Advances in High Energy Physics

Volume 2019, Article ID 5472310, 19 pages

https://doi.org/10.1155/2019/5472310

## EM Duality and Quasinormal Modes from Higher Derivatives with Homogeneous Disorder

Correspondence should be addressed to Jian-Pin Wu; nc.ude.unb.liam@uwnipnaij

Received 14 April 2019; Revised 24 June 2019; Accepted 26 June 2019; Published 21 August 2019

Academic Editor: Sally Seidel

Copyright © 2019 Guoyang Fu and Jian-Pin Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We study the electromagnetic (EM) duality from derivative theory with homogeneous disorder. We find that, with the change of the sign of the coupling parameter of the derivative theory, the particle-vortex duality with homogeneous disorder holds better than that without homogeneous disorder. The properties of quasinormal modes (QNMs) of this system are also explored. When the homogeneous disorder is introduced, some modes emerge at the imaginary frequency axis for negative but not for positive . In particular, with an increase in the magnitude of , new branch cuts emerge for positive . These emerging modes violate the duality related to the change of the sign of . With the increase of , this duality is getting violated more.

#### 1. Introduction

The electrical conductivity is one of the important features of real materials. For the weakly coupled systems, the optical conductivity exhibits Drude-like peak at low frequency, which can be described by the quantum Boltzmann theory [1]. For the strongly coupled systems, due to the absence of the picture of the quasiparticle, the Boltzmann theory is usually invalid. The quantum critical (QC) dynamics described by CFT [2] is just such strongly coupling systems. A novel mechanism to deal with this kind of problem is the anti-de Sitter/conformal field theory (AdS/CFT) correspondence [3–6]. In holographic framework, remarkable progress has been made [7–18].

Due to the electromagnetic (EM) self-duality, the optical conductivity in the neutral plasma dual to the Schwarzschild-AdS (SS-AdS) black brane is trivial [19]. To break the EM self-duality, an extra four-derivative interaction, the Weyl tensor coupled to Maxwell field over the Schwarzschild-AdS (SS-AdS) black brane is introduced. Since the EM self-duality is broke now, the electrical conductivity is frequency dependent [7]. The low frequency conductivity exhibits a Drude-like peak for the coupling parameter , which is similar to the particle excitation [7]. While for , a dip in low frequency conductivity is observed, which resembles the vortex case [7]. It is a possible way to address the CFT of the superfluid-insulator quantum critical point (QCP) though there is still a degree of freedom of the sign of . In addition, the particle-vortex duality related by the change of the sign of in the dual field theory is also found.

We can also introduce higher derivative (HD) terms. When higher derivative (HD) terms are introduced, an arbitrarily sharp DS peak can be observed at low frequency in the optical conductivity depending on the coupling parameters, and the bound of conductivity in derivative theory is violated such that a zero DC conductivity can be obtained at specific parameter [14]. In particular, its behavior is quite closely analogous to that of the superfluid-insulator model in the limit of large- [1]. Another progress is the construction of neutral scalar hair black brane by coupling Weyl tensor with neutral scalar field, which provides a framework to describe QC dynamics and one away from QCP [17, 18, 20, 21].

Also, we can introduce a mechanism of microscopic translational symmetry broken, for example [22–32], which leads to the momentum dissipation of the dual field theory and is also referred as homogeneous disorder, into the holographic QC system [7–18], such that we can study the effects from homogeneous disorder on these systems [33–36]. We find that for derivative theory studied in [33], the homogeneous disorder drives the incoherent metallic state with DS peak into the one with a dip for and an opposite scenario is found for . But for the derivative theory explored in [34], the homogeneous disorder cannot make the peak (gap) transform into its contrary. Another interesting phenomena is that for derivative theory there is a specific value of , for which the particle-vortex duality, corresponding to EM duality in bulk, almost exactly holds with the change of the sign of [33].

