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Advances in High Energy Physics
Volume 2019, Article ID 5472310, 19 pages
https://doi.org/10.1155/2019/5472310
Research Article

EM Duality and Quasinormal Modes from Higher Derivatives with Homogeneous Disorder

1Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China
2School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China

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 SCOAP3.

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 [36]. In holographic framework, remarkable progress has been made [718].

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 [2232], which leads to the momentum dissipation of the dual field theory and is also referred as homogeneous disorder, into the holographic QC system [718], such that we can study the effects from homogeneous disorder on these systems [3336]. 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.

Figure 1: The real part (the plot above) and the imaginary part (the plot below) of the optical conductivity as the function of for and various values of . The solid curves are the conductivity of the original EM theory (4) and the dashed curves display the conductivity of the EM dual theory (7).

Next, we explore the EM duality for derivative theory when the homogeneous disorder is introduced. At the first sight, we find that for the specific value of , the optical conductivity of the original theory is almost (approximate but not exact) the same as that of the dual one when the sign of changes, i.e., the relation (15) holds very well. It is similar to that from derivative theory [33]. For other values of , the relation (15) also approximately holds. It seems that the homogeneous disorder makes the pseudogap-like behavior in the low frequency conductivity of the original theory become small dip, while suppressing the sharp peak of the EM dual theory.

It is hard to obtain an analytical understanding on this phenomenon. But we can follow the method of [33] and attempt to taste kind of analytical similarity between the EOM (B.8) and its dual one, which can be explicitly wrote down: Since for any , the difference between (B.8) and (16) is in the coefficients of , which are And then, we explicitly plot the ration as the function of with different in Figure 3. For , the value of is maximum departure from near the horizon, which indicates a most deviation between (B.8) and (16). Once the homogeneous disorder is introduced, this deviation becomes small. Specially, for , this value approaches to near the horizon, which implies atmost similarity between (B.8) and (16). On the other hand, it is well known that the low frequency behavior of the conductivity is mainly controlled by the near horizon geometry. Therefore, to some extent, the above comparison in the original EOM and its dual one helps us understand the phenomenon shown in Figure 2.

Figure 2: The real part (the plot above) and the imaginary part (the plot below) of the optical conductivity as the function of for and various values of . The solid curves are the conductivity of the original EM theory (4) and the dashed curves display the conductivity of the EM dual theory (7).
Figure 3: as the function of with different . Here we set .

4. The Properties of QNMs

By definition, QNMs are the poles of the Green’s function and are directly related to the conductivity. Summing an increasing number of QNMs and the corresponding residues, one should obtain a better and better approximation of the conductivity itself. In this section, we shall study the properties of QNMs of the transverse gauge mode along direction, i.e., , from derivative term with homogeneous disorder. In particular, we shall also study the particle-vortex duality in the complex frequency panel in the holographic CFT.

It is highly efficient to use the pseudospectral methods outlined in [37] to solve the QNM equations and obtain the QNMs. To this end, we shall work in the advanced Eddington-Finkelstein coordinate, which is Correspondingly, the EOM (B.8) can be changed as For the dual theory, the corresponding EOM can be obtained by letting and . Imposing the ingoing boundary at the horizon, we numerically solve (20) to obtain QNMs.

We first study the case without homogeneous disorder. Figures 4 and 5 show QNMs of the gauge mode with for and , respectively. The left panels are the QNMs of the gauge mode and the right ones are that of the dual gauge mode. We find that for small , for example, in Figure 4, the qualitative correspondence between the poles of and the ones of holds well at low frequency. But for the large frequency region, there are some modes emerging at the imaginary frequency axis for negative , which results in the violation of the particle-vortex duality with the change of the sign of . But with the increase of , this correspondence begins to violate in complex frequency panel even in the low frequency region (see Figure 5 for ), as that in real frequency axis. We also plot the QNMs for in Figure 6. We see that another branch cuts, which are closer to the imaginary frequency axis, emerge in the complex frequency panel.

