Gradient resummation for nonlinear chiral transport: an insight from holography

Nonlinear transport phenomena induced by chiral anomaly are explored within a 4D field theory defined holographically as U(1)V×U(1)A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U(1)_V\times U(1)_A$$\end{document} Maxwell–Chern–Simons theory in Schwarzschild-AdS5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AdS_5$$\end{document}. In presence of weak constant background electromagnetic fields, the constitutive relations for vector and axial currents, resummed to all orders in the gradients of charge densities, are encoded in nine momenta-dependent transport coefficient functions (TCFs). These TCFs are first calculated analytically up to third order in gradient expansion, and then evaluated numerically beyond the hydrodynamic limit. Fourier transformed, the TCFs become memory functions. The memory function of the chiral magnetic effect (CME) is found to differ dramatically from the instantaneous response form of the original CME. Beyond hydrodynamic limit and when external magnetic field is larger than some critical value, the chiral magnetic wave (CMW) is discovered to possess a discrete spectrum of non-dissipative modes.


Introduction
In this paper we continue exploring hydrodynamic regime of relativistic plasma with chiral asymmetries. We closely follow previous works [1,2] focusing on massless fermion plasma with two Maxwell gauge fields, U (1) V × U (1) A . Dynamics of hydrodynamic theories is governed by conservation equations (continuity equations) of the currents. As a result of chiral anomaly, which appears in relativistic QFTs with massless fermions, global U (1) A current coupled to external electromagnetic (e/m) fields is no longer conserved. The continuity equations turn into where J μ , J μ 5 are vector and axial currents and κ is an anomaly coefficient (κ = eN c /(24π 2 ) for SU (N c ) gauge theory with a massless Dirac fermion in fundamental representation and e is electric charge, which will be set to unit from now on). E and B are external vector electromagnetic fields.
The continuity equations could be regarded as time evolution equations for the charge densities ρ (ρ 5 ) sourced by three-current J ( J 5 ). However, these equations cannot be solved as an initial value problem without additional input, the currents J and J 5 . In hydrodynamics, the currents are expressed in terms of thermodynamical variables, such as the charge densities ρ and ρ 5 themselves, temperature T , and the external e/m fields E and B if present. These are known as constitutive relations, which generically take the form J = J [ρ, ρ 5 , T, E, B]; J 5 = J 5 [ρ, ρ 5 , T, E, B]. ( The constitutive relations should be considered as "off-shell" relations, because they treat the charge density ρ (ρ 5 ) as independent of J ( J 5 ). Once (1) is imposed, the currents' constitutive relations (2) are put "on-shell". In addition to the charge current sector discussed above, one has to simultaneously consider energy-momentum conservation. In general, these two dynamical sectors are coupled. However, in the discussion below, we will ignore back-reaction of the charge current sector on the energymomentum conservation. This will be referred to as probe limit.
In the long wavelength limit, the constitutive relations are usually presented as a (truncated) gradient expansion. At any given order, the gradient expansion is fixed by thermodynamic considerations and symmetries, up to a finite number of transport coefficients (TCs). The latter should be either computed from the underlying microscopic theory or deduced experimentally. Diffusion constant, DC conductivity or shear viscosity are examples of the lowest order TCs.
It is well known, however, that in relativistic theory truncation of the gradient expansion at any fixed order leads to serious conceptual problems such as violation of causality. Beyond conceptual issues, causality violation results 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. Below we will refer to such case as all order resummed hydrodynamics [3][4][5][6][7][8]. The first completion of the type was originally proposed by Müller, Israel, and Stewart [9][10][11][12] who introduced retardation effects in the constitutive relations for the currents. Formulation of [9][10][11][12] is the most popular scheme employed in practical simulations. Essentially, all order resummed hydrodynamics is equivalent to a non-local constitutive relation of the type (here we take the charge diffusion current as an example): whereD is the memory function of the diffusion function D(ω, q 2 ) [13], which is generally non-local both in time and space. Causality implies thatD(t) has no support for t < 0. In practice, the memory function is typically modelled: Müller-Israel-Stewart formulation [9][10][11][12] models the memory functions with a simple exponential in time parametrised by a relaxation time. Chiral plasma plays a major role in a number of fundamental research areas, historically starting from primordial plasma in the early universe [14][15][16][17][18]. During the last decade, macroscopic effects induced by the chiral anomaly were found to be of relevance in relativistic heavy ion collisions [19][20][21], and have been searched intensively at RHIC and LHC [22][23][24][25][26]. Finally, (pseudo-)relativistic systems in condensed matter physics, such as Dirac and Weyl semimetals, display anomaly-induced phenomena, which were recently observed experimentally [27][28][29][30][31][32][33] and can be studied via similar theoretical methods [34][35][36][37].
