Anomalous transport from holography: Part II

This is a second study of chiral anomaly induced transport within a holographic model consisting of anomalous $U(1)_V\times U(1)_A$ Maxwell theory in Schwarzschild-$AdS_5$ spacetime. In the first part, chiral magnetic/separation effects (CME/CSE) are considered in presence of a static spatially-inhomogeneous external magnetic field. Gradient corrections to CME/CSE are analytically evaluated up to third order in the derivative expansion. Some of the third order gradient corrections lead to an anomaly-induced negative $B^2$-correction to the diffusion constant. We also find non-linear in $B$ modifications to the chiral magnetic wave (CMW). In the second part, we focus on the experimentally interesting case of the axial chemical potential being induced dynamically by a constant magnetic and time-dependent electric fields. Constitutive relations for the vector/axial currents are computed employing two different approximations: (a) derivative expansion (up to third order) but fully nonlinear in the external fields, and (b) weak electric field limit but resuming all orders in the derivative expansion. A non-vanishing non-linear axial current (CSE) is found in the first case. Dependence on magnetic field and frequency of linear transport coefficient functions (TCFs) is explored in the second.


Introduction and summary
Chiral anomalies emerge and play an important role in relativistic QFTs with massless fermions. The anomaly is reflected in three-point functions of currents associated with global symmetries. When the global U (1) currents are coupled to external electromagnetic fields, the triangle anomaly renders the axial current into non-conserved, where J µ /J µ 5 are vector/axial currents, and κ is an anomaly coefficient. For SU (N c ) gauge theory with a massless Dirac fermion in fundamental representation, κ = eN c /(24π 2 ) , and e is an electric charge which below will be set to unit. Here E, B are external electromagnetic fields.
The chiral separation effect (CSE) [37,38] is another interesting phenomenon induced by the anomalies. It is reflected in separation of chiral charges along external magnetic field at finite density of vector charges. Chiral charges can be also separated along external electric field, when both vector and axial charge densities are nonzero, the so-called chiral electric separation effect (CESE) [39,40]. (1.4) where ρ/ρ 5 are vector/axial charge densities, the underlined terms in J i /J i 5 are the chiral magnetic/separation effects. G i , H i contain derivatives of ρ, ρ 5 , B. It is important to stress that (1.3,1.4) are exact, without any approximations for ρ, ρ 5 , B. The nonlinearity of the CME/CSE in external magnetic field B is completely accounted for by the chemical potentials µ, µ 5 . The non-derivative part of (1.3) is consistent with the "non-renormalisability" of CME [59,60]. However, as will be clear from (1.6,1.7), the derivative corrections introduce new effects, which do modify the original CME. Particularly, the currents along the direction of B get affected.
When ρ, ρ 5 , B vary slowly from point to point, G i , H i can be calculated order-by-order within derivative expansion. Let us introduce a scaling parameter λ: (1.5) Then, derivative counting is by powers of λ. Up to second order in the derivative expansion, G i , H i are (throughout this work, the electromagnetic fields are thought of as of first order in derivative counting) (1.6) π + 3 log 2 κ∂ t ρB i + 36 (1 − 2 log 2) κ 2 ρρ 5 ijk ∂ j B k + 18 (2 − 3 log 2) κ 2 ijk (ρ 5 ∂ j ρB k + ρ∂ j ρ 5 B k ) + O(∂ 3 ). (1.7) Up to O(∂ 2 ), the chemical potentials are At third order O(∂ 3 ), for G i , H i we calculated only terms that are linear in ρ, ρ 5 , see (3.22,3.23) for a complete listing. Among these third order terms, the diffusion constant D 0 (i.e., the DC limit of the diffusion function D) gets a negative correction To the best of our knowledge, this is the first anomaly-induced correction to the diffusion constant and being negative it happens to violate the universal form of [61].
