Quasi-normal modes of holographic system with Weyl correction and momentum dissipation

We study the charge response in complex frequency plane and the quasi-normal modes (QNMs) of the boundary quantum field theory with momentum dissipation dual to a probe generalized Maxwell system with Weyl correction. When the strength of the momentum dissipation $\hat{\alpha}$ is small, the pole structure of the conductivity is similar to the case without the momentum dissipation. The qualitative correspondence between the poles of the real part of the conductivity of the original theory and the ones of its electromagnetic (EM) dual theory approximately holds when $\gamma\rightarrow -\gamma$ with $\gamma$ being the Weyl coupling parameter. While the strong momentum dissipation alters the pole structure such that most of the poles locate at the purely imaginary axis. At this moment, the correspondence between the poles of the original theory and its EM dual one is violated when $\gamma\rightarrow -\gamma$. In addition, for the dominant pole, the EM duality almost holds when $\gamma\rightarrow -\gamma$ for all $\hat{\alpha}$ except for a small region of $\hat{\alpha}$.


I. INTRODUCTION
The quantum critical (QC) dynamics is a long-standing important issues in strongly coupling condensed matter systems, in which the perturbative techniques in traditional field theory unfortunately lose its power. An alternative method is the AdS/CFT correspondence [1][2][3][4], which maps a strongly coupled quantum field theory to a weakly coupled gravitational theory in the large N limit. By this way, the holographic QC dynamics at zero density, dual to a probe Maxwell field coupled to the Weyl tensor C µνρσ in the Schwarzschild-AdS (SS-AdS) in bulk, has been widely studied in [5][6][7][8][9][10][11][12]. Since the Weyl tensor is taken into account, it exhibits non-trivial frequency dependent conductivity. In particular, depending on the sign of the coupling parameter γ, a Damle-Sachdev (DS) peak [13] resembling the particle response or a dip resembling the vortex response is observed in optical conductivity 1 [5]. It is analogous to that of the superfluid-insulator quantum critical point (QCP) [5][6][7]. Further, the charge response from higher derivative (HD) theory is studied, in which we have an arbitrarily sharp Drude-like peak or vanishing DC conductivity depending on the coupling parameter [16]. Of particular interest is that the behavior of the holographic HD system is very similar to the O(N ) N LσM model for large-N [13]. Another important progress is the construction of neutral scalar hair black brane in bulk by introducing the coupling between Weyl tensor and neutral scalar field, which is dual to QC dynamics and the one away from QCP in the boundary field theory [17,18].
Also, we can introduce the momentum dissipation, implemented by a pair of axionic fields linearly dependent on spatial coordinates [34], into the holographic QC systems studied in [5,6,[8][9][10][11][12], which is away from QCP, to explore the corresponding effects. We observe that for the 4 derivative theory, the momentum dissipation drives the peak and dip into each other [19], while for the 6 derivative theory, similar behaviors are not observed [20].
Another appealing phenomena for 4 derivative theory is that there is a specific value of the momentum dissipation strength, i.e.,α = 2/ √ 3, for which the particle-vortex duality exactly remains [19]. In this paper, we want to extend the previous works [19] to the charge response 1 Those kind of peak and dip features have also been observed in probe branes setups [14] and just higher terms in F 2 with F being the Maxwell field strength [15]. Although the role and the presence of a Drudelike peak in the optical conductivity have been partially explained for the probe brane case [14], it is still not so clear why there is a Drude-like peak and to what it is connected. More efforts and attempts are deserved to further pursuit in future.
in complex frequency plane. We shall particularly focus on the properties of the quasinormal modes (QNMs), which corresponds to the poles of the retarded Green's function for the dual boundary CFT (see [21][22][23][24][25][26][27][28][29][30][31][32][33] and references therein). Our paper is organized as follows. In Section II, we introduce the holographic framework, including the Einsteinaxions (EA) theory, which is the background geometry dual to a specific thermal excited state with momentum dissipation, and the 4 derivative theory without electromagnetic (EM) duality, which is regarded as the probe on top of the EA background geometry. And then, we calculate the conductivity in the complex frequency plane in Section III. We are in particular interested in the QNMs of our present models, which are presented in Section IV.
The conclusions and discussions are summarized in Section V.

