Momentum dissipation and holographic transport without self-duality

We implement the momentum dissipation introduced by spatial linear axionic fields in a holographic model without self-duality, broke by Weyl tensor coupling to Maxwell field, and study its response. It is found that for the positive Weyl coupling parameter $\gamma>0$, the momentum dissipation characterized by parameter $\hat{\alpha}$ drives the boundary conformal field theory (CFT), in which the conductivity exhibits a peak at low frequency, into the incoherent metallic phase with a dip, which is away from CFT due to the introduction of axionic fields. While for $\gamma<0$, an oppositive scenario is found. Our present model provides a possible route toward the problem that which sign of $\gamma$ is the correct description of the CFT of boson Hubbard model. In addition, we also investigate the DC conductivity, diffusion constant and susceptibility. We find that for each of these observables there is a specific value of $\hat{\alpha}$, for which these observables are independent of $\gamma$. Finally, the electromagnetic (EM) duality is also studied and we find that there is also a specific value of $\hat{\alpha}$, for which the particle-vortex duality related by the change of the sign of $\gamma$ in the boundary theory holds better than for other values of $\hat{\alpha}$.


I. INTRODUCTION
The transport properties, such as the electrical conductivity, heat conductivity and thermoelectric transport, are great important features of real materials. For the weakly coupled systems, the frequency dependent conductivity exhibits Drude-like peak at low frequency.
Their collective dynamics is well described by the quantum Boltzmann theory of the quasiparticles with long-lived excitations [1]. While for the strongly coupled systems, the picture of the quasi-particle is absent and the Boltzmann theory is usually invalid 1 .
The anti-de Sitter/conformal field theory (AdS/CFT) correspondence [5][6][7][8] provides a powerful tool and novel mechanism to study the transport of the strongly coupled systems. 1 When quasi-particle excitations are only weakly broken, the perturbative method in the Boltzmann framework is developed to deal with such systems, see for example, [1][2][3][4].
proposed in [32], where the momentum dissipation is implemented by a pair of massless field, Φ I with I = 1, 2, which are spatial linear dependent in bulk. It is also referred as "mean-field disordered" [41] due to the homogeneous background geometry. Note that Φ I correspond to turning on spatial linear sources in the dual boundary theory, i.e., φ (0) with α being constant. This nonuniform source means that a dimensionful parameter, i.e., α, is introduced into the dual boundary theory and so the physics we studying is that away from QCP. We hope that our present model provides wider route to address whether the excitation of the CFT of the superfluid-insulator QCP described by the boson Hubbard model is particle-like or vortex-like and also toward the problem that which sign of γ is the correct description of this CFT. In addition, the proximity effect in QCP, which also alters some observables such as the optical conductivity, is also important and has been explored in [1,4,[9][10][11][12][13][14][15]42]. Our present work will also provide some insight into the transport properties away from QCP in holographic framework.
Our paper organizes as follows. We begin with a review of the holographic framework without EM self-duality in Section II. We then introduce a neutral axionic theory, which is responsible for the momentum dissipation in Section III. The optical conductivity of the boundary field theory dual to the Maxwell-Wely system in the neutral axionic geometry is studied in IV. We mainly focus on the role the momentum dissipation plays in the transport propertie in our present model. We also study the diffusion constant and susceptibility of the dual boundary field theory in Section VII. In Section VI, we discuss the EM duality. We conclude with a brief discussion and some open questions in Section VII. In Appendix A, we discuss the constraints imposing on the Weyl coupling parameter γ in the neutral axionic geometry due to the causality and the instabilities.

II. HOLOGRAPHIC FRAMEWORK WITHOUT EM SELF-DUALITY
The optical conductivity in the neutral plasma dual to the standard Maxwell theory in four dimensional AdS spacetimes is frequency independent due to the EM self-duality [23].
To have a frequency dependent optical conductivity in the neutral plasma, we need to break the EM self-duality. A simple way is to introduce the Weyl tensor C µνρσ coupled to gauge field as [16][17][18][19][20][21][22]43] where F = dA is the curvature of gauge field A and g 2 F is an effective dimensionless gauge coupling, which shall be set g F = 1 in the numerical calculation. In this theory, there is a crucial dimensionless coupling parameter γ, which controls the coupling strength of the Maxwell-Weyl term. The Weyl term is a specific combination of some four-derivative interaction term [16], which can be expected to emerge as quantum corrections in the low energy effective action in string theory context [44,45].