The constitutive relations (2) are well known to receive contributions induced by the chiral anomaly. The most familiar example is the chiral magnetic effect (CME) [38][39][40]: a vector current is generated along an external magnetic field when a chiral imbalance between left-and right-handed fermions is present ( J ∼ ρ 5 B). Another important transport phenomenon induced by the chiral anomaly is the chiral separation effect (CSE) [41,42]: left and right charges get separated along an applied external magnetic field ( J 5 ∼ ρ B). Combined, CME and CSE lead to a new gapless excitation called chiral magnetic wave (CMW) [43]. This is a propagating wave along the magnetic field. There is a vast literature on CME/CSE and other chiral anomaly-induced transport phenomena, which we cannot review here in full. We refer the reader to recent reviews [20,21,34,44,45] and references therein on the subject of chiral anomaly-induced transport phenomena.
Beyond naive CME/CSE, there are (infinitely) many additional effects induced or affected by chiral anomaly. Particularly, transport phenomena nonlinear in external fields were realised recently [46] to be of critical importance in having a self-consistent evolution of chiral plasma. This argument, together with the causality discussions mentioned earlier, would lead to the conclusion that the constitutive relations (2) should contain infinitely many "nonlinear" transport coefficients in order to guarantee applicability of the constitutive relations in a broader regime. Recently, this triggered strong interest in nonlinear chiral transport phenomena within chiral kinetic theory (CKT) [47][48][49][50]. Previous works on the subject of nonlinear chiral transport phenomena include [51] based on the notion of entropy current, and [52] based on the fluidgravity correspondence [53].
The objective of present work is to explore all order gradient resummation for nonlinear transport effects induced by the chiral anomaly, 1 further extending the results of Refs. [1,2,58].
Just like in Refs. [1,2,58], our playground will be a holographic model, that is U (1) V × U (1) A Maxwell-Chern-Simons theory in Schwarzschild-Ad S 5 [59,60] to be introduced in Sect. 3, for which we know to compute a zoo of transport coefficients exactly. Hoping for some sort of uni-versality, we could learn from this model about both general phenomena and relative strengths of various effects.
In our recent publication [58], we reviewed all different studies which were performed in [1,2,58]. Those studies and the present one are largely independent even though performed within the same holographic model. For brevity, we will not repeat this review here, but will make connection to these previous works whenever relevant. We refer reader to [58] for the summary of the different approximations which employed in these series of works and the present work. The comparison of the resultant constitutive relations and our comments about the total current is also presented there.
Anomalous transport phenomenon is frequently discussed from the viewpoint of its dissipative nature and, equivalently, its contribution to entropy production [51,[61][62][63][64]. CME is well known to be non-dissipative [20,34,65]. What about the dissipative nature of other anomalous transport phenomena, say beyond CME? In [51] the transport coefficients that are odd in κ were identified as anomaly-induced and, based on space parity P arguments, are claimed to be non-dissipative. This is to distinguish from anomaly-induced corrections to normal transports, which appear to be even in κ. While the P-based arguments seem to work perfectly for the second order hydrodynamics [51], a more natural criterion of dissipation seems to be based on time-reversal symmetry T . T -odd transport coefficients describe dissipative currents, whereas T -even ones are non-dissipative [51]. The anomalyinduced phenomena explored below will involve terms both dissipative and not.