II. HOLOGRAPHIC FRAMEWORK
We consider a specific thermal state with homogeneous disorder, which is holographically described by the EA theory [34] (also refer to [35][36][37][38][39][40][41] and references therein) where φ I = αx I with I = x, y and α being a constant. In this action, there is a negative cosmological constant Λ = −6, which supports an asymptotically AdS spacetimes 2 . The EA action (1) gives a neutral black brane solution [34] u = 0 is the asymptotically AdS boundary while the horizon locates at u = 1. Here we have parameterized the black brane solution byα = α/4πT with the Hawking temperature T = p(1)/4π. Note that for the particular way we adopt to break translations, i.e., the axionic fields, the original (spacetime translations) × (internal translations) group is broken to the diagonal subgroup such that the geometry is homogeneous [42,43].
On top of the geometry background (2), we consider the following 4 derivative theory [5] (also see [9-12, 16, 19, 20]) where The new tensor X in the above equation possess the following symmetries When we set X ρσ µν = I ρσ µν , the generalized Maxwell theory (3) reduces to the standard version. And then, we can write down the equation of motion from the action (3), Next, we can construct the corresponding dual EM theory, which is [5] And then the equation of motion of the dual theory (7) can be written as For the standard Maxwell theory in four dimensional bulk spacetimes, X ρσ µν = I ρσ µν , which indicates that both the theories (3) and (7) are identical and so the Maxwell theory is self-dual. The coupling term between the Weyl tensor and the Maxwell field strength breaks such self-duality. But for small γ, since we have an approximate duality between the theories (3) and (7) with the change of the sign of γ.

III. CONDUCTIVITY IN THE COMPLEX FREQUENCY PLANE
In our previous works [19], the conductivity on the real frequency axis was numerically studied. Here, we extend the study from real frequency axis to complex frequency plane, To this end, we turn on the perturbation of the gauge field along y direction like A y (t, u) ∼ e −iωt A y (u) . And then, the perturbative equation can be written down [5,19] A where ty, tu, xy, xu, yu} . Note that X 1 (u) = X 2 (u) and X 5 (u) = X 6 (u) due to the isotropy. In addition, the dimensionless , has been introduced in the perturbative equation (12). Next, we can numerically solve the Eq.(12) with the ingoing condition at horizon on top of complex frequency plan and read off the optical conductivity in terms of By letting A µ → B µ and X i → X i = 1/X i , the corresponding equation of motion for the dual theory can be obtained.
The numerical results are shown in FIG.1, 2, 3 and 4 for the representativeα and the two values of γ satisfying the bound, γ = ±1/12. We find that all the poles of the conductivity are in the lower-half plane (LHP), which indicates that the perturbative modes are stable ones. As a check on our numerics, we show the conductivity σ (left panels) and its dual σ * (right panels) in the LHP for |γ| = 1/12 (the panels above are for γ = 1/12 and the ones below for γ = −1/12) andα = 0 in the complex frequency plane, which have been obtained in [9]. Our results are in excellent agreement with that in [9]. For comparison, we firstly summary the main properties of the pole structure as what follows.
• For γ = 1/12, the dominant pole in Reσ, i.e., the pole closest to the real axis, is purely imaginary Drude-like pole which can be quantified asω = −iΓ with Γ being the dissipation rate. It corresponds to a particle-like transport with a Drude-like peak at low frequency on the real frequency axis which has been shown in [5]. • While for γ = −1/12, the dominant poles in Reσ are off-axis, which corresponds to non-Drude-like transport with a dip at low frequency on the real frequency axis [5].
• As the increase of the frequency, some satellite poles in Reσ emerge in pairs both for γ = 1/12 and γ = −1/12, which are off-axis and are obviously non-Drude-like.
• There is a qualitative correspondence between the poles of Reσ(ω; γ) and the ones of And then we explore the properties of the conductivity in the complex frequency plane when the effect of the momentum dissipation is introduced. The main properties are summarized as follows.
• When the strength of the momentum dissipation is small, the pole structure of Reσ is similar with that without the momentum dissipation (see FIG.2 forα = 1/2). And so qualitative correspondence between the poles of Reσ(ω; γ) and the ones of Reσ * (ω; −γ) still holds for smallα (FIG.2), which indicates we have a particle-vortex duality in the dual boundary field theory for smallα.  3 and FIG.4). The particle-vortex duality is violated when the off axis modes in Reσ or Reσ * appear.