Figure 4: QNMs (blue spots) of the gauge mode for (the panels above are for and the ones below for ) and . The left panels are the QNMs of the gauge mode and the right ones are that of the dual gauge mode.
Figure 5: QNMs (blue spots) of the gauge mode for (the panels above are for and the ones below for ) and . The left panels are the QNMs of the gauge mode and the right ones are that of the dual gauge mode.
Figure 6: QNMs (blue spots) of the gauge mode for and . The left panels are the QNMs of the gauge mode and the right ones are that of the dual gauge mode.

Now we turn to the study of the effects of homogeneous disorder. Figures 7, 8, and 9 show the QNMs with representative for , , and (from left to right) in the complex frequency plane. The panels above are the QNMs of the gauge mode, while the ones below are that of the dual gauge mode. Rich information and insights on the pole structures are exhibited. At the first sight, we find that all the poles are in the lower-half plane (LHP). It indicates that the gauge mode is stable. Further analysis on stability in terms of QNMs is presented in Appendix A.

Figure 7: QNMs (blue spots) of the gauge mode for . The first, second, and third columns are for , , and , respectively. The panels above are for the gauge mode, while the ones below are for the dual gauge mode.
Figure 8: QNMs (blue spots) of the gauge mode for . The first, second, and third columns are for , , and , respectively. The panels above are for the gauge mode, while the ones below are for the dual gauge mode.
Figure 9: QNMs (blue spots) of the gauge mode for . The first, second, and third columns are for , , and , respectively. The panels above are for the gauge mode, while the ones below are for the dual gauge mode.

Next, we shall study the main properties of the QNMs when the homogeneous disorder is introduced, in particular, the duality between the poles of and the ones of . They are briefly summarized as what follows for .(i)For , some modes emerge at the imaginary frequency axis for negative as that without homogeneous disorder but not for positive .(ii)For , there are still the modes locating in the imaginary frequency axis for negative but not for positive . In particular, new branch cuts emerge for positive .(iii)For , the pole structures for are similar with those for .(iv)These emerging modes violate the duality between the poles of and the ones of . With an increase in the magnitude of , this duality is getting violated more.

We also exhibit the QNMs for with homogeneous disorder (the third columns in Figures 7, 8, and 9). When the homogeneous disorder is introduced, the new branch cuts attach to the imaginary frequency axis. Such pole structures are interesting and are worthy of further study such that we can understand the physics of these pole structures.

The dominant poles reveal the late time dynamics of the system. So, we shall study the dominant pole behaviors such that we have a well understanding on the effect of the homogeneous disorder.

In our previous work [34], we have studied the conductivity at the low frequency in real axis for with homogeneous disorder. For small , it exhibits a sharp DS peak. Though it is not the Drude peak, we can phenomenologically describe this peak by the Drude formula (see Figure 5 in [34]) where is the relaxation rate, which relates the relaxation time as . This peak in the real axis corresponds to a purely imaginary mode in complex frequency panel (see the left column in Figure 10). We also locate the position of the purely imaginary mode, which is listed in Table 1. We find that the relaxation rates fitted by the Drude formula (21) are in agreement with that by locating the position of the purely imaginary mode.

Table 1: The relaxation rate of the holographic system with for different . is fitted by the Drude formula (21) and is obtained by locating the position of the purely imaginary mode in complex frequency panel.
Figure 10: Evolution of the dominant QNMs with for . The left column is the QNMs for the original theory, and the right one is that for its dual theory.

More detailed evolution of the dominant QNMs with for is presented in Figure 10. Most of the dominant QNMs for the original theory, except for that approximately in the region of , are the purely imaginary modes. Correspondingly, the dominant QNMs for the dual theory, except for that approximately in the region of , are off axis.

