Chiral transport in strong fields from holography

Anomaly-induced transport phenomena in presence of strong external electromagnetic fields are explored within a 4D field theory defined holographically as $U(1)_V\times U(1)_A$ Maxwell-Chern-Simons theory in Schwarzschild-$AdS_5$. Two complementary studies are reported. In the first one, we present results on the Ohmic conductivity, diffusion constant, chiral magnetic conductivity, and additional anomaly-induced transport coefficients as functions of the external e/m fields. Next, gradient resummation in a constant background magnetic field is performed. All-order resummed constitutive relations are parameterised by four momenta-dependent transport coefficient functions (TCFs). A highlight of this part is a thorough study of {\it non-dissipative} chiral magnetic waves (CMW) in strong magnetic fields.

Beyond conceptual issues, truncations of the gradient expansion result in numerical instabilities rendering the entire framework unreliable. Causality is restored when all order gradient terms are included, in a way providing a UV completion to the "old" hydrodynamic effective theory. The resummation generalises the concept of TC to transport coefficient functions (TCFs), which are functionals of ∂ t and ∇ 2 (or equivalently functions of frequency ω and three-momentum squared q 2 in Fourier space). Therefore, TCFs contain information about infinitely many gradients and they extend the applicability of the effective theory beyond hydrodynamic limit of small frequency and momenta. When inverse Fourier transformed, the TCFs correspond to memory functions [14,15]. Below, an effective theory based on TCFs will be refered to as all order resummed hydrodynamics [14,[16][17][18][19][20].
The goal of the present work is to explore the generic structure of the currents (1) for external e/m fields, beyond the weak field limit explored by us in [3][4][5][6]. Our study will be split into two complimentary directions. In the first one, we compute various TCs as functions of constant e/m fields, denoted as E, B, up to first order in the gradient expansion.
In the second part, we consider gradient resummation in the presence of constant external magnetic field B only.
where λ is introduced via the replacement ∂ µ → λ∂ µ , and it counts the order in the gradient expansion.
Apart of the last term in J [0] 5 , which predicts separation of chiral charge along the Poynting vector S = E × B, all the terms in (4,5) have already appeared in the literature, particularly in our previous publications [3,5]. The main novelty of the present work is that we consider all the TCs in (4,5) as scalar functions of external e/m fields without assuming any weak field approximation, in contrast to what has been done in the past.
One might prefer an alternative representation of (4, 5) reflecting apparent anisotropy induced by the external fields.
with the longitudinal Ohmic and CME conductivities The constitutive relations (4,5) are in a sense "off-shell" since the charge densities ρ and ρ 5 are treated as independent of the three-currents J and J 5 . Imposing the continuity equations (2), the constitutive relations (1) are put "on-shell". To the leading order in spatial momentum, this results in a dispersion relation for the chiral plasma, There are three types of gapless modes propagating in the chiral plasma: the chiral magnetic wave (CMW) [27], the chiral electric wave (CEW) [23] and the chiral Hall density wave (CHDW) [5,6]. The external field dependent TCs σ 0 χ , σ 0 χe , σ 0 χH reflect speeds of these modes. Note that, in contrast to CMW and CEW both propagating in two opposite directions, CHDW is directed along the Poynting vector only.
Our goal is to compute the dependence of the TCs above on the external fields. The only analytic result that has been already known in the literature is for σ 0 where Γ[x] is the Gamma function and all the units are rescaled by the temperature, πT = 1.
Small and large field limits are Thus, in the strong magnetic field limit, the speed of CMW v χ goes to that of light [27].
While we were not able to obtain any new analytical insights, all the TCs in (4,5) were computed numerically and the results are presented in section III A. Here we quote some asymptotic behaviours for large-κB and large-κE At large κE, σ 0 χ [κB = 0] decays much faster than any other TCs, and asymptotically does not scale as a power function of κE. σ 0 e has certain asymptotic behaviour at large values of e/m fields only when κE = κB First order (n = 1). We consider the cases of either E = 0 or B = 0 separately: In [4,5], the diffusion constant D 0 was shown to receive negative (perturbatively small) The TC τχ is a relaxation time for CME/CSE conductivity σ 0 χ , see (4,5). The anomalous chiral Hall conductivity σ aχH [5,30] depends on the external electric field E. D 0 B and D E are new TCs. Both contribute to the longitudinal parts of the diffusion tensor, see (70, 82).