It is more convenient for subsequent calculations and discussions to write down the action (2) in a general form [16] (also see [17][18][19][20][21]) A new tensor X is introduced in the above equation as with an identity matrix acting on two-forms It is easy to find that the X tensor possess the following symmetries When we set X ρσ µν = I ρσ µν , the theory (2) reduces to the standard Maxwell theory. And then, from the action (3), we have the equation of motion as In presence of the Weyl term, the EM self-duality breaks down [16]. However, we can still construct the dual EM theory for the gauge theory (2). For more details, we can see [16]. Here we directly write down the corresponding dual EM theory In addition, the tensor X is defined by where ε µνρσ is volume element and X −1 is defined by Also we can derive the equation of motion of the dual theory (8) as For the standard four-dimensional Maxwell theory, X ρσ µν = I ρσ µν and therefore the theory (3) and (8) are identical, which means that the Maxwell theory is self-dual. When the Weyl term is introduced and for small γ, we find that which implies that the self-dual is violated for the theory (2) but for small γ, there is a duality between the actions (3) and (8) with the change of the sign of γ.

III. A NEUTRAL AXIONIC THEORY
We intend to implement the momentum dissipation in the Maxwell-Weyl system and study its response. The simplest way is to introduce a pair of spatial linear dependent axionic fields [32], in which the action is 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 5 . When the momentum dissipation is weak, the standard Maxwell theory with action (14) describes coherent metallic behavior [32,46,47]. Conversely once the momentum dissipation is strong, we have an incoherent metal [32,46,47]. 5 Here, without loss of generality we have set the AdS radius L = 1 for simplify.
Since the Einstein-Maxwell-axion-Weyl (EMA-Weyl) theory (Eqs. (14) and (2)) involves solving a set of third order nonlinear differential equations, it is hard to solve them even numerically. As an alternative method, we can construct analytical background solutions up to the first order of the Weyl coupling parameter γ [48][49][50][51][52][53][54]. But it is still hard to obtain the frequency dependent conductivity and only the DC conductivity is worked out in [54]. As the first step, here we shall follow the strategy in [16][17][18][19][20][21][22] and turn to study the transports of the Maxwell-Weyl system (2) in the neutral plasma dual to the Einstein-axions (EA) theory (14).
The neutral black brane solution of the EA action (14) can be written down as [32] where u = 0 is the asymptotically AdS boundary while the horizon locates at u = 1. And then, the Hawking temperature can be expressed as Since the black brane solution (15) with (16) is only parameterized by one scaling-invariant parameterα = α/4πT , for later convenience, we reexpress the function p(u) as Further, the energy density , pressure p and entropy density s of the dual boundary theory can be calculated as [32] = 2 1 − α 2 2 , p = 1 + α 2 2 , s = 4π .
At this moment, there are some comments presenting in order. First, it is easy to see that from (17) at zero temperature (α = √ 6), the IR geometry is AdS 2 × R 2 , which is similar to the Reissner-Nordström-AdS (RN-AdS) black brane. Second, there is special value of α = √ 2 where the energy density vanishes. At this point, there is self-duality in Maxwell equations and the AC heat conductivity is frequency-independent [47]. Third, the axionic fields φ I in bulk correspond to turning on sources in the dual boundary theory which is linearly dependent of the spatial coordinate. Such sources result in the momentum dissipation, which is controlled by the parameterα. At the same time, as pointed out in the introduction, Φ I introduces the nonuniform source with a dimensionful parameter α in the dual boundary theory such that the system we are studying is away from QCP.

IV. OPTICAL CONDUCTIVITY
A simple but important transport behavior is the electrical optical conductivity at zero momentum. We mainly study it in this paper. The other transport properties, such as the thermal conductivity, the optical conductivity at finite momentum, will be studied elsewhere.
In holographic framework, the optical conductivity along y-direction can be calculated A y (u,ω,q = 0) is the perturbation of the gauge field at zero momentum along y-direction in Appendix A.