Further, we study the evolution of the dominant QNMs with for , which is shown in Figure 11. We describe the properties as what follows.(i)For , all the dominant QNMs for the original theory are purely imaginary modes. With the increase of , these modes firstly migrate downwards and then migrate upwards, finally approaching to a certain value. The evolution of the dominant QNMs for the dual theory with is similar with that for the original theory with . Such evolution is also similar with that for derivative theory studied in [36]. For small , the poles for the dual theory are closer to the real frequency axis than that for the original theory.(ii)For , the dominant poles for the original theory are off axis for small . With the increase of , the poles migrate downwards and are closer to the real axis. When reaches the value of , the poles merge into one purely imaginary pole. And then, when is beyond , the purely imaginary pole splits into two off-axis modes. While for , the evolution of the dominant poles for the dual theory is similar with that for the original theory with . But when , the pole for the dual theory with becomes a purely imaginary one.(iii)In summary, the correspondence between the evolution of the poles as the function of for the original theory for and that for its dual theory for is violated.

Figure 11: Evolution of the dominant QNMs with for (the first two rows are for and the last two rows for ). The left columns are the QNMs for original theory, and the right ones are that for its dual theory.

5. Conclusion and Discussion

In this paper, we extend our previous work [34], which studied the optical conductivity from derivative theory on top of EA-AdS geometry, to the holographic response of its EM dual theory. In particular, we explore thoroughly the EM duality. Also we study the QNMs and the EM duality in complex frequency panel.

In absence of the homogeneous disorder, with the change of the sign of , the particle-vortex duality only holds for small . With the increase of , this duality is violated. When the homogeneous disorder is introduced, as found in derivative theory, we find that for the specific value of , the optical conductivity of the original theory is almost the same as that of the dual one when the sign of changes. For other values of , the particle-vortex also approximately holds with the change of the sign of . Therefore, we can conclude that the homogeneous disorder makes the pseudogap-like of the low frequency conductivity of the original theory become small dip, while suppressing the sharp peak of the EM dual theory such that we have an approximate particle-vortex duality with the change of the sign of .

The properties of the QNMs are also analyzed. In absence of the homogeneous disorder, the qualitative correspondence between the poles of and the ones of holds well at low frequency only for small . When becomes large, this correspondence is also violated even at the low frequency region. For , new branch cuts of QNMs are observed.

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 between the poles of and the ones of . With the increase of , this duality is getting violated more. In addition, for , the homogeneous disorder drives the off-axis new branch cuts for to the purely imaginary modes.

The evolution of the dominant QNMs with is also explored. We find that all the dominant QNMs for the original theory with and that for the dual theory with are purely imaginary modes. Their evolutions are also similar. However, for the evolution of the dominant QNMs for the original theory with and that for the dual theory with , the case is somewhat different. The main difference is that the modes are not again the purely imaginary ones except for some specific . In summary, the correspondence between the evolution of the poles as the function of for the original theory and that for its dual theory is violated.

There are lots of open questions worthy of further study.(i)We would like to carry out an analytical study on the complex frequency conductivity by matching method developed in [38]. Also we can analytically work out the QNMs by WKB method as [11, 14]. These analytical analyses can surely provide more physical insight and understanding into our present observation.(ii)At present, the perturbative black brane solution to the first order of the coupling parameter from derivative theory has been worked out in [35, 3945] as well as the related exploration, including holographic metal-insulator transition, holographic entanglement, and holographic thermalization, at finite charge density on top of the perturbative background with HD corrections. In future, it is also interesting to further study the QNMs, including scalar, vector, and tensor modes, from HD theory at finite charge density (the QNMs of massless scalar field over the perturbative black hole with spherical symmetry horizon have been studied in [41]).(iii)The dispersing QNMs with finite momentum deserve further studying. It surely reveals richer physics of the system.(iv)The superconducting phase from HD theory has been widely explored in [4655] and references therein. It is also interesting to study the QNMs in the superconducting phase of these models such that we can get richer insight and understanding on the HD theory.(v)It would be interesting to simultaneously investigate the effects of both and derivative terms on the conductivity and the QNMs. In this way, the main effect of higher derivative terms on EM duality could be revealed. We shall continue to explore this project in future.

Appendix

A. Brief Analysis on the Stability from Derivative Term

In [34], by analyzing the instabilities of gauge mode and the causality in CFT, in addition to the constraint from , we confirm that the constraint , which has been obtained over SS-AdS geometry in [14], also holds over EA-AdS geometry. In this section, we further examine this constraint by studying the QNMs. Indeed, we find that when , all the QNMs are in the LHP.