In the next section, we will review our results including connections to the previous works [1,2,58]. The following sections present details of the calculations.

Summary of the results
The objective of [1,2,58] and of the present work is to systematically explore (2) under different approximations. Following [1,2,58], the charge densities are split into constant backgrounds and space-time dependent fluctuations whereρ,ρ 5 , E and B are the constant backgrounds, while δρ, δρ 5 , δ E, δ B stand for the fluctuations. Here is a formal expansion parameter to be used below. Furthermore, being most of the time unable to perform calculations for arbitrary background fields, we introduce an expansion in the field strengths where α is the corresponding expansion parameter. Below we will introduce yet another expansion parameter λ, which will correspond to a gradient expansion. For the purpose of the gradient counting, e/m fields will be considered as O(λ 1 ). Throughout this work, the e/m backgrounds E and B are treated as weak. The constitutive relations (2) can be formally expanded both in and α where the first superscript denotes order in and the second in α.
and J (1)(0) 5 were derived in [1]. The goal of present paper is to extend the work initiated in [1] by computing J (1) (1) . Particularly, we will evaluate transport coefficients functions (TCFs) associated with relevant nonlinear transport phenomena discovered in [58] via a fixed order gradient expansion. For simplification, we turn off the fluctuations of the external e/m fields, δ E = δ B = 0. At O 1 α 1 , the currents take the following forms where all the coefficients are scalar functionals of the deriva- Thanks to the linearisation, the constitutive relations (7), (8) could be conveniently presented in Fourier space. Then, the functionals (9) are turned into functions of frequency and spatial momentum, (∂ t , ∇) → (−iω, i q), which we refer to as TCFs [6]. TCFs contain information about infinitely many derivatives and associated transport coefficients. In practice, they are not computed as a series resummation of order-by-order hydrodynamic expansion, and are in fact exact to all orders. TCFs go beyond the hydrodynamic low frequency/momentum limit and contain collective effects of non-hydrodynamic modes. Fourier transformed back into real space, TCFs turn into memory functions, cf. (3). Except for theσ aχ H -term, all the rest of the terms in (7), (8) have already appeared in our previous publication [58] at a fixed order in the gradient expansion. The novelty of present study is to consistently generalise many of the TCs of [58] into TCFs, guaranteeing applicability of the constitutive relations (7), (8) in a broader regime.
To the best of our knowledge, the TCF σχ is introduced here for the first time and will play a crucial role below, see (11). It is important to stress the difference between σχ and σ χ of [1,59,66]. Both TCFs generalise CME/CSE. Yet, while the latter is induced by spacetime variation of the magnetic field, the former is due to inhomogeneity of the charge densities ρ, ρ 5 . One might naively expect that both TCFs are equal. In fact they are not, as we demonstrate below. For comparison, here we quote the hydrodynamic expansion of the CME TCF σ χ which was calculated in [1] As seen from (10), (17) the first order gradient corrections to CME/CSE (i.e., the relaxation time corrections) are different depending on if it is the magnetic field or the charge density that varies with time. In addition, while σ χ depends on ρ, ρ 5 nonlinearly, σχ does not depend on ρ, ρ 5 at all. The TCF σχ enters the dispersion relation of CMW: which is exact to all orders in q 2 . In the hydro limit, using (44), (17), the dispersion relation can be solved analytically with the most comprehensive result reported in [58]. Yet, we have discovered a set of solutions with purely real ω in the present work. That is, for some (continuum set of) values of magnetic field B, there is a discrete density wave mode (ω B , q B ), which propagates without any dissipation (Fig. 21). This is a quite intriguing result, which originates solely from the all order resummation procedure. The details about the non-dissipative discrete density wave mode are deferred to Sect. 4.4. As mentioned in the Introduction, TCFs could be Fourier transformed into memory functions, for an extensive discussion see e.g. [7,13]. The CME current with retardation effects is Via inverse Fourier transform, the CME/CSE memory function is (we focus on the case q = 0), The memory functionσχ is displayed in Fig. 1. An important feature of this function is that it has no support at negative times, which is nothing but manifestation of causality. Another very interesting observation is that rather than having an instantaneous response picked at the origin, like in original CME, the actual response is significantly delayed and picked at a finite time of order temperature. This behaviour ofσχ is quite distinct from diffusion memory func-tionD(t) and shear viscosity memory functions computed previously in [7,13], which are picked at the origin. Let us briefly comment on the remaining terms. D H generalises the Hall diffusion D 0 H [47,48,58] into a TCF of ω, q 2 ; D H is just its axial analogue. So, we will refer to D H and D H as Hall diffusion functions. σ aχ H is a TCF extending the anomalous chiral Hall conductivity σ 0 aχ H [47,48,58].σ aχ H could be considered as an axial analogue of σ aχ H . However, as will be clear later,σ aχ H has an overall factor q 2 so that it will be non-vanishing starting from fourth order in the gradient expansion only. σ 1,2,3 andσ 3 are TCFs of the third order derivative operators (we remind the reader that the e/m fields are counted as of first order). σ 1 , σ 2 correspond to rotor of Hall diffusion [58], and σ 3 ,σ 3 are rotors of anomalous chiral Hall effect [58].