A. Optical conductivity
In [16][17][18][19][20][21], the optical conductivity of the boundary field theory dual to SS-AdS geometry has been explored. For γ > 0, it exhibits a peak at low frequency 7 , which qualitatively resembles the Boltzmann transport of particles. While for γ < 0, a dip appears, which is similar to the excitation of vortices. It provides a possible route to resolve whether the excitation of the CFT of the superfluid-insulator QCP described by the boson Hubbard model is particle-like or vortex-like. Before proceeding, let us briefly address this problem. 6 Due to the symmetry between x and y directions, we only need calculate the conductivity along either x or y direction. 7 There is a deviation from the standard Drude formula for γ ∈ S 0 . We shall illustrate this point below.
For the details, we can refer to [24]. In the insulating phase of the boson Hubbard model, it is the excitation of the particle and hole and so we can infer that the conductivity at low frequency should exhibit a peak if we approach the QCP from the insulator side. However, if we approach the QCP form the superfluid side, which described by the excitation of the vortices, the conductivity at low frequency should be a dip. Until now, we do not know which of the two qualitatively distinct results is correct. The Maxwell-Weyl system in the SS-AdS geometry provides a possible description for the CFT of the boson Hubbard model in holographic framework though we still have a degree of freedom of the sign of γ. Here, we hope that implementing the momentum dissipation, which is also equivalent to the "meanfield disordered" effect [41], will provide more clues in addressing this problem. That is to say, asα increases, the dip in optical conductivity at low frequency gradually upgrades and eventually develops into a peak. It indicates that if the CFT is described by Maxwell-Weyl system with negative γ, then the disorder drives it into the metallic phase characterized by a peak. Though the present model still cannot give a definite answer for which sign of γ being the correct description of the CFT of boson Hubbard model, it indeed provide a route toward this problem, which can be detected in future condensed matter and ultracold atomic gases experiments or solved in theory by introducing the disorder effect into the boson Hubbard model. In holographic framework, we can introduce different types of disorder, for example the Q-lattice [29,30], in the Maxwell-Weyl system to see how universal results presented here are. We shall address this problem in near future. Next, we present more details on the optical conductivity of our present model, in particular its low frequency behavior.

B. The low frequency behavior of the optical conductivity
In this subsection, we intend to study the low frequency behavior of the optical conductivity and try to give some insights into the coherent/incoherent behavior from the momentum dissipation and the Weyl term.
In the boundary field theory dual to the Maxwell-Weyl system in SS-AdS geometry, although a peak emerges in the optical conductivity at low frequency for γ > 0 [16], there is in fact a deviation from the standard Drude formula if we require γ ∈ S 0 . Inspired by the incoherent metallic phase studied in [55] (also see [46,[56][57][58][59][60] for the related studies), we use the following modified Drude formula to fit the data for γ = 1/12, where K is a constant, τ the relaxation time and σ Q characters the incoherent degree. The Quantitatively, using the modified Drude formula (21), we fit the low frequency behavior of the optical conductivity for γ = 1/12 and differentα (FIG.3). Also, we list the characteristic quantity of incoherence σ Q in Table I. Two illuminating results are summarized as what follows. First, σ Q increases with the increase ofα in our present model (Table I), which indicates that the incoherent behavior becomes more evident. But we also note that for small momentum dissipation (α < 0.06), σ Q is smaller than that in SS-AdS. It is because the small momentum dissipation produces coherent contribution, which reduces the incoherent part from the Weyl term. While with the increase ofα, the momentum dissipation becomes strong, combining with that from the Weyl term, and so the system exhibits more prominent incoherent behavior. Second, the fit by Eq.(21) for largeα is better than that for smallα (FIG.3). It is because for smallα, the incoherent contribution mainly comes from the Weyl term and also implies that we need a new non-Drude formula beyond the simple case (21) to depict meticulously the incoherent contribution from the Weyl term. We shall further explore this question in future.  We are also interested in the low frequency behavior for γ < 0 and largeα, in which a peak exhibits (the bottom plots in FIG.1). From FIG.4, we see that the non-Drude behavior can be well fitted with differentα for γ = −1/12 by the modified Drude formula (21). Quantitatively, we fit σ Q with differentα for γ = −1/12 in Table II. Again, it confirms that the modified Drude formula is more suitable to describe the incoherent transport from the momentum dissipation than that from the Weyl term.   There are many ways to calculate these quantities. Here we shall use the membrane paradigm approach [61,62]. The key point of the membrane paradigm is to define the membrane current on the stretched horizon u H = 1 − with 1 where n ν is a unit radial normal vector. By the Ohm's law, it is straightforward to write down the expression of the DC conductivity in our present framework [16,43] Moreover, following [61,62], it is also easy to obtain the diffusion constant [16,43]  And then, using the Einstein relation D = σ 0 /χ, the susceptibility can be expressed as [61,62] For the details, we can refer to [16,43].
