Transport phenomena and Weyl correction in effective holographic theory of momentum dissipation
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Abstract
We construct a higher derivative theory involving an axionic field and the Weyl tensor in four dimensional spacetime. Up to the first order of the coupling parameters, the charged black brane solution with momentum dissipation in a perturbative manner is constructed. Metal–insulator transitions are implemented when varying the system parameters at zero temperature. Also, we study the transports including DC conductivity and optical conductivity at zero charge density. We observe the exact particle–vortex duality for some specific momentum dissipation strength.
1 Introduction
The quantum critical (QC) system has long been a central and challenging subject in condensed matter physics [1]. It is believed to account for the most interesting phenomena, such as the strange metal and pseudogap phase, in strongly correlated quantum materials. The QC system is associated with a QC phase transition and a QC phase. Since the QC system is strongly correlated, the conventional perturbative tools in traditional field theory, unfortunately, lose their power. We need to develop novel nonperturbative techniques and methods.
The AdS/CFT correspondence [2, 3, 4, 5], mapping a strongly coupled quantum field theory to a weakly coupled gravitational theory in the large N limit, provides a powerful tool to the study of QC physics and has led to great progress. Especially, the metal–insulator transition (MIT), a special example of the QC phase transition, has been widely studied in the holographic framework; for instance see [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] and the references therein. To implement an MIT in a holographic framework, the key point is to deform the infrared (IR) geometry to a new fixed point by the introduction of momentum dissipation [6, 7].
Holographic QC phase at zero density has also been intensely explored in [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. By studying transport phenomena, in particular the optical conductivity, from a probe Maxwell field coupled to the Weyl tensor \(C_{\mu \nu \rho \sigma }\) on top of the Schwarzschild–AdS (SS–AdS) black brane background [20, 21, 22, 23, 24, 25, 26, 27, 28], one observed a nontrivial frequency dependent conductivity attributed to the introduction of the Weyl tensor. It exhibits a peak, which resembles the particle response and we refer to this as the Damle–Sachdev (DS) peak [32], or a dip, which is similar to the behavior of the vortex response, and is analogous to the one in the superfluid–insulator quantum critical point (QCP)^{1} [20, 21, 22].
But the peak is not the standard Drude peak and the DC conductivity has a bound which cannot approach zero. When higher derivative (HD) terms are introduced, an arbitrarily sharp Drudelike peak can be observed at low frequency in the optical conductivity and the bound of conductivity is violated such that a zero DC conductivity can be obtained at a specific parameter^{2} [27]. Another step forward is the construction of a neutral scalar hair black brane by coupling the Weyl tensor with a neutral scalar field, which provides a framework to describe the QC phase and a transition away from QCP [30, 31].
In this paper, we shall construct a higher derivative theory including the four derivative terms, a simple summation of the Weyl tensor as well as a term from the trace of axions coupling with the gauge field, and a six derivative term, a mixed term of the product of the Weyl tensor and the axionic field coupling with the gauge field, and we obtain a charged black brane solution in a perturbative manner. By using a perturbative method, some charged black brane solutions from higher derivative gravity theory have been constructed; for instance see [14, 38, 39, 40, 41, 42] and the references therein. Especially, in [14], it is the first time that an MIT is realized in the framework of higher derivative gravity. Along the line of [14], we shall study the MIT physics of our present model. Also, we explore the QC phase of this model at zero charge density.
We organize this paper as follows. In Sect. 2, we construct the higher derivative model coupling axionic field and Weyl tensor with the gauge field. Then the perturbative black brane solution is obtained in Sect. 3. In Sect. 4, we calculate the DC conductivity at finite charge density and study the MIT at zero temperature. The conductivity at zero charge density is explored in Sect. 5. A brief discussion is presented in Sect. 6. The constraint on the coupling parameters is obtained in Appendix A.
2 Holographic model
When \(X_{\mu \nu }^{\ \ \rho \sigma }=I_{\mu \nu }^{\ \ \rho \sigma }\), the modified Maxwell theory (1b) is reduced the standard Maxwell one. In this case, one can easily deduce that \(X^{1}=X\) and so \(\widehat{X}_{\mu \nu }^{\ \ \rho \sigma }=I_{\mu \nu }^{\ \ \rho \sigma }\) from Eqs. (7) and (6). Hence, the actions (1b) and (5) are identical, which demonstrates that the standard Maxwell theory is selfdual.
3 Black brane solution
4 DC conductivity at finite density
4.1 The derivation of the DC conductivity

