the DC transport coefficients in the dual theory and we discuss the influence from Born–Infeld parameter. Also, we numerically study the AC electric conductivity and analyze its low frequency behavior via (modified) Drude formula. Finally, we analyze the nonlinear currentvoltage behavior with BI correction in probe limit.
2 Einstein–Born–Infeldaxions theory
Since the Born–Infeld Anti deSitter (BIAdS) geometry and its extensions have been explored in detail in [55, 69, 70, 71, 72, 73] and references therein, here, we only give a brief review on the BIAdS geometry related with our present study.
3 Electric, thermal and thermoelectric DC conductivity

The electric DC conductivity \(\sigma _{DC}\) is temperature dependent for the fixed \(\hat{\beta }\) (Fig. 1). It is the key difference comparing with that in 4 dimensional RNAdS black brane, in which the DC conductivity is the temperature independence.

At \(\hat{T}=0\), \(\sigma _{DC}\) is a positive constant (Fig. 1), which implies that the dual system is an metal. While \(\bar{\kappa }=0\) for \(\hat{T}=0\) (Fig. 3) as we expect because for \(\hat{T}\rightarrow 0\), we have \(\kappa (\hat{T})\sim \hat{T}\rightarrow 0\). It comes just from the scalings on the near horizon geometry.

For the fixed finite temperature \(\hat{T}\), with the increase of \(\gamma \), \(\sigma _{DC}\), \(\alpha _{DC}/\bar{\alpha }_{DC}\) and \(\kappa _{DC}\) increase (Figs. 1, 2 and 4), but \(\bar{\kappa }_{DC}\) decreases (Fig. 3).
4 Optical electric conductivity
We will explore the AC electric conductivity from nonlinear BI electrodynamics with momentum dissipation. We shall fix the temperature \(\hat{T}\) to study the effects of \(\hat{\beta }\) and \(\gamma \). Figure 5 exhibits the electric conductivity as the function of the frequency \(\hat{\omega }\) for different \(\hat{\beta }\) and \(\gamma \). Similarly with the standard Maxwell theory, for the fixed \(\gamma \), with the increase of \(\hat{\beta }\), the Drudelike peak gradually reduces and a transition from coherent phase to incoherent phase happens. For the fixed \(\hat{\beta }\), the peak seems to augment when \(\gamma \) increases. But the quantitative analysis later indicates that although with the increase of \(\gamma \) the peak augments, the degree of deviation from the Drude one becomes grave. Note that as a quick check on the consistency of our numerics, we denote the DC electric conductivity analytically calculated by Eq. (25) (red points) in Fig. 5, which match very well with the numerical results.
The momentum relaxation rate \(\Gamma \) fitted by the standard Drude formula (38) for different \(\gamma \) with fixed \(\hat{\beta }=0.25\)
\(\gamma \)  0  1  2 

\(\Gamma \)  0.0165  0.00827  0.00630 
The momentum relaxation rate \(\Gamma \) fitted by the modified Drude formula (39) for different \(\gamma \) with fixed \(\hat{\beta }=0.5\)
\(\gamma \)  0  1  2 

\(\Gamma \)  0.0490  0.0261  0.0206 
\(\sigma _Q\)  0.528  0.696  0.821 
The momentum relaxation rate \(\Gamma \) fitted by the modified Drude formula (39) for different \(\gamma \) with fixed \(\hat{\beta }=1\)
\(\gamma \)  0  1  2 

