# AC charge transport in holographic Horndeski gravity

## Abstract

In this paper, we investigate the AC charge transport in the holographic Horndeski gravity and identify a metal-semiconductor like transition that is driven by the Horndeski coupling. Moreover, we fit our numeric data by the Drude formula in slow relaxation cases.

## 1 Introduction

Understanding the dynamics of various exotic quantum matters, including quark-gluon plasma as well as strongly coupled electronic systems, has long been a core topic in modern physics [1, 2, 3, 4]. These systems are, in general, hard to deal with by conventional perturbative theories due to the existence of strong interactions between the microscopic degrees of freedom. The string theory inspired holographic duality, as a strong/weak duality, offers a powerful tool to touch the anomalous phenomena in strongly coupled systems, such as the linear dependance of DC resistivity on temperature in a wide class of strange metals.

Physically, symmetry of spatial translation implies the conservation law of momentum. For finite density system without momentum dissipations, the DC conductivity is divergent because of a non-trivial overlap between the electric current and the conserved momentum. However, in most of realistic materials, such a symmetry is broken for the presence of lattice, impurities or defect, which results in the finite conductivity. Then, realizing effects of momentum dissipation in holography has been extensively studied in recent years [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

In this paper, we shall study the holographic charge transport of Horndeski gravity model. The Horndeski gravity was first constructed in 1970s [24, 25], and was recently rediscovered in cosmology [26, 27]. The theory has a remarkable property that it contains higher than two derivatives at the level of Lagrangian, however, each field has at most two derivatives at the level of equation of motion. This property is similar to that of Lovelock gravity [28], and thus the theory can be ghost free. The black hole solutions of the theory were constructed in [29, 30] and their thermodynamics were carried out in [31, 32]. The stability and causality were studied in [33, 34, 35]. It was showed in [36] that the theory admits the holographic a-theorem in a critical point of the parameter space, which suggests the theory may have a holographic dual field theory. Further holographic properties and applications of Horndeski theory were investigated in [37, 38, 39, 40]. The DC limit of holographic transports of Horndeski gravity models were extensively analytically studied in [41, 42], especially it was pointed out that the thermal conductivity bound [43] was broken by a type of Horndeski model [44]. Here, we want to go a step further to study the properties of the AC electric conductivity of the Horndeski theory through numerical methods.

This paper is organized as follows: in Sect. 2, we introduce the black hole background in the Horndeski theory. In Sect. 3, we calculate the electric conductivity by turning linearized perturbations of the bulk fields and show the numeric results. In Sect. 4, we conclude. Finally, we show the details of calculating AC conductivity in the appendix.

## 2 Holographic setup

*a*-theorem [36]. For simplicity, we shall set \(\kappa =\alpha =1\) from now on. \(\phi ^i\) are massless axion fields which are coupled to the Einstein tensor. Thus \(\gamma \) should be regarded as the Horndeski coupling strength. If we set \(\gamma =0\), it returns to Einstein–Maxwell theory with two free axions [18]. From the above action (2), the Einstein equation, Klein–Golden equation as well as Maxwell equation are given by

*k*is a constant, and \(x^i\) denotes the boundary spatial coordinates. In our convention, the AdS boundary is located at \(u=0\). The Horndeski axion fields provide the momentum relaxation in the dual boundary field theories. The Einstein and Maxwell equations then turn out to be

*q*are integration constants which should be identified as the chemical potential and the charge density of the dual field theory, respectively. On the horizon, the regularity of the gauge field \(A_\mu \) requires that these two quantities should obey the following relation

## 3 Electric conductivity

*T*is shown in Fig. 1. As seen in Fig. 1, when the Horndeski coupling is vanishing, \(\gamma =0\), the conductivity declines monotonically from a non-zero initial value as the temperature increases, behaving like a metal. When \(\gamma \) is at the critical value 1 / 3, the conductivity increases monotonically from non-zero initial value as the temperature increases behaving like a semiconductor. When \(\gamma \) is below the critical value, the conductivity firstly increases and then declines as the temperature goes up. Note that for most of semiconductors, their conductivities obeys the Steinhart–Hart equation [49]