A metal–insulator transition (MIT) happens at zero temperature for a given nonzero \(\gamma \) when we change the axionic charge \(\bar{\alpha }\).
 There is a mirror symmetry at zero temperature^{4}$$\begin{aligned} \frac{\partial \sigma _0}{\partial \bar{T}}(\gamma ,\bar{\alpha })= \frac{\partial \sigma _0}{\partial \bar{T}}(\gamma ,\bar{\alpha }). \end{aligned}$$(27)

Metallic phase: \(\partial _T\sigma _0<0\).

Insulating phase: \(\partial _T\sigma _0>0\).

Critical point (line): \(\partial _T\sigma _0=0\).
4.2 DC conductivity without Weyl term
In Appendix A, we analyze the causality and instabilities of the vector modes at zero density. When we only consider the \(\gamma _{1,0}\) term, the analysis and the requirement of the positive DC conductivity indicate \(3/40\le \gamma _{1,0}\le 1/40\). But it is hard to analyze the causality and instabilities of the vector modes at finite density even if we have an analytical perturbative black brane solution. We shall leave this problem for future study. Here, we only approximately impose a further constraint from the requirement of the positive DC conductivity at finite density.

At zero temperature, the DC conductivity monotonously decreases in terms of \(\bar{\alpha }\).

At finite temperature, the DC conductivity is qualitatively similar to that at zero temperature when \(\gamma _{1,0}>0\). Meanwhile for \(3/40\le \gamma _{1,0}<0\), the DC conductivity no longer monotonously decreases but has a minimum at some finite value of \(\bar{\alpha }\).

When \(\bar{\alpha }\) is fixed, the DC conductivity monotonously decreases in terms of \(\bar{T}\) for \(\gamma _{1,0}>0\), which demonstrates a metal phase. When the sign of \(\gamma _{1,0}\) changes, an opposite behavior is found, which is an insulator phase.

Different from that for the four derivative Weyl term, no MIT happens for a given nonzero \(\gamma _{1,0}\) when changing \(\bar{\alpha }\) (see Fig. 3). But the mirror symmetry on \(\frac{\partial \sigma _0}{\partial \bar{T}}(\bar{\alpha })\) (27) at zero temperature holds when the sign of \(\gamma _{1,0}\) changes.

Equation (26) holds when the sign of \(\gamma _{1,0}\) changes and \(\bar{\alpha }\) is fixed.
4.3 DC conductivity from four derivative theory
When only the four derivative Weyl term \(\gamma \) is involved, an MIT occurs at zero temperature by varying the axionic charge \(\bar{\alpha }\). In particular, the quantum critical line is independent of the coupling parameter \(\gamma \) [14].

Equations (26) and (27) hold for fixed \(\bar{\alpha }\) and changing the signs of \(\gamma \) and \(\gamma _{1,0}\) (Fig. 4 and left plot in Fig. 5).