With the third order results for J µ and J µ 5 , we computed the dispersion relation for a free mode that can propagate in the medium: (1.10) The first term in (1.10) represents the chiral magnetic wave (CMW) [59]. Interestingly, we see nonlinear in B corrections to both the speed of CMW and its decay rate. Note that we also expect emergence in (1.10) of the following terms ( q · B) 2 , q 2 ( q · B), ( q · B) 3 , q 2 ( q · B) 2 and ( q · B) 4 . However, our ability to determine coefficients of these terms is limited by the undertaken approximations.
In the second part of this work, we focus on a special setup which is experimentally accessible in condensed matter systems 2 . CME emerges from a nonzero axial chemical potential µ 5 , which is usually assumed to have some background profile. It is, however, possible to induce ρ 5 (and thus µ 5 ) dynamically through interplay between the electric and magnetic fields, as clear from the continuity equation (1.1). Specifically, we are ready to consider a constant magnetic field B and a time-dependent but spatially-homogeneous electric field E(t). For simplicity the charge densities ρ, ρ 5 will be assumed to be spatiallyhomogeneous too 3 . From (1.1), ρ could be set to zero. The constitutive relations for the vector/axial currents are where V j (1), A j (1), G i and H i depend on ρ 5 , E and B nonlinearly and will be computed below. Our study is further split into two parts. In section 4.1, V j (1), A j (1), G i and H i will be evaluated perturbatively within the gradient expansion (1.5). In section 4.2, we will consider another approximation-linearisation of the constitutive relation in the external electric field. Within the gradient expansion, up to the order O ∂ 3 , (1.11,1.12) are (1.13) (1.14) where C is a Catalan constant and # 1 is known numerically only Up to second order in derivatives O(∂ 2 ), the chemical potentials are 4 We thank Dmitri Kharzeev for proposing us this study. 3 In principle it is not excluded that the charge densities ρ, ρ 5 could be spatially-inhomogeneous. Yet such spatial inhomogeneity would render the derivative resummation highly complicated. 4 While we suspect that the chemical potential µ is zero to all orders in the gradient expansion, we have not been able to prove that.
Evaluated on shell via (1.1), the axial current J i 5 is fully nonlinear in the amplitude of the electric field E(t), as clear from (1.14).
In the linearised regime, we assume the following scaling for the fields ρ 5 , E and B which will be referred to as amplitude expansion. To linear order in , the vector/axial currents are where σ e is a q 2 = 0 limit of the electric conductivity introduced in (1.2), while τ 1,2 are new TCFs. As with other TCFs, they are functionals of time derivative operator and become functions of frequency ω in Fourier space, Imposing the continuity equation (1.1), the electric current is put on-shell, where the transverse conductivity σ T is not affected by the magnetic field in contrast to the longitudinal conductivity σ L which gets corrected by the magnetic field via the chiral anomaly. Particularly, in the DC limit where σ 0 e = 1, and τ 0 1 and τ 0 2 are where Γ[z] is a Gamma function, and # 2 is known numerically only τ 1 1 is the coefficient of iω in hydrodynamic expansion of τ 1 . When the magnetic field is very strong, τ 0 1 and τ 0 2 behave similarly The result for τ 0 1 is in agreement with [62,63]. Numerically computed B-dependence of τ 1 1 and τ 0 2 and ω-dependence of τ 1 , τ 2 will be presented in section 4.2.
In our calculation, Re σ 0 L acquires negative correction due to magnetic field and eventually vanishes when the magnetic field gets large, see Figure 1. This is in contrast with many related studies of negative magnetoresistivity, the phenomenon of enhancement of longitudinal DC conductivity due to magnetic field [64][65][66][67][68][69]. However, taking strict DC limit in σ 0 L is problematic due to the explicit 1/ω divergence. The latter is frequently regularised by introduction of axial charge dissipation effects via shifting the frequency ω → ω + i/τ 5 , where τ 5 corresponds to some relaxation time. The physics of this axial charge relaxation is beyond the scope of the present work. It was addressed within the holographic approach in [62,[70][71][72]. These studies primarily rely on the Kubo formula. Utilising the weak electric field approximation (1.17), [70] analytically evaluated the magnetic field dependence of the longitudinal conductivity σ L in DC limit, while [62] calculated its ω-dependence. Backreaction effects on σ L were considered in [72]. Ref. [63] performed similar study focusing on time evolution of the induced vector current, given some specially chosen initial profile for the electric field.