First, it has been shown in Figures 10 and 11 that for , , and , all the poles and zeros, which are the poles of the dual theory, are in the LHP. In addition, we also see that when , the imaginary modes approach to a constant or even continue to migrate downwards. Therefore, we can conclude that for the selected , the modes are stable for all .

And then, for representing , we show the poles and zeros for larger region of . As that on top of SS-AdS geometry in [14], the dominant poles from EA-AdS geometry are purely imaginary modes and also asymptotically approach the real -axis as . In addition, the data can be well fitted by a power-law formula as . The coefficients and are listed in Table 2. We can see that at least over the decades we have studied, the fit is very well as shown in Figure 12. We would like to point out that this result is consistent with the peak of the conductivity at low frequency becoming sharper with the decrease of and finally approaching a delta function as . For the zeros, we also see that they all locate at the LHP at least over the decades we have studied here (see right plots in Figure 12). Therefore, by the analysis of QNMs, we again confirm that the gauge mode is stable over EA-AdS geometry when the coupling parameter is confined to .

Table 2: The coefficients of the power-law fit of the form for the dominant poles for different .
Figure 12: Left plot: location of the dominate QNM for different . Red dots are the numerical data and the blue line is the power-law fit of the form . Right plot: location of the next QNM closet to the real axis, a zero, for different .

B. Constraint on the Coupling Parameter

In this section, we derive the constraint on the coupling parameter over the EA-AdS background (1) by examining the bound from the positive definiteness of the DC conductivity, the instabilities and the causality of gauge mode, and also the QNMs.

B.1. Bound from DC Conductivity

The DC conductivity for our present model can be expressed as It can be explicitly worked out as up to derivatives with Here we separately consider the term. So we set and . In the limit of , the DC conductivity reduces to Immediately, from the positive definiteness of the DC conductivity, we obtain the bound of as in the SS-AdS background.

Further, we examine if this bound of is violated when the background is EA-AdS. To this end, we plot the DC conductivity as the function of for some fixed (left plot) and as the function of for some fixed (right plot). From Figure 13, we clearly see that, for the positive , when , DC conductivity increases with the increase of . Once is beyond the value of , DC conductivity decreases with the increase of ; but it tends to certain positive constant value when . When is negative, DC conductivity is larger than or equal to unity for all . In summary, the constraint of also holds in the EA-AdS background.

Figure 13: Left plot: the DC conductivity versus for some fixed . Right plot: the DC conductivity versus for some fixed . Here we have set other coupling parameters vanishing.
B.2. Bound from Anomalies of Causality

In this subsection, we examine the bound from the instabilities and the causality of the perturbative gauge modes. To this end, we turn on the perturbations of the gauge field as , where . Without loss of generality, we choose and fix the gauge as . And then, we have a group of perturbative equations as [33]: where . Equations (B.5) and (B.6) result in a decoupled equation for as Letting and in the above equations, we can obtain the equations for the dual EM theory.

Further, we recast (B.9) and (B.8) into the following Schrdinger form: The potential is where Note that we have made the change of variables and wrote with .

When the potentials satisfy the modes are stable and also satisfy the causality of the dual boundary theory [5658]. If violates the lower bound, the stability of the modes is determined by the zero energy bound state of the potential.

In the limit of the large momentum , is the dominant terms. Figures 14 and 15 give and as the function of for sample and , respectively. By careful analysis, we give the allowed range of parameter as .

Figure 14: The potentials for sample and , which constrain in the region of .
Figure 15: The potentials for sample and , which constrain in the region of .

In the limit of the small momentum (), becomes important. We examine the potentials and in the region of . In Figure 16, we exhibit for sample and . We find that the upper bound of (Eq. (B.15)) is satisfied. But we note that there is a negative minimum in . Therefore, we need to further examine the zero energy bound state of the potentials, which is [58] where is a positive integer and the interval defines the negative well. Both and as the function for sample are shown in Figure 17. The detailed analysis indicates that when , are smaller than unit and no unstable modes are developed. Therefore, the constraint on is

Figure 16: The potentials (plots above) and (plots below) for sample and .
Figure 17: Both and as the function for sample .
B.3. Examining QNMs

In this subsection, we examine the constraint (B.17) by studying the QNMs. Figure 18 exhibits the dominate QNMs of the gauge modes and the dual ones. We find that all the poles are in the LHP. In particular, when , the imaginary modes continue to migrate downwards. Therefore, the modes are stable for all in the region of (B.17). In Appendix C,we shall study the conductivity, QNMs, and the EM duality in the region of (B.17).