In the following, we shall explicitly discuss these quantities. First, evaluating Eq. (23), we obtain the explicit expression of the DC conductivity It is obvious that the DC conductivity is linear dependence on γ for givenα (Eq. (26)  The diffusion constant and the susceptibility with Weyl correction in the boundary field theory dual to the SS-AdS geometry can be analytically worked out [16,43]. However, for the EA-AdS geometry (15) and (16), it is difficult to analytically derive their expressions be found for the susceptibility. Third, the momentum dissipation suppresses the diffusion constant and the inverse of the susceptibility, regardless of the sign of γ. It implies that the momentum dissipation has similar effect on the diffusion constant or the susceptibility regardless of the excitation being particles or vortices. We shall further understand and explore such phenomenon and their microscopic mechanism in future.

VI. EM DUALITY
In Section II and Appendix A (also see [16]), it has been illustrated that for the metric of the background geometry being diagonal, the EM duality holds and for very small γ, the original EM theory relates its dual theory by changing the sign of γ. In the AdS/CFT correspondence, the bulk EM duality corresponds to the particle-vortex duality in the boundary field theory, in which the optical conductivity of the dual theory is the inverse of that of its original theory [16,17] σ * (ω;α, γ) = 1 σ(ω;α, γ) .
From Eqs. (27) and (28), we can conclude the following relation The above equation indicates that the optical conductivity of the dual EM theory is approximately equal to that of its original theory for the oppositive sign of γ. It also have been explicitly illustrated for γ = ±1/12 in Figure 5 in [16], from which we can obviously see that the conductivity of the dual EM theory is not precisely equal to that of its original theory for the oppositive sign of γ except forω → ∞. Next, we shall explore the effect of the momentum dissipation on the EM duality by explicitly presenting the frequency dependent conductivity of the original theory and its dual theory.
First, we focus on the DC conductivity, which can be analytically derived (see Eq. (26)).
Subsequently, we turn to study the optical conductivity. FIG.9 displays the real and imaginary part of the optical conductivity of the bulk EM theory and its dual EM theory as the function of the frequencyω for γ = ±1/12 and various values ofα. As expected, an oppositive picture appears in the dual EM theory. That is to say, for smallα, a peak at small frequency in optical conductivity occurs for γ = −1/12 while a dip exhibits for γ = 1/12 and for largeα, the case is opposite. In addition, FIG.9 also further illustrates that the relation (28) or (29) holds only for |γ| 1. But for the specific value ofα = 2/ √ 3, which is previously found in DC conductivity, the optical conductivities of the original EM theory and its dual EM theory are almost exactly inverses of each other for either value of γ, i.e., the relation (29) holds very well. It indicates that the particle-vortex duality exactly holds forα = 2/ √ 3. Note that our result is numerically worked out. The analytical derivation and understanding deserves pursuing in future.
Finally, we would like to point out that the specific pointα = 2/ √ 3 is not the self-duality point of our present model because we don't haveX = X at this point such that the dual theory (8) is not identical with it original theory (3). Therefore, this point is different from the point found in the thermal conductivity of the Maxwell theory in EA-AdS geometry, at which point the thermal conductivity is independent of the frequency [47].

VII. DISCUSSIONS AND OPEN QUESTIONS
In this paper, we have studied the transports, in particular the electric conductivity, in a neutral plasma with momentum dissipation dual to the Maxwell-Weyl system in EA-AdS geometry. In previous studies [16][17][18][19][20][21], they find that the optical conductivity in the neutral system at finite temperature dual to the Maxwell-Weyl system in SS-AdS geometry exhibits a peak or a dip depending on the sign of the Weyl coupling parameter γ. It provides a possible description of the CFT of the boson Hubbard model in holographic framework [24]. But there is still a degree of freedom of the sign of γ. Our main results for the optical conductivity without self-duality but with the momentum dissipation are displayed in FIG.1 and the corresponding physical interpretation is presented in Section IV.
