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

Authors: Guoyang Fu; Jian-Pin Wu
Journal: Advances in High Energy Physics
 (2019)

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## Abstract

We study the electromagnetic (EM) duality from6 derivative theory with homogeneous disorder. We find that, with the change of the sign of the coupling parameter γ1 of the 6 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 γ1 but not for positive γ1. In particular, with an increase in the magnitude of α^, new branch cuts emerge for positive γ1. These emerging modes violate the duality related to the change of the sign of γ1. With the increase of α^, this duality is getting violated more.


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## Body

## 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 Cμνρσ 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 γ>0, which is similar to the particle excitation [7]. While for γ<0, 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 γ. In addition, the particle-vortex duality related by the change of the sign of γ 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 in4 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 O(N) superfluid-insulator model in the limit of large-N [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 4 derivative theory studied in [33], the homogeneous disorder drives the incoherent metallic state with DS peak into the one with a dip for γ>0 and an opposite scenario is found for γ<0. But for the 6 derivative theory explored in [34], the homogeneous disorder cannot make the peak (gap) transform into its contrary. Another interesting phenomena is that for 4 derivative theory there is a specific value of α^=2/3, for which the particle-vortex duality, corresponding to EM duality in bulk, almost exactly holds with the change of the sign of γ [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 of4/6 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 4 derivative theory is introduced, more rich pole structures are exhibited [36]. In this paper, we shall study the EM duality and the QNMs of 6 derivative theory with homogeneous disorder. Our paper is organized as follows. In Section 2, we present a brief introduction on the 6 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 6 derivative term and its properties of conductivity and QNMs, respectively.


## 2. Holographic Setup

We consider the following neutral Einstein-axions (EA) theory [32]: (1)S0=∫d4x-gR+6L2-12∑I=x,y∂ϕI2. In the above action, we have introduced a pair of spatial linear dependent axionic fields, ϕI=αxI with I=x,y and α being a constant, which are responsible for dissipating the momentum of the dual boundary field. L is the radius of the AdS spacetimes.From the EA action (1), we have a neutral black brane solution [32] (2)ds2=L2u2-fudt2+1fudu2+dx2+dy2,where (3)fu=1-upu,pu=1+6α^2-2α^2-1α^2u2+u+1.The AdS boundary is at u=0 and the horizon locates at u=1. α^≡α/4πT with the Hawking temperature T=p(1)/4π. Since the translational symmetry breaks, the momentum dissipates. But the geometry is homogeneous and so we refer to this mechanism as homogeneous disorder and α^ 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(4)SA=∫d4x-g-18gF2FμνXμνρσFρσ,where the tensor X is an infinite family of HD terms [14] (5)Xμνρσ=Iμνρσ-8γ1,1L2Cμνρσ-4L4γ2,1C2Iμνρσ-8L4γ2,2CμναβCαβρσ-4L6γ3,1C3Iμνρσ-8L6γ3,2C2Cμνρσ-8L6γ3,3Cμνα1β1Cα1β1α2β2Cα2β2ρσ+…. Note that Iμνρσ=δμρδνσ-δμσδνρ is an identity matrix and Cn=Cμνα1β1Cα1β1α2β2…Cαn-1βn-1μν. In the above equations, the factor of L is introduced such that the coupling parameters gF and γi,j are dimensionless. But for later convenience, we work in units where L=1 in what follows and set gF=1 in the numerical calculation. When Xμνρσ=Iμνρσ, the theory reduces to the standard Maxwell theory. For convenience, we denote γ1,1=γ and γ2,i=γi(i=1,2). In the main body of this paper, our main focus is the γ1 derivative terms (the main effect of both γ1 and γ2 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 γ2 higher derivative term in Appendixes B and C.). Note that when other parameters are turned off, γ1 is confined in the region γ1≤1/48 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 Reσ(ω)≥0, they confirm that the constraint γ1≤1/48 also holds over EA-AdS geometry.) [14]. And then, from the action (4), we can write down the equation of motion (EOM) as (6)∇νXμνρσFρσ=0.As [7], the dual EM theory of (4) can be constructed, which is (7)SB=∫d4x-g-18g^FGμνX^μνρσGρσ.Gμν≡∂μBν-∂νBμ is the dual field strength and g^F is the coupling constant, which relates gF as g^F2≡1/gF2. X^ is the dual tensor, which satisfies (8)X^μνρσ=-14εμναβX-1αβγλεγλρσ,12X-1μνρσXρσαβ≡Iμναβ. And then, the EOM of the dual theory (7) can be wrote down as (9)∇νX^μνρσGρσ=0.For the four dimensional standard Maxwell theory,X^μνρσ=Iμνρσ. 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 γ1 is small, we have (10)X-1μνρσ=Iμνρσ+4γ1C2Iμνρσ+Oγ12,(11)X^μνρσ=X-1μνρσ+Oγ12.It indicates that, for small γ1, there is an approximate duality between both theories (4) and (7) with the change of the sign of γ1.