IV. QUASI-NORMAL MODES
QNMs are by definition the poles of the Green's function and are directly related to the conductivity. Comparing with that of direct solution of conductivity on complex frequency, the method of QNMs has a wider range of applicability and is numerically more stable and high efficient such that we can carry out the quantitative study on the pole structure and obtain more insight.
To calculate the QNMs, we shall follow the method outlined in [32] and work in the advanced Eddington-Finkelstein coordinate, which is And then, the perturbative equation (12) can be changed correspondingly as Again, the equation for the dual theory can be obtained by letting A µ → B µ and X i → We briefly present the novel observations as follows. • Smallα (includingα = 0) 1. The qualitative correspondence between the poles of Reσ(ω; γ) and the ones of Reσ * (ω; −γ) approximately holds. But we also note that with the increase of the frequency, the pole moves more quickly away from imaginary frequency axis for γ < 0. So the particle-vortex duality holds better for small frequency than for high frequency.
2. There are some poles quite away from the generic structure for γ > 0. In particular, these special poles seem to emerge periodically (see the plots above in FIG.5, 6). 2. For γ < 0, there are some off-axis modes at high frequency region, which emerges periodically. With the increase ofα, the period of off-axis modes becomes large. 3. Therefore, the correspondence between the poles of Reσ(ω; γ) and the ones of The above pole structures are intriguing and are worthy of further exploration to reveal the physics of these pole structure.
The dominant pole describes the late time dynamical behavior of the dual system. So, we shall deeply explore the dominant pole evolution withα for |γ| = 1/12 to understand the effect of the momentum dissipation. We find that the motion of the dominant pole along the imaginary axis has the common tendency that with the increase ofα the pole firstly migrates downwards, and then migrates upwards, finally approaches to a certain value. FIG.9 shows this evolution. The more detailed properties are summarized as what follows.
• For γ = 1/12, when the momentum dissipation is introduced, out of the range ofα ∈ (1.35, 1.83), all the dominate poles in the original theory locate at the purely imaginary axis. But as revealed in [19], the low frequency conductivity in real frequency axis is a dip for largeα. This gap between them calls for deeper understanding. We also need to understand the behavior of the QNMs belong to the region ofα ∈ (1.35, 1.83), in which the pole is off-axis although the distance away from real axis is small. The correspondence between the poles of Reσ(ω; γ) and the ones of Reσ * (ω; −γ) holds except for the range ofα ∈ (1.35, 1.83).
• For γ = −1/2, the dominant pole in the original theory is off-axis for smallα. With the increase ofα, the poles migrate downwards and is close to the real axis. When α reaches the value ofα ≈ 0.77, the poles merge into one pole locating at imaginary axis. After that, the pole migrates upwards along the imaginary axis asα increases and approaches a fixed value for largeα. This picture is consistent with the observation in [19] that asα increases the dip develops into a peak in the conductivity in real frequency axis. The correspondence between the poles of Reσ(ω; γ) and the ones of Reσ * (ω; −γ) holds very well for almost allα.

V. CONCLUSION AND DISCUSSION
In this letter, we study the charge response in complex frequency plane and the QNMs of the boundary quantum field theory with momentum dissipation dual to a probe generalized Maxwell system with Weyl correction. When the strength of the momentum dissipationα is small, the pole structure of the conductivity is similar to the case without the momentum dissipation. The qualitative correspondence between the poles of Reσ(ω; γ) and the ones of Reσ * (ω; −γ) approximately holds, which indicates that the particle-vortex duality in the dual boundary field theory. But we also note that there are some poles quite away from the generic structure for γ > 0. In particular, these special poles seem to emerge periodically.
While the strong momentum dissipation alters the pole structure such that most of the poles locate at the purely imaginary axis. But there are a pair of poles of Reσ * (ω) are off-axis for γ > 0. With the increase ofα, this pair of poles migrate downwards along but close the imaginary axis. For γ < 0, there are some off-axis modes at high frequency region, which emerges periodically. With the increase ofα, the period of off-axis modes becomes large.
Therefore, the qualitative correspondence between the poles of Reσ(ω; γ) and the ones of Reσ * (ω; −γ) is violated when the momentum dissipation is large.
We are also interested in the dominant pole. With the increase ofα, the dominant pole firstly migrates downwards along the imaginary axis and then migrates upwards, finally approaches to a certain value. For γ = 1/12, the correspondence between the poles of Reσ(ω; γ) and the ones of Reσ * (ω; −γ) holds except for a small region ofα. But for γ = −1/2, the correspondence between the poles of Reσ(ω; γ) and the ones of Reσ * (ω; −γ) holds very well for almost allα.
There are lots of questions worthy of further study.
• An urgent task is to carry out an analytical study on the complex frequency conductivity on top of our present model by matching method developed in [44], which can provide more physical insight and understanding, in particular, for the gap between the conductivity in complex frequency and the one in real frequency observed for γ = 1/12.
• In this letter, we only study the QNMs at zero density. It would be worth to consider the model at finite charge density and compute the QNMs beyond the probe limit. But when we take into account the full backreaction of our present model, the equations of motion become a set of third order differential equations with high nonlinearity, which is very hard to solve analytically so far. The nonlinearity problem could be attacked in the small limit of γ by expanding the equations of motions up to the first order of γ. However, even if we have obtained the perturbative black brane solution to the first order of γ in [40], we still need to solve the linear perturbative differential equations beyond the second order to obtain the QNMs. It is a hard task and so we shall deeply explore it in future.
• We can also study the QNMs of our present model with finite momentum, which reveals richer physics of the system.