*A*,

*B*, and

*C*are so-called Steinhart–Hart coefficients to be determined. We can find that the DC conductivity of our model fits this equation quite well by subtracting the universal unit part which comes from the absence of Galilean invariance in critical systems. Thus in order to analyze the problem qualitatively and fit the conductivity curve, the zero temperature conductivity contribution should be ignored. We draw the figure of conductivity with temperature by using formula (19) and show the comparison with the conductivity curve of the critical coupling \(\gamma =1/3\) in Fig. 2. According to this figure, the behaviors of these two conductivities are in qualitative agreement. In other words, when \(\gamma \) is at the critical value, the conductivity increases monotonically with the temperature increasing, behaving like a semiconductor. Then, it seems that there exists a semiconductor-metal phase transition driven by the Horndeski coupling term at some certain critical temperature \(T_c\) when \(\gamma \) is below the critical value. On top of this, we will further explore how the AC conductivity behaves in both phases.

^{1}This in general, happens in many semiconducting systems whose DC conductivity obeys \(d\sigma _{DC}/dT>0\) but still satisfies the Drude formula. For more details, one refers to [53, 54].

*T*is shown in Fig. 5. The real parts of the AC conductivity are also shown in Fig. 5. The behaviors of AC conductivity imply this holographic model could provide a mechanism of semiconductor-metal transition driven by the Horndeski coupling term.

*k*is small. In this case \(\tau _{rel.}\) can be calculated in a hydrodynamical approach. The inverse of the relaxation time is also called the relaxation rate. And, it takes the following form [20]

*s*is given by [42]

*T*limit, \(\tau _{rel.}^{-1}\) becomes

The plots of the AC conductivity fitted by Drude formula are also shown in Fig. 7. From this figure, it can be seen the numerical AC conductivity results can be fitted well by Drude formula.

## 4 Conclusion

In this paper, we study the charge transport in a simple holographic Horndeski model with momentum relaxation. Firstly, we go over the computation of the DC conductivity and show its relation with the temperature. We find that when the Horndeski coupling is zero the DC conductivity declines monotonically with the increase of the temperature, behaving like a metal. While, when \(\gamma \) reaches a critical value 1 / 3, the conductivity always increases monotonically as the increase of the temperature, which approximately obeys the Steinhart-Hart equation for a semiconductor. When \(\gamma \) is below the critical value, there always exists a transition point from the normal metal to the semiconductor like phase. Then, this holographic model may provide a metal-semiconductor transition that is driven by the Horndeski coupling.

To deepen our understanding on this transition, we further compute the AC conductivity by numerical method and obtain the dependance of the real part of AC conductivity on the frequency. We choose a special coupling to investigate the AC conductivity at both sides of the transition point. However, we do not find an off-axis peak in the low frequency region which usually happens in metal-insulator transitions. Moreover, we also find that our numeric data of the AC conductivity fits well with the Drude formula for a semiconductor in slow relaxation cases.

## Footnotes

- 1.
Such peaks can be realized by considering some higher derivative terms which make the UV behavior of the axions different from the canonical case [51, 52]. However, in our model, since \(G^{\mu \nu }\propto g^{\mu \nu }\) in the UV, the Horndeski term does not change the UV expansion of the axions at all.

## Notes

### Acknowledgements

We would like to thank M. Baggioli and Y. Liu for helpful discussions! X. J. Wang and W. J. Li are supported by NFSC Grant No. 11905024 and DUT19LK20. H. S. Liu is supported in part by NSFC Grants No. 11475148 and No. 11675144. X. J. Wang also would like to thank Y. Ling and M. H. Wu for the stimulating discussions and warm hospitality during the completion of part of this work.

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