## 3. Conductivity and EM Duality

Turning on the perturbationAy(t,u)~e-iωtAy(u), we can write down the EOM of gauge field [7, 33, 34] (12)Ay′′+f′f+X6′X6Ay′+p2ω^2f2X2X6Ay=0, where Xi, i=1,…,6, are the components of XAB defined as XAB={X1(u),X2(u),X3(u),X4(u),X5(u),X6(u)}, with A,B∈{tx,ty,tu,xy,xu,yu}. Due to the isotropy of background, X1(u)=X2(u) and X5(u)=X6(u). In addition, we have defined the dimensionless frequency in the above equation ω^≡ω/4πT=ω/p, with p≡p(1)=4πT. Letting Xi→X^i=1/Xi, 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 (13)σω^=∂uAyu,ω^iω^pAyu,ω^.It has been illustrated in the last section that, for very smallγ1, there is an approximate duality between both the original EM theory and its dual theory with the change of the sign of γ1 as that for 4 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]: (14)σ∗ω^;α^,γ1=1σω^;α^,γ1, where σ∗ is the conductivity of the dual theory. Therefore, for small γ1, we conclude that [33] (15)σ∗ω^;α^,γ1≈σω^;α^,-γ1,γ1≪1. It indicates that the dual optical conductivity is approximately equal to its original one with the change of the sign of γ1. For 4 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 γ [7]. But when the homogeneous disorder is introduced, we find that this relation holds better for a specific value of α^=2/3 than other values of α^. Here, we shall study the EM duality from 6 derivative theory.Figure1 exhibits the optical conductivity as the function of ω^ for α^=0 and various values of γ1. We find that for small |γ1| (γ1=0.001), with the change of the sign of γ1, the relation (15) holds very well. With the increase of γ1, the relation (15) violates. In particular for γ1=0.02, the real part of the conductivity at low frequency is a pseudogap-like behavior. Correspondingly, its dual conductivity Reσ∗ at low frequency becomes sharper, which is Drude-like. But for γ1=-0.02, 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 ofω^ for α^=0 and various values of γ1. The solid curves are the conductivity σ of the original EM theory (4) and the dashed curves display the conductivity σ∗ of the EM dual theory (7).Next, we explore the EM duality for6 derivative theory when the homogeneous disorder is introduced. At the first sight, we find that for the specific value of α^=2/3, 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 γ1 changes, i.e., the relation (15) holds very well. It is similar to that from 4 derivative theory [33]. For other values of α^, 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: (16)Ay′′+f′f-X6′X6Ay′+p2f2X6X2ω^2Ay=0. Since X2/X6=X6/X2 for any α^, the difference between (B.8) and (16) is in the coefficients of Ay′, which are (17)X1γ1≡X6′X6=-16/3γ1u3f′′u2-8/3γ1u4f3uf′′u1-4/3γ1u4f′′u2,(18)X2γ1≡-X6′X6=--16/3γ1u3f′′u2-8/3γ1u4f3uf′′u1-4/3γ1u4f′′u2. And then, we explicitly plot the ration (X1/X2)γ1=0.02 as the function of u with different α^ in Figure 3. For α^=0, the value of X1/X2 is maximum departure from 1 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 α^=2/3, this value approaches to 1 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 ofω^ for γ1=0.02 and various values of α^. The solid curves are the conductivity σ of the original EM theory (4) and the dashed curves display the conductivity σ∗ of the EM dual theory (7).Figure 3
X 1 ( γ 1 ) / X 2 ( - γ 1 ) as the function of u with different α^. Here we set γ1=0.02.