The first order TCs give rise to the decay rates of the chiral plasma modes (10): When B = 0 there is no propagating mode. At E = 0, CMW is a propagating dissipative density wave. One may ask the question if at some external magnetic field this mode becomes fully non-dissipative (real ω). In our previous publication [6] we were able to find such a mode in a weak external magnetic field limit and for finite momenta (beyond n = 1 approximation). It is obviously interesting to explore the effect more rigorously, and confirm the result beyond the weak field approximation. This is one of the subjects of our second study.
All TCs in (16,17,18,19) are computed numerically, see section III B and III C. Here, we present their large-κB and large-κE behavior: Just like σ 0 χ [κB = 0], at large-κE, D 0 [κB = 0] decays much faster than other TCs and asymptotically does not scale as a power function of κE.
C. Summary of the results: Part II In this part, we perform the gradient resummation taking into account inhomogenuity and time-dependence of the charge densities ρ, ρ 5 . That is, the constitutive relations will be constructed to all orders in λ. The resummation technique was developed in [14,[16][17][18][19][20]. To this goal we neglect the terms nonlinear in ρ, ρ 5 . Consequently, the currents are parameterised by four TCFs (only the case E = 0 will be considered here), Beyond some critical value of κB 0.5, these TCFs display singular behavior at certain values of real ω, identified with quasi-normal modes (QNMs). These QNMs become real at large B [29,31], their B dependence is that of a Landau level. These real modes lead to a phenomenon of anomalous resonance [32].
The resummed constitutive relations (24,25) give rise to exact dispersion relations for the CMW, beyond small frequency/momenta approximations of (10,20,21), Since the TCFs are complex functions of the frequency/momenta, (26) is expected to have infinitely many solutions, including many gapped modes. In section IV, we will demonstrate that there are purely real, and thus non-dissipative solutions to (26), similar to the ones discovered by us in [6] based on weak magnetic field analysis.
The rest of this paper is structured as follows. Section II presents the holographic setup.
Section III introduces calculational details for Part I and displays numerical results for the TCs mentioned above. Section IV presents the results for Part II. Section V contains concluding remarks.
The bulk action is where and the counter-term action S c.t. is Above, (F V ) M N and (F a ) M N denote the field strengths for the vector V and axial A gauge fields in the bulk, respectively. κ is Chern-Simons coupling. The terms proportional to κ mimic the chiral anomaly of the boundary field theory.
In the ingoing Eddington-Finkelstein coordinate, the line element of the metric of Schwarzschild-AdS 5 is where the blackening factor is f (r) = 1 − 1/r 4 . The Hawking temperature identified as a temperature of the boundary theory is normalised to πT = 1.
The bulk equations of motion are where the radial and boundary components of (31, 32) correspond to dynamical and constraint equations, respectively. The boundary currents read The radial gauge V r = A r = 0 is imposed. The boundary currents (33) can be determined by solving the dynamical equations only, leaving the constraints aside. The dynamical equations are sufficient to fix all the TCs/TCFs while the constraints translate into the continuity equations for the boundary theory currents (2). We use the following ansatz for the bulk gauge fields [3,15], Here V µ is a gauge potential of the external e/m fields E and B, Both V µ and ρ, ρ 5 are assumed to be known functions of the boundary coordinates, while the dynamical equations of motion are solved for V µ , A µ as functionals of V µ and ρ, ρ 5 .
Boundary conditions are classified into three types. First, V µ and A µ are regular over the domain r ∈ [1, ∞). Second, at the boundary r = ∞, we require Additional integration constants will be fixed by employing the Landau frame convention for the boundary currents According to the holographic dictionary, the boundary currents are determined in terms of near boundary (r = ∞) pre-asymptotic expansion of the bulk gauge fields: µ are the coefficients of r −2 terms in the near boundary expansion of the bulk vector V µ and axial A µ gauge fields, respectively.
Thus, our program boils down to integrating the dynamical bulk equations, from the horizon to the boundary, for the bulk gauge fields and determining their near boundary asymptotic behaviour encoded in the coefficients V (2) µ and A (2) µ . This has to be done for our specific setup of constant e/m fields corresponding to linear potential V of arbitrary strength.

III. PART I: GRADIENT EXPANSION IN EXTERNAL E/M FIELDS
In this section we study the non-perturbative E-, B-dependencies of the TCs introduced in Section I B, (up to first order in the gradient expansion). The background fields E and B are treated as zeroth order in the gradient expansion, as opposed to our previous studies [3,6]. Introducing λ as a gradient expansion parameter (by ∂ µ → λ∂ µ ), the bulk fields V µ and A µ are expandable in powers of λ, Hence the currents J and J 5 are expanded in λ too, see (3). In what follows, we compute A.