Each TCF in (7), (8) can be split into real part (even powers of frequency) and imaginary part (odd powers of frequency). Based on the time reversal criterion, we conclude that the real parts of σχ ,1,2 , D H ,D H and imaginary parts of σ aχ H ,σ aχ H ,σ 3 ,σ 3 are non-dissipative; all the rest do lead to dissipation of the currents. It is interesting to notice that there are points in the (ω, q) phase space, where some of the dissipative terms vanish. Particularly, this happens to Re[D] and Im [σχ ]. This feature leads to presence of a non-dissipative discrete density wave mode which is mentioned earlier.
The constitutive relations (7), (8) could be re-written in a more compact way, where σ 1,2,3 andσ 3 constitute corrections to CME/CSE and, through spatial inhomogeneities of ρ, ρ 5 , influence the Ohmic conductivity. The scalar diffusion function D [13] now becomes tensor TCFs D 1 i j and (D χ ) 1 i j , linearly depending on E and B because of the weak field approximation adopted here.
In the hydrodynamic limit ω, q 1, the TCFs in (7), (8) are expandable (below we set π T = 1 for convenience and the dimensionful frequency and momentum are π T ω and π T q): σ aχ H = κ 6 log 2 + iω σ 1 = 162κ 2μμ 5 6 + log 2(5 log 2 − 12) + · · · , (22) where · · · denotes higher powers in ω, q 2 and C ≈ 0.915966 is the Catalan's constant. Here,μ =ρ/2,μ 5 =ρ 5 /2 are backgrounds for vector/axial chemical potentials. While each term in (17)- (25) have been computed as individual TC in [58], the resummation procedure here collects all relevant TCs into a single TCF and determines the most general structure of currents, valid to all orders. Beyond the hydrodynamic limit, the TCFs are computed numerically. The results are presented and discussed in Sect. 4.3. We observe a relatively weak dependence on q 2 while ω-dependence is more profound: damped oscillations towards asymptotic regime around ω 5. We remark that none of the TCFs survives beyond asymptotically large ω 5. For the details about each TCF we refer to Sect. 4.3.
It is interesting to explore dependence of the TCFs on the chemical potentials. Of special interest is the case of zero background axial charge density,ρ 5 = 0, which is the most realistic scenario for any conceivable experiment. However, even in this case, μ 5 could be nonzero and would be proportional to E · B due to the chiral anomaly (1). Because of the linearisation approximation, the TCFs σ 1 ,σ 3 vanish in the limitρ 5 = 0. We expect them to be nonzero beyond the current approximation. At q = 0, for the remaining TCFs we discover some universal dependence:σ aχ H vanishes; σ aχ H , D H ,D H do not depend on the chemical potentials at all; σ 1 is linear in κ 2μμ 5 ; σ 3 is linear in κμ; similarly,σ 3 is linear in κμ 5 ; σ 2 has a normal component independent of the chemical potentials and anomaly induced correction which is linear in κ 2 (μ 2 +μ 2 5 ). All these features can be derived from the underlying equations (see Appendix A1 for relevant ODEs).