For positive (negative) small \(\gamma _{1,0}\), an MIT can be observed for negative (positive) \(\gamma \) (see right plot in Fig. 5). But different from the case only involving the four derivative term in [14], the quantum critical line is dependent on \(\gamma \) (Fig. 6). It provides a new platform of QCP such that we can study the holographic entanglement entropy and the butterfly effect close to QCP as in [13, 14, 51]. We shall explore them in our present model in the future.
Before proceeding, we present some comments on the phase diagram for the MIT from four derivative theory at zero temperature (Fig. 6). For \(\gamma _{1,0}<0\) and \(\gamma >0\), with the increase of the strength of momentum dissipation, there is a phase transition from metallic phase to insulating one. This phenomenon is consistent with that of the usual charged particle excitations. On the other hand, for \(\gamma _{1,0}>0\) and \(\gamma <0\), we find that with the increase of the strength of momentum dissipation, the phase transition is opposite, i.e., the stronger momentum dissipates, the more insulating is the material. A better description of this phenomenon is provided by considering the excitations of vortices. Just as described [20], the EM duality of the bulk theory, which is related by changing the sign of \(\gamma \), corresponds to the particle–vortex duality in the dual holographic CFT. Figure 6 shows such a duality; when we change the sign of \(\gamma \), there is a duality between metallic and insulating phase. In fact, the phenomena can be easily concluded from Eq. (26). Finally, we would like to mention two corresponding examples. One is the transition observed in [20] from the Drudelike peak at low frequency optical conductivity, which is interpreted as the charged particle excitations, to the dip, which resembles the excitations of vortices. Another one is the observation in [43] that the momentum dissipation drives the Drudelike peak into the dip of the low frequency optical conductivity for \(\gamma >0\). Meanwhile for \(\gamma <0\), the opposite scenario appears. When the sign of \(\gamma \) changes, an approximate duality in optical conductivity is also observed for fixed strength of momentum dissipation. This duality is also observed in the next section.
4.4 DC conductivity from six derivative theory
5 Transports at zero density
5.1 Four derivative theory
As revealed in [43], particle–vortex duality is recovered with the change of \(\gamma \rightarrow \gamma \) for a specific value of \(\hat{\alpha }=2/\sqrt{3}\). Now we want to explore if this phenomenon is generic when a new higher derivative coupling term \(\gamma _{1,0}\) is taken into account. Figure 10 shows the DC conductivity \(\sigma _0\) as a function of \(\hat{\alpha }\) for the representative \(\gamma \) and \(\gamma _{1,0}\). We find that, for a given \(\gamma _{1,0}\), all the lines of \(\sigma _0(\hat{\alpha })\) with different \(\gamma \) intersect at one point \(\hat{\alpha }=2/\sqrt{3}\), which is similar to that found for only the Weyl term \(\gamma \) being involved. It indicates that \(\sigma _0(\hat{\alpha })\) is independent of \(\gamma \) for \(\hat{\alpha }=2/\sqrt{3}\), which can also be deduced from the expression for DC conductivity (A16). But we note that the value of \(\sigma _0(\hat{\alpha }=2/\sqrt{3},\gamma )\) is not equal to unity. Also, the relation \(\sigma _0(\hat{\alpha }=2/\sqrt{3},\gamma )=\frac{1}{\sigma _0(\hat{\alpha }=2/\sqrt{3},\gamma )}\) does not hold. It indicates the exact duality of the DC conductivity only with the Weyl term for \(\hat{\alpha }=2/\sqrt{3}\) is violated when the \(\gamma _{1,0}\) term is taken into account. Furthermore, we study the optical conductivities of both the original EM theory and its dual theory for the specific value of \(\hat{\alpha }=2/\sqrt{3}\), shown in Fig. 11, and we find that the exact particle–vortex duality is indeed violated when \(\gamma \rightarrow \gamma \) and \(\gamma _{1,0}\rightarrow \gamma _{1,0}\). It is easy to check that if we fix \(\gamma _{1,0}\), the particle–vortex duality is also violated when \(\gamma \rightarrow \gamma \).
5.2 Six derivative theory