## 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 alongy direction, i.e., Ay, from 6 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 (19)ds2=1u2-fudt2-2dtdu+dx2+dy2. Correspondingly, the EOM (B.8) can be changed as (20)Ay′f′f+2ipω^f+X6′X6+ipω^AyX2′fX6+Ay′′=0. For the dual theory, the corresponding EOM can be obtained by letting Aμ→Bμ and Xi→X^i=1/Xi. Imposing the ingoing boundary at the horizon, we numerically solve (20) to obtain QNMs.We first study the case without homogeneous disorder. Figures4 and 5 show QNMs of the gauge mode with α^=0 for γ1=0.001 and 0.02, 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 |γ1|, for example, γ1=0.001 in Figure 4, the qualitative correspondence between the poles of Reσ(ω^;γ1) and the ones of Reσ∗(ω^;-γ1) holds well at low frequency. But for the large frequency region, there are some modes emerging at the imaginary frequency axis for negative γ1, which results in the violation of the particle-vortex duality with the change of the sign of γ1. But with the increase of γ1, this correspondence begins to violate in complex frequency panel even in the low frequency region (see Figure 5 for γ1=0.02), as that in real frequency axis. We also plot the QNMs for γ1=-1 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|γ1|=0.001 (the panels above are for γ1=-0.001 and the ones below for γ1=0.001) and α^=0. 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|γ1|=0.02 (the panels above are for γ1=-0.02 and the ones below for γ1=0.02) and α^=0. 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 forγ1=-1 and α^=0. 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. Figures7, 8, and 9 show the QNMs with representative α^ for γ=0.02, -0.02, and -1 (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 forα^=1/2. The first, second, and third columns are for γ1=0.02, -0.02, and -1, 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 forα^=2/3. The first, second, and third columns are for γ1=0.02, -0.02, and -1, 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 forα^=2. The first, second, and third columns are for γ1=0.02, -0.02, and -1, 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 ofReσ(ω^;γ1) and the ones of Reσ∗(ω^;-γ1). They are briefly summarized as what follows for |γ1|=0.02.(i)
Forα^=1/2, some modes emerge at the imaginary frequency axis for negative γ1 as that without homogeneous disorder but not for positive γ1.(ii)
Forα^=2/3, there are still the modes locating in the imaginary frequency axis for negative γ1 but not for positive γ1. In particular, new branch cuts emerge for positive γ1.(iii)
Forα^=2, the pole structures for α^=2 are similar with those for α^=2/3.(iv)
These emerging modes violate the duality between the poles ofReσ(ω^;γ1) and the ones of Reσ∗(ω^;-γ1). With an increase in the magnitude of α^, this duality is getting violated more.We also exhibit the QNMs forγ1=-1 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 γ1=-1 with homogeneous disorder. For small α^, 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]) (21)σω^=KΓ-iω^, where Γ is the relaxation rate, which relates the relaxation time τ as Γ=1/τ. 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 rateΓi of the holographic system with γ1=-1 for different α^. Γ1 is fitted by the Drude formula (21) and Γ2 is obtained by locating the position of the purely imaginary mode in complex frequency panel.
α ^ 0 0.01 0.1 0.5 Γ 1 0.0114 0.0115 0.0120 0.0284  Γ 2 0.0114 0.0115 0.0122 0.0370Figure 10
Evolution of the dominant QNMs withα^ for γ1=-1. 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 withα^ for γ1=-1 is presented in Figure 10. Most of the dominant QNMs for the original theory, except for that approximately in the region of α^∈(1,2/3), are the purely imaginary modes. Correspondingly, the dominant QNMs for the dual theory, except for that approximately in the region of α^∈(1,2/3), are off axis.Further, we study the evolution of the dominant QNMs withα^ for γ1=0.02, which is shown in Figure 11. We describe the properties as what follows.(i)