For γ > 0, the strong disorder drive the peak in the low frequency optical conductivity into a dip. While for γ < 0, an oppositive scenario happens. That is to say, the dip in optical conductivity at low frequency gradually upgrades and eventually develops into a peak with the increase of the disorder. Our present model provides a route toward the problem that which sign of γ is the correct description of the CFT of boson Hubbard model. Also we have quantitatively studied the low frequency behavior of the optical conductivity by using the modified Drude formula (21) to fit our numerical data. It provides a hint regarding the coherent or incoherent contribution from the Weyl term and the momentum dissipation and deserves further studying. Further, we find that there is a specific value of the momentum dissipation constantα = 2/ √ 3, for which the DC conductivity σ 0 is independent of γ and the particle-vortex duality related by the change of the sign of γ holds very well. Aside from the conductivity, there is also a specific value ofα for the diffusion constant and susceptibility, for which these quantities are independent of the Weyl coupling parameter γ.
But these specific values ofα are different from each other and so they are not universal in this Maxwell-Weyl system in EA-AdS geometry.
In addition, we also present several comments on the physics of our present model as follows. References [16][17][18][19][20][21] introduce an extra Weyl tensor C µνρσ coupling to Maxwell field in the SS-AdS black brane to study the QC transports in the neutral plasma at finite temperature. Another way to study the QC physics in holographic framework is that by incorporating a neutral bulk scalar field interacting with gravity, the corresponding scalar operator has an expected value [63]. Setting the source of scalar field in the dual boundary theory to zero, we can study the QC physics [63]. If we set the source of scalar field in the dual boundary field theory nonvanishing, the model [63] also provides a starting point to study the physics away from QCP. In our present model, Φ I correspond to turning on sources being spatial linear in the dual boundary theory, which introduces a dimensionful parameter α into the dual boundary theory and so the physics we studied is that away from QCP. Therefore, our results also provide some insight into the proximity effect in QCP. More comprehensive analysis on this point will be discussed elsewhere.
Finally, we comment some open questions deserving further exploration.
1. In this paper, we mainly study the optical conductivity at the zero momentum, which is relatively easy to calculate since the equations of motion simplify to a great extent. But the transports at the finite momentum and energy in condensed matter laboratories have been obtained now or shall be given in near future [64][65][66][67][68], which reveal more information of the systems. On the other hand, in [18] they have also studied the responses of Maxwell-Weyl system in SS-AdS geometry at the finite momentum and found that it indeed provided far deeper insights into this system than that at the zero momentum. Therefore it is interesting and important to further study the responses of our present model with the momentum dissipation at full momentum and energy spaces.
2. The spatial linear dependent axionic fields are the simplest way to implement the momentum dissipation, or say disorder. We can also introduce the momentum dissipation by incorporating the higher order terms of axions [69][70][71][72][73] to study the properties of transport of the Maxwell-Weyl system. The higher order terms of axions induce metalinsulator transition (MIT) [72,73] and provide a way to study the properties of solid in the holographic framework [69,70]. Also an insulating ground state can be obtained in this way [71]. In addition, another mechanism of momentum dissipation in holographic framework is the Q-lattice [29,30], by which various type of holographic MIT model have been built [29,30,60,74]. It is certainly interesting and valuable to incorporate Q-lattice responsible for the momentum dissipation into the Maxwell-Weyl system and study its transport behavior, in particular to see how universal our results presented in this paper are.
3. Another important transport quantity is the magneto-transport, which has been studied when the momentum dissipation is presented in [75][76][77][78]. It is interesting to study the magneto-transport property in our framework and explore the meaning of EM duality.
4. We would like to study the holographic superconductor in our present framework. In [79], the holographic superconductor with Weyl term is constructed. A main result is that the ratio of the gap frequency over the superconducting critical temperature ω g /T c runs with the Weyl parameter γ. Subsequently, a series of works study such holographic superconducting systems with Weyl term, see for example [80][81][82][83][84][85][86][87]. It would be interesting and useful to study the holographic superconducting systems without self-duality but with the momentum dissipation and further reveal the role that the momentum dissipation play in the Maxwell-Weyl system.

5.
A challenging question is to obtain a full backreaction solution for the EMA-Weyl system. As has been pointed out in [16,54], we need to develop new numerical technics to solve differential equations beyond the second order with high nonlinearity.