While there is some overlap between our results and the literature, differences between the present study and those of [62,[70][71][72] must be clarified. All the studies [62,[70][71][72] focused on a weak electric field, in which the axial current vanishes. So, our nonlinear results and particularly the axial charge separation current (1.14) appear as new. As for the linearised setup (1.17), [62,[70][71][72] imposed the continuity equation and replaced the axial charge density ρ 5 in favour of the external electric and magnetic fields, so the vector current there is on-shell. This is in contrast to our off-shell formalism. As we argued in our previous publications [51,[53][54][55][56], only off-shell construction reveals transport properties of the system in full. Particularly, there are three independent TCFs (σ e and τ 1,2 ) in the constitutive relation (1.18), all of which we are able to determine separately, compared to only two independent conductivities in (1.20).
Another difference worth mentioning is that we explicitly trace all the effects in the induced current that arise from the relative angle between E(t) and B fields. This is in contrast to [62,63], which limited their study to the case of parallel fields only, primarily focusing on the longitudinal electric conductivity σ L . By varying the relative angle between E(t) and B fields, one can separate the anomaly induced effects (parametrised by τ 1 and τ 2 ) from the ones that are not related to the anomaly (σ e ).
The paper is structured as follows. In section 2 we present the holographic model and outline the strategy of deriving the boundary currents from solutions of the anomalous Maxwell equations in the bulk. Section 3 presents the first part of our study: CME/CSE with static but varying in space magnetic field. In section 4, CME/CSE in the presence of constant magnetic and time-varying electric fields are analysed. This study is further split into two subsections. Exploration of nonlinear phenomena in the induced vector/axial currents is done in 4.1. In section 4.2 we focus on the linearised regime (1.17) and calculate the dependence of AC conductivity on magnetic field. The last section 5 presents the conclusions.
2 The holographic model: The holographic model is the U (1) V × U (1) A theory in the Schwarzschild-AdS 5 . The chiral anomaly of the boundary field theory is modelled via the gauge Chern-Simons terms in the bulk action M N P QR is the Levi-Civita symbol with the convention rtxyz = +1, and the Levi-Civita 3) is based on minimal subtraction, that is the counter-term does not make finite contribution to the boundary currents.
In the ingoing Eddington-Finkelstein coordinates, the spacetime metric is where f (r) = 1 − 1/r 4 , so that the Hawking temperature (identified as temperature of the boundary theory) is normalised to πT = 1. On the constant r hypersurface Σ, the induced metric γ µν is Equations of motion for V and A fields are The boundary currents are defined as which, in terms of the bulk fields, are where n M is the outpointing unit normal vector on the slice Σ, and ∇ is compatible with the induced metric γ µν . The currents (2.10) are defined independently of the constraint equations (2.7). Throughout this work, the radial gauge V r = A r = 0 will be assumed. Consequently, in order to completely determine the boundary currents (2.11) it is sufficient to solve the dynamical equations (2.6) for the bulk gauge fields V µ , A µ only, leaving the constraints aside. The constraint equations (2.7) give rise to the continuity equations (1.1). In this way, the currents' constitutive relations to be derived below are off-shell.
It is useful to reexpress the currents (2.11) in terms of the coefficients of near boundary asymptotic expansion of the bulk gauge fields. Near r = ∞, In (2.12) the constant term for A µ is set to zero given that axial external fields are turned off in our present study. The holographic dictionary implies that V µ is a gauge potential of external electromagnetic fields E and B, (2.14) When obtaining (2.12,2.13), only the dynamical equations (2.6) were utilised. The nearboundary data V (2) µ and A (2) µ have to be determined by completely solving (2.6) from the horizon to the boundary. The currents (2.11) become The remainder of this section is to outline the strategy for deriving the constitutive relations for J µ and J µ 5 . To this end, consider finite vector/axial charge densities exposed to external electromagnetic fields. Holographically, the charge densities and external fields are encoded in asymptotic behaviors of the bulk gauge fields. In the bulk, we will solve the dynamical equations (2.6) assuming some charge densities and external fields, but without specifying them explicitly.