The rest of this paper is structured as follows. In Sect. 3, we present the holographic model briefly. For more details about holographic model we refer to [58]. Section 4 contains the main part of the study: gradient resummation for nonlinear chiral transport. It is further split into four subsections. In Sect. 4.1, the constitutive relations (7), (8) are derived from the dynamical components of the bulk anomalous Maxwell equations near the conformal boundary. In Sect. 4.2, the TCFs are analytically computed in the hydrodynamic limit. Section 4.3 numerically extends the results beyond this limit. Section 4.4 focuses on the CMW dispersion relation beyond hydrodynamic limit. Section 5 concludes our study. Appendices supplement calculational details for Sect. 4.

Holographic setup: U(1) V × U(1) A
The bulk action is [59,60] where and the counter-term action S c.t. is The gauge Chern-Simons terms (∼ κ) in the bulk action mimic the chiral anomaly of the boundary field theory. Note M N P Q R is the Levi-Civita symbol with the convention rt x yz = +1, while the Levi-Civita tensor is M N P Q R / √ −g.
In the ingoing Eddington-Finkelstein coordinate, the metric of Schwarzschild-Ad S 5 is where f (r ) = 1−1/r 4 . Here we have normalised the Hawking temperature (identified as the temperature of the boundary theory) to π T = 1. The bulk equations of motion read where EV μ = EA μ = 0 and EV r = EA r = 0 correspond to dynamical and constraint equations, respectively. The boundary currents are defined as Employing the radial gauge V r = A r = 0, it is sufficient to solve the dynamical equations only to determine the boundary currents (32), leaving constraints aside. Indeed, the constraint equations give rise to continuity equations of currents (1). Thus, without imposing the constraint equations, the currents to be constructed are off-shell. For practical purpose, it is useful to express the currents in terms of the coefficients of near boundary (r = ∞) preasymptotic expansion of the bulk gauge fields: where F V μν is field strength of the external e/m potential (2) μ and A (2) μ are the coefficients of 1/r 2 in the near boundary expansions of bulk fields V μ and A μ , respectively. Note V (2) μ and A (2) μ have to be determined by fully solving the dynamical equations from the horizon to the boundary.
As the remainder of this section, we outline the strategy for deriving the constitutive relations for J μ and J μ 5 . To this end, we turn on finite vector/axial charge densities for the dual field theory, which are also exposed to external e/m fields V μ . Holographically, the charge densities and external fields are encoded in the asymptotic behaviors of the bulk gauge fields. In the bulk, we will solve the dynamical equations assuming the charge densities and external fields as given, but without specifying them explicitly. For more details, we refer the reader to our previous publications [1,2,58].
We start with the ansatz where V μ (x) is the external gauge potential, and ρ, ρ 5 are vector and axial charge densities of the boundary theory. V μ and A μ will be determined by solving dynamical equations. Appropriate boundary conditions are classified into three types. First, V μ and A μ are regular over the domain 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 field). Additional integration constants will be fixed by the Landau frame convention, The Landau frame convention corresponds to a residual gauge fixing for the bulk fields.
To facilitate the exchange between charge density and chemical potential, we define Generically, μ, μ 5 are nonlinear functionals of densities and external fields. For generic profiles of V μ (x), ρ(x), ρ 5 (x), it is impossible to solve dynamical components of (30), (31). As announced in Sect. 3, we employ the approximation schemes (4), (5). Consequently, the corrections V μ and A μ are first expanded in powers of , and then each order in is further expanded in powers of α: where will act as sources in the dynamical equations at O( 1 α 1 ), we summarise them below (the notations here will be slightly different from [1]).