Constitutive relations at zeroth order
The dynamical equations for V where the Landau frame convention (37) has been used to fix one integration constant both t . The dynamical equations for the remaining components V i take the form: where the decomposition coefficients C i (i = 1 − 6) are functions of r only, satisfying a system of coupled ordinary differential equations (ODEs). These ODEs could be grouped into two partially decoupled sub-sectors: When the solutions (44, 45) are substituted into holographic expression (38) for the currents, one obtains the zeroth order constitutive relations (4,5) with the TCs given by the near boundary expansion of the decomposition coefficients C Here c When was found analytically in [4,28,29], yielding an analytic expression (11) for σ 0 χ . We are able to compute the remaining TCs in (4,5) numerically only. Prior to demonstrating new results for the field dependent TCs, we quote their values at vanishing e/m fields (see [5] and the references therein): where C ≈ 0.915966 is the Catalan's constant.
In Figure 1, 3D plots for all the TCs in (4,5) are shown as functions of κE and κB, first focusing on the case of E B. Influence of the relative angle θ between E, B will be discussed later (see Figure 4). Figure 1a displays κE-and κB-dependence of the Ohmic conductivity σ 0 e . In our previous publications [3][4][5][6]15], due to weak field assumption, the Ohmic conductivity σ 0 e did not depend on external e/m fields at all. Here, we observe that σ 0 e gets enhanced when E · B = 0. While the Ohmic conductivity dependence on external electric fields was already considered in holography [33][34][35], to the best of our knowledge, anomaly-induced corrections to σ 0 e found here have not been reported before. In [33], the nonlinear conductivity emerged from a gravitational back-reaction effect, while in [34,35] it emerged from a coupling between the bulk gauge fields and an additional charged scalar.  Overall, the nonlinear conductivities of [33][34][35] decrease with strengthening of the electric field, while, as demonstrated in the present work, the anomaly-induced effect has totally opposite signature.
All the other TCs decrease dramatically with increasing κE, κB, and vanish asymptotically. The results are presented normalised with respect to their values at vanishing κE and κB. The TC δσ 0 χ turns negative in a certain region of κE and κB (Figure 1c). Figure 2 helps to extract the asymptotic behaviour of the TCs when either κE or κB is very large. The TCs are rescaled so that the asymptotic scaling summarised in (13,14) becomes transparent. Figure 3 displays rescaled σ 0 e on (κE = κB)-slice of Figure 1a. Finally, in Figure 4, we examine the effect of the relative angle θ between E and B on the TCs in (4,5). In Figure 4, the TCs are normalised with respect to their values when χe and σ 0⊥ s . Two values of the e/m fields are considered: κB = κE = 0.25 ( Figure 4a) and κB = κE = 0.5 (Figure 4b). We observe that the stronger the fields the more pronounced the dependence of the TCs on cos 2 θ. It is also very clear that while some TCs become weaker others become stronger.
Constitutive relations for J and J 5 at first order are derived in next two Subsections III B and III C. To simplify the algebra, two cases are explored separately, either E = 0 or B = 0.
Furthermore, only the terms linear both in ρ and ρ 5 are considered. It is important to stress that smallness of the charge densities is a self-consistent approximation: when E · B = 0, there is no pumping of the axial charge into the system through the continuity equation (2). µ are The solutions are The decomposition coefficients C (1) i obey partially decoupled ODEs, which we group into two decoupled sub-sectors: where F 1 parameterises solutions to V t (see (40, 41)): Near the boundary (r = ∞), the asymptotic expansion of the decomposition coefficients Plugging (66) into (38), the first order constitutive relations (16,17) are obtained with the TCs related to the boundary data as Alternatively, (16,17) can be re-expressed by collecting terms involving ∇ρ, ∇ρ 5 into transverse and longitudinal parts (with respect to B): where For generic values of κB, we are able to solve the ODEs (60-65) numerically only. The results are presented in Figure 5. The TCs are normalised to their values at vanishing e/m fields (see [5]) quoted below and also rescaled by their asymptotic behaviour: Obviously, all the TCs in (16,17) decrease when κB becomes stronger, and vanish asymptotically. Large-κB asymptotic behavior for all the TCs in (16,17) is summarised in (22). The equations for V The solutions are where the decomposition coefficients satisfy coupled ODEs: Near r = ∞, the asymptotic expansion for C (1) i is, which via (38) helps to derive the first order constitutive relations (18,19) with the TCs given by As done for (16,17), we combine the ∇ρ, ∇ρ 5 -terms in (18,19) into longitudinal and transverse parts (with respect to E), where For generic value of κE, the ODEs (75-77) are solved numerically and the results are displayed in Figure 6. Analytical results for each TCs when E = B = 0 are quoted for completeness (see [5]): From Figure 6, one can read off the large-κE behavior for the TCs in (18,19), as summarised in (23). We note that large-κE behavior of D 0 does not scaled as a power function and it decays faster.