On the other hand, to study the holographic QC dynamics, a good method we can use is the quasinormal modes (QNMs) of a gravitational theory on the bulk AdS spacetime. Recently, the structure of the QNMs of derivative theory over SS-AdS has been explored [11, 14]. In particular, the particle-vortex duality has also been discussed. Further, when the homogeneous disorder in derivative theory is introduced, more rich pole structures are exhibited [36]. In this paper, we shall study the EM duality and the QNMs of derivative theory with homogeneous disorder. Our paper is organized as follows. In Section 2, we present a brief introduction on the derivative theory with homogeneous disorder. EM duality is discussed in Section 3. In Section 4, we study the properties of the QNMs. A brief discussion is presented in Section 5. In Appendix A, we also give a brief discussion on the instabilities of gauge mode by the QNMs. Appendixes B and C present the constraint on another derivative term and its properties of conductivity and QNMs, respectively.

#### 2. Holographic Setup

We consider the following neutral Einstein-axions (EA) theory [32]: In the above action, we have introduced a pair of spatial linear dependent axionic fields, with and being a constant, which are responsible for dissipating the momentum of the dual boundary field. is the radius of the AdS spacetimes.

From the EA action (1), we have a neutral black brane solution [32] where The AdS boundary is at and the horizon locates at . with the Hawking temperature . Since the translational symmetry breaks, the momentum dissipates. But the geometry is homogeneous and so we refer to this mechanism as homogeneous disorder and denotes the strength of disorder. Therefore, the background describes a specific thermal excited state with homogeneous disorder.

Over this background, we study the following HD action for gauge field coupling with Weyl tensor where the tensor is an infinite family of HD terms [14] Note that is an identity matrix and . In the above equations, the factor of is introduced such that the coupling parameters and are dimensionless. But for later convenience, we work in units where in what follows and set in the numerical calculation. When , the theory reduces to the standard Maxwell theory. For convenience, we denote and . In the main body of this paper, our main focus is the derivative terms (the main effect of both and terms is similar. But there is still some difference between them. We shall present a brief discussion on the properties of the conductivity and the QNMs from higher derivative term in Appendixes B and C.). Note that when other parameters are turned off, is confined in the region over SS-AdS black brane background (In [34], they also study the instabilities of the gauge mode and the causality in CFT; in addition to the constraint from , they confirm that the constraint also holds over EA-AdS geometry.) [14]. And then, from the action (4), we can write down the equation of motion (EOM) as

As [7], the dual EM theory of (4) can be constructed, which is is the dual field strength and is the coupling constant, which relates as . is the dual tensor, which satisfies And then, the EOM of the dual theory (7) can be wrote down as

For the four dimensional standard Maxwell theory, . At this moment, both theories (4) and (7) are identical and so the standard Maxwell theory is self-dual. Once the HD terms are introduced, such self-duality is broken. However, if is small, we have It indicates that, for small , there is an approximate duality between both theories (4) and (7) with the change of the sign of .

#### 3. Conductivity and EM Duality

Turning on the perturbation , we can write down the EOM of gauge field [7, 33, 34] where , , are the components of defined as , with Due to the isotropy of background, and . In addition, we have defined the dimensionless frequency in the above equation , with . Letting , we can obtain the dual EOM. Solving the EOM (12) or the dual EOM with ingoing boundary condition at the horizon, we can read off the conductivity in terms of

It has been illustrated in the last section that, for very small , there is an approximate duality between both the original EM theory and its dual theory with the change of the sign of as that for derivative theory [7, 33]. At the same time, there is an inverse relation for the conductivity of the original theory and its dual theory (this relation has been proved in [7, 11]. Also it has been derived for a specific class of CFTs in [19]) [7, 11]: where is the conductivity of the dual theory. Therefore, for small , we conclude that [33] It indicates that the dual optical conductivity is approximately equal to its original one with the change of the sign of . For derivative theory, it has been studied and found that the conductivity of the dual EM theory is not precisely equal to that of its original theory with the change of the sign of [7]. But when the homogeneous disorder is introduced, we find that this relation holds better for a specific value of than other values of . Here, we shall study the EM duality from derivative theory.

Figure 1 exhibits the optical conductivity as the function of for and various values of . We find that for small (), with the change of the sign of , the relation (15) holds very well. With the increase of , the relation (15) violates. In particular for , the real part of the conductivity at low frequency is a pseudogap-like behavior. Correspondingly, its dual conductivity at low frequency becomes sharper, which is Drude-like. But for , the real parts of the conductivity of the original theory and its dual one at low frequency are only peak and dip, respectively. At this moment, the relation (15) is strongly violated.