Also, we note that, for the specific value of \(\hat{\alpha }=2/\sqrt{3}\), the DC conductivity \(\sigma _0=1\) and is independent of \(\gamma _{1,1}\) (see Fig. 23), which is similar to that with only the Weyl term [43]. Furthermore, we study the particle–vortex duality of this case, shown in Fig. 13. It is obvious that, for small \(\gamma _{1,1}\), the particle–vortex duality approximately holds. Meanwhile, for the specific value of \(\hat{\alpha }=2/\sqrt{3}\), the duality exactly holds. Though here we do not work out the analytical understanding on the particle–vortex duality for the specific value of \(\hat{\alpha }=2/\sqrt{3}\), it seems to originate from the Weyl term. The additional \(\gamma _{1,0}\) term violates this exact duality. Further, we examine the duality from another six derivative term with \(X_{\mu \nu }^{\ \ \rho \sigma }=4\gamma _1C^2I_{\mu \nu }^{\ \ \rho \sigma }\), of which the original theory has been studied in our previous work [44]. Again, the particle–vortex duality exactly holds for \(\hat{\alpha }=2/\sqrt{3}\) when \(\gamma _1\rightarrow \gamma _1\) (see Fig. 14). In future, we will further test the robustness of this phenomenon by exploring that with the higher order terms of the Weyl coupling.
6 Discussions
In this work, we extend our previous work [14, 43] to constructing a higher derivative theory including the coupling among the axionic field, the Weyl tensor and the gauge field. To be more specific, we construct four derivative terms, a simple summation of the Weyl term \(C_{\mu \nu \rho \sigma }\) coupling with the gauge field, as well as a term from the trace of axions coupling with the gauge field, and a six derivative term, a mixed term by the product of Weyl tensor and the axionic field, coupling with the gauge field.
Following the strategy in [14], we construct the charged black brane solution with momentum dissipation in a perturbative manner up to the first order of the coupling parameters. We study the QCP from 4 and six derivative theory, respectively. For four derivative theory, because of the introduction of \(\gamma _{1,0}\), the quantum critical line is independent of \(\gamma \), which is different from the case only involving the 4 derivative term in [43]. It provides a new platform of QCP such that we can study holographic entanglement entropy and the butterfly effect close to QCP, which may inspire new insight. For six derivative theory, the quantum critical line is independent of the coupling parameter \(\gamma _{1,1}\), which is similar that in [14].
Also, we study the transport phenomena including DC conductivity and optical conductivity at zero charge density, which is away from the QC phase. For four derivative theory, the momentum dissipation makes the transition from peak (dip) to dip (peak) easier, comparing with that in our previous work [43]. In addition, we find that for the specific value of \(\hat{\alpha }=2/\sqrt{3}\), the exact particle–vortex duality, holding for only the \(\gamma \) term, survives [43] and is violated when the \(\gamma _{1,0}\) term is turned on. For the six derivative theory, particle–vortex duality exactly holds for \(\hat{\alpha }=2/\sqrt{3}\). Meanwhile the effect of the momentum dissipation on the transition between the gap and the dip is similar to that in four derivative theory.
It is definitely a novelty and an interesting matter to compute the optical conductivity at finite chemical potential \(\mu \). However, even if we have obtained the perturbative black brane solution to the first order of \(\gamma \) in Sect. 3, we still need to solve the linear perturbative differential equations beyond the second order to obtain the optical conductivity. It is a hard task and so we shall leave it for the future. In addition, this simple model including the mixed terms between the Weyl tensor and the axions can be straightforwardly generalized to include the charge complex scalar field such that we can study the superconducting phase. It is also interesting and valuable to further explore the transport of our present model at full momentum and energy spaces, which certainly will reveal more information of the systems. This work deserves further study and we plan to publish our results in the near future.
Footnotes
Notes
Acknowledgements
We in particular thank Zhenhua Zhou for the very helpful discussion on the calculation of the DC conductivity at finite density. We are grateful to Peng Liu and WeiJia Li for helpful discussions. We also thank the anonymous referee for his/her very valuable suggestions, greatly improving our manuscript. This work is supported by the Natural Science Foundation of China under Grant nos. 11775036, 11305018 and the Natural Science Foundation of Liaoning Province under Grant no. 201602013.
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