Following [51] we start with the most general static and homogeneous profiles for the bulk gauge fields which solve the dynamical equations (2.6), where V µ , ρ, ρ 5 are all constants for the moment. Regularity requirement at r = 1 fixes one integration constant for each V i and A i . As explained below (2.13), the constant in A µ is set to zero. Through (2.15), the boundary currents are Hence, ρ and ρ 5 are identified as the vector/axial charge densities.
Next, following the idea of fluid/gravity correspondence [57], we promote V µ , ρ, ρ 5 into arbitrary functions of the boundary coordinates Then, (2.16) ceases to be a solution of the dynamical equations (2.6). To have them satisfied, suitable corrections in V µ and A µ have to be introduced: where V µ , A µ will be determined from solving (2.6). Appropriate boundary conditions have to be specified. First, V µ and A µ have to be regular over the whole integration interval of r ∈ [1, ∞]. Second, at the conformal boundary r = ∞, we require which amounts to fixing external gauge potentials to be V µ and zero (for the axial fields).
Additional integration constants will be fixed by the Landau frame convention for the currents, The Landau frame choice can be identified as a residual gauge fixing for the bulk fields. The vector/axial chemical potentials are defined as Generically, µ, µ 5 are nonlinear functionals of densities and external fields. In terms of V µ and A µ , the dynamical equations (2.6) are (2.26) In the following sections we will present solutions to the dynamical equations (2.23-2.26) under two setups discussed in the Introduction.

CME/CSE with time-independent inhomogeneous magnetic field
In this section we consider the case in which the magnetic field is the only external field that is turned on. The magnetic field is assumed to be varying in space, but it should be time-independent to avoid creating an electric field. There is no restriction on charge densities ρ, ρ 5 . From the general results (2.12,2.13), In obtaining large r estimates for V t and A t , the frame convention (2.21) was used to fix the coefficients of 1/r 2 in near-boundary expansion for V t , A t (thus those of V t and A t ).
The dynamical equations (2.23-2.26) could be put into integral forms

4)
where µ and µ 5 are the chemical potentials defined in (2.22). The frame convention (2.21) was utilised to fix integration constants, one for V t and one for A t . The functions G i (x) µ and A [n] µ up to n = 2 are listed below.

13)
A [2] t = V [2] t (ρ ↔ ρ 5 ) , (3.14) Substituting the first order solutions (3.10-3.12) into (3.6,3.7), we obtain the results (1.6,1.7). Meanwhile, the second order results (3.13,3.14) give rise to the expansion of the chemical potentials (1.8). In principle, the second order results (3.13-3.16) could be inserted into (3.6,3.7), producing derivative expansion for G i (x = ∞) and H i (x = ∞) up to third order. However, at third order O(∂ 3 ), computing G i , H i becomes quite involved. So, at third order O(∂ 3 ) we decided to track only linear in ρ, ρ 5 terms. As a result, we are able to identify the first anomalous correction to the diffusion constant D 0 due to magnetic field. The final expressions are G [3] i (x = ∞) =
Our results for J µ and J µ 5 can be used to explore dispersion relations for free modes propagating in the chiral medium. We consider a constant magnetic field only. Let us take a plane wave ansatz for the vector/axial charge densities ρ = δρ exp (−iωt + q · x) , ρ 5 = δρ 5 exp (−iωt + q · x) . which has a nontrivial solution when and only when

CME/CSE with constant magnetic and time-dependent electric fields