Forγ1=-0.02, all the dominant QNMs for the original theory are purely imaginary modes. With the increase of α^, 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 γ1=0.02 is similar with that for the original theory with γ1=-0.02. Such evolution is also similar with that for 4 derivative theory studied in [36]. For small α^, the poles for the dual theory are closer to the real frequency axis than that for the original theory.(ii)
Forγ1=0.02, the dominant poles for the original theory are off axis for small α^. With the increase of α^, the poles migrate downwards and are closer to the real axis. When α^ reaches the value of α^≃0.8, the poles merge into one purely imaginary pole. And then, when α^ is beyond α^≃1.4, the purely imaginary pole splits into two off-axis modes. While for γ1=-0.02, the evolution of the dominant poles for the dual theory is similar with that for the original theory with γ1=0.02. But when α^>3.5, the pole for the dual theory with γ1=-0.02 becomes a purely imaginary one.(iii)
In summary, the correspondence between the evolution of the poles as the function ofα^ for the original theory for γ1=-0.02 and that for its dual theory for γ1=0.02 is violated.Figure 11
Evolution of the dominant QNMs withα^ for |γ1|=0.02 (the first two rows are for γ1=-0.02 and the last two rows for γ1=0.02). 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 6 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 ofγ1, the particle-vortex duality only holds for small γ1. With the increase of γ1, this duality is violated. When the homogeneous disorder is introduced, as found in 4 derivative theory, we find that for the specific value of α^=2/3, the optical conductivity of the original theory is almost the same as that of the dual one when the sign of γ1 changes. For other values of α^, the particle-vortex also approximately holds with the change of the sign of γ1. 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 γ1.The properties of the QNMs are also analyzed. In absence of the homogeneous disorder, the qualitative correspondence between the poles ofReσ(ω^;γ1) and the ones of Reσ∗(ω^;-γ1) holds well at low frequency only for small γ1. When γ1 becomes large, this correspondence is also violated even at the low frequency region. For γ1=-1, new branch cuts of QNMs are observed.When the homogeneous disorder is introduced, some modes emerge at the imaginary frequency axis for negativeγ1 but not for positive γ1. In particular, with an increase in the magnitude of α^, new branch cuts emerge for positive γ1. These emerging modes violate the duality between the poles of Reσ(ω^;γ1) and the ones of Reσ∗(ω^;-γ1). With the increase of α^, this duality is getting violated more. In addition, for γ1=-1, the homogeneous disorder drives the off-axis new branch cuts for α^=0 to the purely imaginary modes.The evolution of the dominant QNMs withα^ is also explored. We find that all the dominant QNMs for the original theory with γ1=-0.02 and that for the dual theory with γ1=0.02 are purely imaginary modes. Their evolutions are also similar. However, for the evolution of the dominant QNMs for the original theory with γ1=0.02 and that for the dual theory with γ1=-0.02, the case is somewhat different. The main difference is that the modes are not again the purely imaginary ones except for some specific α^. In summary, the correspondence between the evolution of the poles as the function of α^ 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 from4 derivative theory has been worked out in [35, 39–45] 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 [46–55] 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 both4 and 6 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.



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