\(\Gamma \)  0.116  0.070  0.0589 
\(\sigma _Q\)  0.837  0.842  0.855 
Further, we compare the results in the standard linear axions theory [5, 32] to those of the present study. First, we analyze \(\Delta \Gamma (\gamma )\equiv \Gamma (\gamma =0)\Gamma (\gamma )\) as the function of \(\gamma \) at \(\hat{T}=10^{3}\) (Fig. 8). As expected, the difference becomes large with the increase of \(\gamma \) and the change of \(\Delta \Gamma (\gamma )\) is nonlinear. Then, we study \(\Delta \Gamma \) as the function of the temperature \(\hat{T}\) for sample \(\gamma \) and \(\hat{\beta }\) (Fig. 9). The difference \(\Delta \Gamma \) is suppressed with the increase of the temperature. In addition, \(\Delta \Gamma \) becomes negative when the temperature goes beyond certain value, which means that in this case, \(\Gamma (\gamma )>\Gamma (\gamma =0)\). However, we would like to point out that with the increase of the temperature, the fitting of conductivity by Eq. (39) becomes ill. Therefore, we only focus the cases at low temperature.
Before closing this section, we shall present some comments on the conductivity in nonlinear Maxwell theory. As addressed in [78, 79, 80, 81, 82, 83, 84], the nonlinear Maxwell theory, including DBI deformation and the coupling between Weyl tensor and Maxwell term, produces a Drudelike peak contribution. This outcome is different from that in the axions models [5, 27, 28, 29, 30, 32], in which the Drude peak attributes to the approximate conservation of momentum [85, 86].
To reveal the origin of the Drudelike peak from nonlinear Maxwell theory, the analytical derivation is needed. One way is to decouple the perturbation equations following the strategy of [85, 86], which can surely provide some novel insights and understanding into this question. Another way is to obtain a formula for the low frequency conductivity using matching method as done in the standard linear axions model [5]. However, we find the origin of the Drudelike peak of nonlinear Maxwell theory is difficult to reveal due to the complexity and we will leave these for future work. Note that the authors of [87] have attempted to explain the origin of the Drudelike peak from DBI deformation. Specially, they show that the momentum relaxation rate \(\Gamma \) is proportional to \(T^2\). However, we find in the present study, \(\Gamma \) does not follow the \(T^2\) law (see Fig. 10). We hope to further explore the law of the temperature dependence of \(\Gamma \) in the BI model at finite charge density in near future.
5 Nonlinear electric conductivity in probe limit
As is discussed in [68], the usual way to study nonlinear conductivity is to show the nonlinear current–voltage diagram, from which we may see the nonlinear behavior of the electric conductivity. To get analytical control, we will work in the probe limit, i.e., we ignore the mixture with the metric perturbation and keep the nonlinear self couplings of the Maxwell field as done in the references [88, 89].
The current voltage behavior from Eq. (42) is shown in Fig. 11. \(\gamma =0\) corresponds to the standard Maxwell theory, so that the current–voltage relation is linear and means \(J_x=E_x\) as we expect. When \(\gamma \ne 0\), the nonlinear behavior is observed. For \(\gamma > 0\), the curve is above the linear case and \(dJ_x/dE_x\) is stronger than 1 as that happened in DBI model [68], however, it is enhanced as the applied field \(E_x\) increases which is different from that in DBI model. For \(\gamma <0\), the nonlinear \(dJ_x/dE_x\) is always lower than 1. As the applied field increases, it approaches to be vanished, but this will not happen because backreacion should be involved when \(E_x\) is very large . The unstable case \(dJ_x/dE_x<0\) shown in iDBI model is not observed in our model. We notice that when we continue to lower \(\gamma \), the current become complex, this deserves further study.
6 Conclusions
In this paper, we introduced the Maxwell field with Born–Infeld correction into the Einsteinaxions theory and studied a new charged BIAdS black hole. Then we analytically calculated various DC transport coefficients of the dual boundary theory. We found that the DC electric conductivity depends on the temperature of the boundary theory, which is a novel property comparing to that in RNAdS black hole. At zero temperature, The DC electric conductivity are positive while the thermal conductivity vanishes as we expect. With the increase of Born–Infeld parameter, the electric conductivity, electricthermo conductivity and thermal conductivity at zero increases at finite fixed temperature.
We also studied the AC electric conductivity of the theory. When the momentum dissipation is weak, the low frequency AC conductivity behaves as the standard Drude formula and the electric transport is coherent for various correction parameter. When the momentum dissipation is stronger, the modified Drude formula is applied and a crossover from coherent to incoherent phase was observed. Also, we found that the Born–Infeld correction makes the incoherent behavior more explicit. We notice that here we only numerically compute the AC electric conductivity dual to its simply. It would be very interesting to further study the AC thermal and electricthermo conductivity which are related with boundary data of both Maxwell perturbation and gravitational perturbation [15]. We hope to show the results elsewhere soon.
Finally, we analyze the nonlinear current–voltage behavior with BI correction in probe limit. The curve with \(\gamma > 0\) is above the linear case and \(dJ_x/dE_x\) is always bigger than 1. Different from that happened in DBI model [68], the slope is enhanced as \(E_x\) increases. For \(\gamma <0\), the nonlinear \(dJ_x/dE_x\) is always lower than 1 and it tends to be zero as \(E_x\) go to infinity in which case the backreaction should be considered. For more negative \(\gamma \), the current become complex and further study is called for.
In future, there are many interesting questions deserving further exploration. First of all, we can study the holographic anomalous transport from BI electrodynamics as [90, 91]. In [92, 93], they study the thermal transport and quantum chaos in the EMDA theory with a small Weyl coupling term. In particular, in [92], they find that the Weyl coupling correct the thermal diffusion constant and butterfly velocity in different ways, hence resulting in a modified relation between the two at IR fixed points. It is interesting to further explore this relation in present of BI correction. We shall come back these topics in near future.
Footnotes
 1.
More forms for nonlinear Maxwell theory, such as power Maxwell theory, logarithmic Maxwell theory were also proposed in [48, 63, 64]. Also, the magnetotransport in holographic DBI (BI) model has been studied in [65]. In particular, in [66], they find that the inplane magnetoresistivity exhibits the interesting scaling behavior that is compatible to that observed recently in experiments on \(BaFe_2(As_{1x}P_x)_2\) [67].
Notes
Acknowledgements
We are very grateful to Guoyang Fu and Peng Liu for many useful discussions and comments on the manuscript. Also we thank the anonymous referee for lots of useful suggestions. This work is supported by the Natural Science Foundation of China under Grants Nos. 11705161, 11775036 and 11747038. X. M. Kuang is also supported by Natural Science Foundation of Jiangsu Province under Grant No. BK20170481. J. P. Wu is also supported by the Natural Science Foundation of Liaoning Province under Grant No. 201602013.
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