Creating systems with chiral imbalance (µ 5 = 0) experimentally is problematic. In this section we consider a special setup in which the axial chemical potential µ 5 is not imposed externally but rather induced dynamically through chiral anomaly. This setup is of particular interest due to intriguing possibility for it to be realised experimentally in chiral condensed matter systems. Consider a constant magnetic field B and a time-dependent homogeneous electric field E(t). We also assume the charge densities to be spatially-homogeneous as well 5 . The continuity equation (1.1) degenerates to which implies that the vector charge density is constant while the axial charge density has nontrivial time dependence inherited from E(t). The setup under consideration is Under the frame convention (2.21), the corrections V µ and A µ of (2.19) depend on r and t only. As a result, the dynamical equations (2.23-2.26) are reduced to (4.6)

Non-linear phenomena: general analysis and derivative expansion
The dynamical equations (4.3-4.6) can be put into integral forms where µ, µ 5 are defined via (2.22). G i and H i are where V i = V i + E i /r. From (2.15), the near-boundary expansion in (4.7-4.10) is translated into boundary currents (1.11,1.12). For generic boundary conditions, the quantities V i (1), A i (1), G i (x = ∞) and H i (x = ∞) cannot be computed analytically. To proceed, we perturbatively solve the dynamical equations (4.3-4.6) under the derivative expansion (1.5), as done in section 3. Up to second order O(∂ 2 ), the corrections V µ and A µ are 14)

Linear in E phenomena
In the previous subsection we focused on hydrodynamic regime, in which we were able to identify some non-linear phenomena. Below, we proceed with an alternative approximation, that is the weak electric field approximation (1.17): The scaling of ρ 5 follows from the continuity equation (4.1). Both corrections V µ and A µ are of order O( ) too. The dynamical equations (4.3-4.6) get further simplified Integrating (4.21,4.23) over r once, we get where the frame convention (2.21) was used to fix the integration constant. (4.25) makes it possible to decouple V i , A i from V t , A t . Consequently, (4.22,4.24) become Homogeneity property of (4.27), combined with the regularity requirement at r = 1 and vanishing boundary condition at r = ∞ for A i , fixes A i = 0 completely. From (4.25), Therefore, at order O( ), the axial current J 5 = 0. This is in contrast with the nonlinear analysis of section 4.1.
V i is decomposed as 6 where The decomposition coefficients C i 's satisfy partially decoupled ordinary differential equations (ODEs), While C 1 does not feel the effect of magnetic field, C 2,3 have nontrivial dependence on the magnetic field via κ 2 B 2 . Near r = ∞, pre-asymptotic expansions of C i 's are where c i 's are boundary data and have to be fixed through full solution of (4.31-4.33) from the horizon r = 1 to the conformal boundary r = ∞. From (2.15), the conductivities of (1.18) are determined by the boundary data c i 's, The ODE for C 1 was solved in [51]. The conductivity σ e , which is computed from C 1 was completely determined and explored in [51], while only q = 0 limit enters into our current study (the results are quoted below). We therefore focus on the remaining two conductivities τ 1 , τ 2 , both induced by the chiral anomaly. As is obvious from (4.31-4.33), τ 1 , τ 2 depend on the magnetic field via κ 2 B 2 .
Using the continuity equation (4.1), the constitutive relations (1.18) are put into a linear response form, from which on-shell current-current correlators can be read off. Since the electric field is the only external perturbation that is turned on, it is possible to compute only a subset of all two-point correlators in the theory, where J i J j is split into transverse (G T ) and longitudinal (G L ) components with respect to the direction of B.
To determine the remaining current-current correlators we would have to introduce additional field perturbations, particularly an axial external field, which is beyond the scope of this paper.