IV. PART II: GRADIENT RESUMMATION IN EXTERNAL MAGNETIC FIELD
In this section, we focus on all-order resummation of gradient terms that are linear in both ρ and ρ 5 when B is taken as arbitrary in amplitude. External electric field E will be turned off throughout this section. In order to keep terms linear in the charge densities only, we introduce a parameter , We solve the dynamical equations up to O( 1 ): The equations (86, 88) could be integrated over r: where the Landau frame convention (37) has been used to fix the integration constants.
At O( 1 ), the dynamical equations become The corrections V (1) µ and A (1) µ are decomposed in terms of basic structures built from the external magnetic field B and the charge densities ρ, ρ 5 , where, in contrast to our previous publications [3,6], the decomposition coefficients S i ,S i , V i andV i now depend on B non-linearly. Particularly, they are scalar functionals of ∂ t , ∇ 2 and (κ B · ∇) 2 ; and functions of r and (κB) 2 . In Fourier space via (∂ t , ∇) → (−iω, i q), they turn into scalar functions Accordingly, the PDEs (92-95) give rise to a set of partially decoupled ODEs for the decomposition coefficients in (96-99), which we collect as (ii): The sub-sector S 1 , S 2 ,V 1 ,V 2 , V 3 , V 4 satisfies the same equations as the sub-sector (i): S 1 ,S 2 , V 1 , V 2 ,V 3 ,V 4 . Given that they obey the same boundary conditions, we conclude The remaining functionsV 5 , V 6 andV 6 satisfy the same ODEs as V 5 . Note that the ODEs for V 5 , V 6 ,V 5 ,V 6 are homogeneous, they do not have non-trivial solutions due to regularity requirement (at r = 1) and vanishing boundary condition (at r = ∞): Indeed, the conclusion V 5 =V 5 = 0 is in perfect agreement with [5,6], where V 5 ,V 5 were found to depend on ρ, ρ 5 nonlinearly.
Solving (104-109) near the boundary r = ∞ gives rise to currents' constitutive relations (24,25) with the TCFs related to the near boundary expansion of V i 's: where v i 's denote the coefficients of 1/r 2 in the near-boundary expansion of V i 's. In contrast to our previous publications [3,6], the TCFs D, D B ,σχ and D χ also depend on the external magnetic field B non-perturbatively, Determination of these TCFs is a novel result of the present work. Since the external magnetic field is constant, the TCFs in (24) measure a response of the chiral medium to inhomogenuity and time-dependence of the charge densities ρ, ρ 5 .

B. The TCFs
We are not able to solve the ODEs (104-109) analytically and thus resort to numerical techniques. The results are summarised in several plots below. First, we fix the magnetic field B and its relative angle α with respect to q, and study the ω, q 2 -dependence of the TCFs in (24,25). Then, we examine the effects due to the magnetic field B and angle α variations.
In Figures 7 and 8, selecting representative values κB = 0.25 and α = 0, we display 3D plots for all the TCFs in (24,25) as functions of ω and q 2 . All the TCFs exhibit a similar behavior: relatively weak dependence on the spatial momentum squared q 2 , reflecting spatial quasi-locality of relevant transport processes; damped oscillations in the frequency ω, towards asymptotic region where all the TCFs essentially vanish.  results are normalised with respect to their values at α = π/2 denoted as D ⊥ ,σ ⊥ χ , D ⊥ B , D ⊥ χ . In Figure 9, κB = 0.5 and ω = q 2 = 0.1. All the TCFs display a relatively mild dependence on cos 2 (α).
In Figure 10 we show the TCFs as functions of κB while fixing all the rest of the parameters ω, q 2 and α. Since the latter effect is weak, we take α = 0 in the subsequent discussion. Regarding ω and q 2 , we make two different choices to implement a comparative study: ω = q 2 = 0.1 versus ω = q 2 = 2. In Figure 10, the TCFs are normalized with respect to their values when κB = 0 with the same ω and q 2 values (which are obtained from our numerical results). All the TCFs are found to approach zero at large magnetic field. When ω, q 2 are increased (i.e., with more non-hydrodynamic modes included), the asymptotic regime is shifted towards larger values of κB.