We plan to explore these questions and publish our results in the near future.
space as where q · x = −ωt + q x x + q y y. Without loss of generality, we set q µ = (ω, q, 0) and choose the gauge fixed as A u (u, q) = 0. Further, it is convenient to write the tensor X ρσ µν as [16] X with A, B ∈ {tx, ty, tu, xy, xu, yu} . (A3) Next we discuss the bound of γ imposed by the causality and the instabilities. Note that since Eq. (A4) gives the relation between A x and A t , there are only two independent vector modes A t and A y and we only need to consider the corresponding equations (A9) and (A7).
It is convenient to formulate Eqs. (A9) and (A7) into the Schrödinger form. To this end, we make the change of variables dz/du = p/f and write A i (u) = G i (u)ψ i (u) where we denote At(u) := A t (u) and i =t, y. And then we have where V i (u) is the effective potential. We decompose it into the momentum dependent part and the independent one where [20] V There is a simple relation between Vt and V y as Vt = V y | X i → X i and vice-versa [20]. At the same time, from Eq. (12), one has X i ≈ X i | γ→−γ for small γ. Therefore, we mainly focus on the discussion of Vt in what follows.
Subsequently, we mainly examine whether the constraint γ ∈ S 0 in SS-AdS geometry holds when the momentum dissipation is introduced. For the γ beyond S 0 , we present brief comments. First and foremost we consider the case of the limit of large momentum (q → ∞).
In this limit, the constraint on V 0i should be imposed as The upper bound of V 0i (u) comes from the constraint of the causality in the dual boundary theory [88,89], which is a key constraint on the coupling γ. Otherwise, there will be superluminal modes with ω/q > 1 in this neutral plasma. While the lower bound of V 0i (u) is from the requirement of stability of the vector modes since in the WKB limit, a negative potential will results in bound states with a negative effective energy, which corresponds to unstable FIG. 10: The shape of the potentials V 0t of the longitudinal mode A t for various value of γ ∈ S 0 andα is shown. We find that V 0t well belongs to the region (A15). quasinormal modes in the bulk theory [90]. FIG.10 shows the shape of the potential V 0t of the longitudinal mode A t for various value of γ ∈ S 0 andα. It implies that V 0t well belongs to the region (A15). The similar result is found for the potential V 0y of the transverse mode A y . Further careful examination indicates that provided γ ∈ S 0 the constraint (A15) is well satisfied for arbitraryα. In fact, when the momentum dissipation is introduced, the constraint (A15) can be satisfied for wider region of γ beyond S 0 (see FIG.11).
Second we consider the case in the small momentum region, in which V 1i play an important role in the effective potential V i . FIG.12 shows the shape of the potentials V 1t with γ = −1/12 and γ = 1/12 for various value ofα. By careful examination, we find that for γ = −1/12, V 1t (u) develops a negative minimum close to the horizon in the regionα ∈ (0, 0.95), which means some unstable modes. While for γ = 1/12, the negative minimum in V 1t (u) appears inα ∈ (0.95, +∞). However, although in small momentum region, the potential V 1i develops a negative minimum close to the horizon for some regions of γ andα, we find that there are no unstable modes in these regions by analyzing the zero energy bound state in the potential V 1i . We shall demonstrate it below. As analyzed in [90], there is a zero energy bound state in the potential V 1i by the WKB approximation, where n is a positive integer. The integration is over the values of u for which the potential well is negative. Defining I i ≡ n − 1/2 π and introducingñ it = I i /π + 1/2, we plotñ 1t as the functionα for given γ in the region ofα in which a negative potential develops close to horizon (see FIG.13). From this figure, we find thatñ 1t is always less than unit and so no unstable modes appear in the small momentum region.
Finally, after examining the instabilities for small and large momentum limit, we examine the instabilities for some finite momentum. FIG.14 shows the potentials Vt(u) with different γ andα at some finite momentum. We see that the potential is always positive, which indicates that no unstable modes appear even for the finite momentum. It is because the positive contribution of V 0t (u) is larger than the negative one of V 1t (u).
We conclude that the region γ ∈ S 0 is still physically viable even introducing the momentum dissipation. In fact, this physically viable region maybe become larger in this neutral axionic geometry (15) (see for example FIG.11). More detailed exploration will be left for  FIG. 14: The potentials Vt(u) with different γ andα at some finite momentum.
the future and here we only restrict ourself in the region γ ∈ S 0 .