To evaluate the TCFs τ 1 , τ 2 , we have to completely solve ODEs (4.31-4.33). We first analytically solve them when ω = 0. As a result, the DC limits τ 0 1 (for arbitrary B) and τ 0 2 (up to leading B 2 -correction) are known analytically, as quoted in (1.22,1.23). The coefficient # 2 in (1.23) is For illustration, in figure 1 we show κB-dependence of τ 0 1 , τ 0 2 (divided by 5 to match scales), τ 1 1 and Re σ 0 L . The behaviour of Re σ 0 L agrees with that of [62]. Strong magnetic field limit of τ 0 1 , τ 0 2 was also computed, and the results are summarised in (1.25). In the DC limit ω → 0, when the magnetic field is very strong the on-shell vector current behaves as which is in agreement with [63]. When ω → 0 (DC limit), the current-current correlator is dominated by the chiral anomaly induced effects ∼ τ 0 1 . The DC limit is of interest for experiments with electric fields turned on adiabatically, such as the ones considered in [63].
For arbitrary ω, we resort to numerical methods and solve ODEs (4.31-4.33) for representative values of κB. The numerical procedure is identical to that of [52] and for all the numerical details we refer the reader to this publication. In Figure 2 we show ω-dependence for τ 1 and τ 2 for sample choices of κB. In Figure 3 we plot the normalised TCFs τ 1 /τ 0 1 and τ 2 /τ 0 2 . Overall, τ 1 and τ 2 display quite similar dependence on the frequency ω. After some oscillations, both τ 1 and τ 2 approach zero asymtotically.
Approach to the asymptotic regime, however, depends on strength of the magnetic field. When κB is increased, the asymptotic behaviour is delayed towards larger ω. What is more intriguing is that increasing κB renders τ 1 and τ 2 to develop a resonance-like enhancement at finite ω. This could be an interesting experimentally observable feature. For very strong magnetic fields κB → ∞, the chiral anomaly-induced effects would be pushed to the UV, corresponding to early time effects, such as in [63].  In Figure 4 we show two-point correlators G T,L for different choices of κB. However, it is difficult to appreciate the anomaly induced effects from Figure 4 because in the correlators they get mixed with non-anomalous ones. To illuminate κB-correction to G L , in Figure  5 we plot the difference δG L = G L − G T . From these plots, the effect of chiral anomaly on the induced vector current is seen more clearly. We again notice a remarkable relative enhancement at intermediate values of ω.

Conclusions
In this paper we continued explorations of the chiral anomaly induced transport within a holographic model containing two U (1) fields interacting via Chern-Simons terms. For a finite temperature system, we computed off-shell constitutive relations for the vector/axial currents responding to external electromagnetic fields. When a static spatially-inhomogeneous magnetic field is the only external field that is turned on, we showed that both the CME and CSE get corrected by derivative terms, see (1.3,1.4). Within the derivative expansion, we analytically calculated corrections up to third order in the expansion, see (1.6,1.7,3.22,3.23). Apart from the derivative corrections to CME and CSE, the diffusion constant D 0 was found to receive a negative anomaly-induced correction, see (1.9). The dispersion relation of the chiral magnetic wave was also found to be modified.
In the second part of our study, we focused on the case of time-varying electric and constant magnetic fields without any externally enforced axial charge asymmetry, though the E(t) · B term in the continuity equation generates the axial charge density ρ 5 (and thus µ 5 ) dynamically. For such configuration of external fields, we first analysed the most general constitutive relations for the vector/axial currents, see (1.11,1.12). Then, within the derivative expansion, we explicitly calculated the currents up to third order at nonlinear level, see (1.13,1.14). When put on-shell, the axial current J 5 is fully nonlinear in the external electric field.
Employing another approximation, we linearised the constitutive relations assuming the electric field to be weak (1.17). Within this approximation the axial current is zero, while the "off-shell" vector current is parameterised by three frequency-dependent transport coefficient functions: the electric conductivity σ e , and two chiral anomaly-induced conductivities τ 1 , τ 2 , see (1.18). In the DC limit, we analytically computed these conductivities, see (1.22,1.23). Then, for generic ω, the numerical plots were presented in section 4.2. Based on these studies, we notice that the anomaly-induced effects get enhanced at some finite frequency ω, whereas the position of the maximum and strength of the effect depends on the external magnetic field. It might be an effect worth looking for experimentally.