Figure 18: The dominate QNMs of the gauge modes (plots above) and the dual modes (plots below) for sample and .

C. The Properties of Conductivity and QNMs from Higher Derivative Term

In this section, we shall present a brief exploration on the properties of the conductivity and QNMs from the higher derivative term.

C.1. The Properties of Conductivity

Figure 19 exhibits the optical conductivity as the function of for and various values of . When is negative, the low frequency conductivity of the original theory is Drude-like. While for positive , it is a dip. For small , with the change of the sign of , the relation (15) holds very well (see the first column in Figure 19). With the increase of , the relation (15) gradually violates (see the second and third columns in Figure 20). This observation is similar with that from term. Further, we plot the optical conductivity for the case of the lower bound, i.e., (), which is shown in Figure 20. We can see that the Drude-like peak at low frequency becomes sharper. But in the allowed region of , we cannot see a pseudogap-like behavior at low frequency.

Figure 19: The real part (the plot above) and the imaginary part (the plot below) of the optical conductivity as the function of for and various values of . The solid curves are the conductivity of the original EM theory (4) and the dashed curves display the conductivity of the EM dual theory (7).
Figure 20: The real part and the imaginary part of the optical conductivity as the function of for and . The solid curves are the conductivity of the original EM theory (4) and the dashed curves display the conductivity of the EM dual theory (7).

When the homogenous disorder is introduced, the results are similar with that from higher derivative term, that is,(i)for , the optical conductivity of the original theory is almost the same as that of the dual one when the sign of changes (see the second columns in Figures 21 and 22);(ii)for other values of , the relation (15) also approximately holds (the first and the third columns in Figures 21 and 22).

Figure 21: The real part (the plot above) and the imaginary part (the plot below) of the optical conductivity as the function of for and various values of . The solid curves are the conductivity of the original EM theory (4) and the dashed curves display the conductivity of the EM dual theory (7).
Figure 22: The real part (the plot above) and the imaginary part (the plot below) of the optical conductivity as the function of for and various values of . The solid curves are the conductivity of the original EM theory (4) and the dashed curves display the conductivity of the EM dual theory (7).
C.2. The Properties of QNMs

First of all, we work out the QNMs from higher derivative term without homogeneous disorder (Figure 23). In the region of , the result is closely similar with that from term (see Figure 4), that is,(i)the qualitative correspondence between the poles of and the ones of holds well at low frequency;(ii)in the region of large frequency, there are some modes emerging at the imaginary frequency axis for negative , which violates the particle-vortex duality with the change of the sign of .

Figure 23: QNMs (blue spots) of the gauge mode for (the first two panels are for and the second two panels for ) and . The first and third panels are the QNMs of the gauge mode and the second and forth panels are that of the dual gauge mode.

When the homogeneous disorder is introduced, the main results are still similar with that from term. We summarize them as follows:(i)For small (, see Figure 24), the pole structure is qualitatively similar with that without homogeneous disorder (see Figure 23).(ii)With the increase of , the correspondence between the poles of and the ones of is seriously violated (see Figure 25).

Figure 24: QNMs (blue spots) of the gauge mode for (the first two panels are for and the second two panels for ) and . The first and third panels are the QNMs of the gauge mode and the second and forth panels are that of the dual gauge mode.
Figure 25: QNMs (blue spots) of the gauge mode for (the first two panels are for and the second two panels for ) and . The first and third panels are the QNMs of the gauge mode and the second and forth panels are that of the dual gauge mode.

Data Availability

This manuscript has no associated data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the Natural Science Foundation of China under grant nos. 11775036 and 11847313.

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