An interesting new phenomenon emerges at finite κB: when magnetic field is large, say To conclude this part, we present the dependence of the lowest QNM mode on κB, see Figure 12. The numerical results (black dots) are best fitted by: which is plotted as a continuous (blue) curve in Figure 12. The mode with Re(ω) ∝ √ κB while Im(ω) → 0 is a manifestation of the Landau level behavior [29,31]. Magnetic field dependence of the QNMs in a holographic model was first studied in [36,37] but without any anomaly effects included.

C. Non-dissipative CMW modes
The original CMW is a dissipative wave at small momenta. In [6], we asked the question if it can happen that, beyond the hydrodynamic limit, the dissipative (imaginary) part of the CMW vanishes. The answer is within the all order dispersion relation. We indeed found that the CMW possesses a discrete spectrum of non-dissipative modes when the magnetic field is larger than a critical one. Yet, the results of [6] were based on a "weak" magnetic field approximation. Below we reexamine the very same question, but now without any approximations involved.
The constitutive relations (24,25), combined with continuity equations (2), result in the exact CMW dispersion relation The last term is new compared to [6]. Furthermore, all the TCFs in (117) are functions of the magnetic field.
The procedure for finding purely real solution was devised in [6]. To this goal, we first split the dispersion relation (26) into real and imaginary parts (assuming B q) Then we search for a real ω solution of two equations φ R = 0 and φ I = 0.
In Figure 13, the functions φ R and φ I (with upper plus sign in (118)  and φ I = 0 (bold solid red curve) implies that both functions vanish simultaneously. This is the desired solution.
In [6], such a crossing occurred at a "single point" in the (ω, q 2 ) space, for each κB ≥ 0.33, similarly to the the case shown in Figure 13a. However, beyond the weak field approximation, new branches emerge as demonstrated in Figures 13b, 13c and 13d: the contours φ R = 0 and φ I = 0 coincide within a "continuous interval" in the (ω, q 2 ) space, starting from a large enough κB. This implies a continuum of non-dissipative CMW modes. In Figure 13d, it seems like the contours φ R = 0 and φ I = 0 coincide completely, though the Figure is somewhat misleading. Indeed, by zooming in (ω, q 2 ) space, we observe that the solution exists only on piece-wise intervals in the (ω, q 2 ) space. It is also worth adding that there are additional solution branches at larger ω, which we do not display here.
Finally, the results above were obtained for non-dissipative CMW propagating along the magnetic field direction. One can ask the question if similar wave could propagate at an angle α with respect to the magnetic field, or even orthogonal to it. We briefly report that changing α from q B to q ⊥ B leads to separation of φ R = 0 and φ I = 0 contours.

V. CONCLUDING REMARKS
In this work we focused on influence of strong background e/m fields on chiral anomalyinduced transport phenomena for a holographically defined thermal plasma. Constitutive relations for the vector and axial currents J, J 5 were evaluated within two complimentary approximation schemes: a fixed order gradient expansion (up to first order) and an all-order gradient resummation (linear in the charge densities ρ, ρ 5 ). A summary of all results could be found in the introductory section. The main highlights of our study are: • There are three types of gapless modes propagating in the chiral plasma: the chiral magnetic wave (CMW) [27], the chiral electric wave (CEW) [23] and the chiral Hall density wave (CHDW) [5,6], which could be searched experimentally in heavy ion collisions or, more likely, in condensed matter experiments where external fields are under better control.
• While most of the transport coefficients are found to be suppressed by the external fields and vanish at asymptotically strong fields, the Ohmic conductivity gets enhanced in parallel electrical and magnetic fields, which is an experimentally interesting phenomenon to be searched for.
• Some anomaly-induced transport phenomena display noticeable dependence on relative angle between the external fields E and B. This sensitivity could be used in real experiments to zoom into one or another anomaly-induced phenomena.
• When E = 0, the all-order resummed constitutive relations (24,25) are parameterised by four independent TCFs, which are functions of ω, q and B. Intriguingly, these TCFs are found to show a common singularity at certain value of real ω when κB is strong enough.
Moreover, this singularity is identified as QNM frequency and obey Landau level behavior (116) as a function of κB.
• In [6] we discovered a discrete set of entirely non-dissipative and thus long-lived CMW modes emerging in a weak magnetic field. Present work examined CMW exactly without the weak field approximation. We found that, depending on magnetic field, the discrete set extends into several continuous intervals in (ω, q 2 ) space. This effect should have a clear experimental signature worth exploring.
Real experiments involving chiral plasma, such as the one produced in heavy ion collisions or primordial plasma in early Universe, involve strong dynamical fields with nonhomogeneous profiles. While the study reported above was limited to constant background fields, it revealed important new anomaly-induced phenomena and serves as a step towards development of a self-consistent chiral MHD. Any dynamical simulations of the latter are beyond the scope of this paper.