At O( 0 α 1 ), we have where f 1 and f 2 are [1] At O( 1 α 0 ), the corrections are (note δ E = δ B = 0 throughout this work) g 3 and g 4 satisfy coupled ordinary differential equations (ODEs), which were solved both analytically in the hydro limit (ω, q 1) and numerically for generic values of ω, q in Ref. [1]. Below we quote the hydro expansion of diffusion function D [13] (which can be extracted from solution to g 4 ): At O ( 1 α 1 ), the dynamical equations reduce to the following linear partial differential equations for the corrections V In the next Sect. 4, we will solve (45)-(48) by the technique invented in [5,6].

Nonlinear chiral transport and gradient resummation
In this section, we focus on all order gradient resummation. It is split into four subsections. The first one Sect. 4.1 is devoted to derivation of the constitutive relations (7), (8). In the following Sects. 4.2 and 4.3, the TCFs in (7), (8)  are decomposed in terms of basic structures built from the external fields and inhomogeneous parts of the charge densities, where S i ,S i , V i andV i are functionals of the boundary derivative operator ∂ μ and functions of the radial coordinate r . They also depend on the constant valuesμ andμ 5 of the chemical potentials. Fourier transforming δρ and δρ 5 turns all the derivatives into momenta. Thus, in momentum space, these decomposition coefficients become functions of the radial coordinate, frequency ω and spatial momentum squared q 2 : which satisfy partially decoupled inhomogeneous ODEs listed in Appendix A1. The decomposition functions S i ,S i , V i andV i are nothing else but elements of the inverse Green function matrix for the system of ODEs. As discussed in Sect. 3, the boundary conditions for the decomposition coefficients in (49)-(52) are Additional integration constants will be fixed by the Landau frame convention (36).
Solving the ODEs (A1)-(A16) near the boundary r = ∞ reveals the pre-asymptotic behaviour for the corrections, which can be summarised as where s 1,L i , v 1,L i ,s 1,L i ,v 1,L i are fixed uniquely from the nearboundary analysis alone, while the coefficients s i , v i ,s i ,v i can be determined only when the ODEs are fully solved in the entire bulk, from the horizon to the Ad S boundary.
Then, at O( 1 α 1 ) the boundary currents (33) are The Landau frame convention (36) implies Combined with the ODEs (A1), (A16), (62) leads to constraints among the decomposition coefficients in (49)-(52), see (A17), (A18), (A19). Helped by these constraints, (59), (61) can be eventually put into compact form (7), (8). All the TCFs can be identified with the near boundary data v i ,v i : The TCF σχ does not depend onμ,μ 5 at all. The rest of the TCFs bear reminiscence of the axial symmetry. It get reflected in some mirror symmetries with respect to exchange ofρ andρ 5 (or equivalently ofμ ↔μ 5 ). We found some "symmetric relations" among the decomposition coefficients in (49) Instead of the charge densities ρ, ρ 5 , chemical potentials are frequently used as hydrodynamic variables to parameterise the currents' constitutive relations. Up to O( 1 α 1 ), the chemical potentials defined in (37) are where g 3 (r = 1) andS 1 (r = 1) denote horizon values of g 3 [appearing in (42)] andS 1 , respectively. Then, (65) can be where we have utilised the fact that g 3 has non-vanishing value starting from second order in the gradient counting. After some manipulations, the currents (7), (8) turn into where the TCFs with prime are related to those in (7), (8) by and similar equations for the rest.

Beyond the hydrodynamic limit: numerical results
In this section, we present our results for the TCFs in (7), (8) for finite frequency/momentum via solving the ODEs (A1)-(A16) numerically. Pseudo-spectral collation method is employed, which essentially converts the continuous boundary value problem of linear ODEs into that of discrete linear algebra. For more details on the numerical method, we recommend the references [67][68][69]. Thanks to the symmetry relations (64), we plot the TCFs σ aχ H ,σ aχ H , σ 1,2 for κμ κμ 5 only without loss of generality. For D H and σ 3 , this constraint is abandoned so thatD H andσ 3 could be extracted from D H and σ 3 via the exchangeμ ↔μ 5 .
First, consider TCF σχ , which generalises the original CME (CSE) and measures the response to inhomogeneity of charge density ρ (ρ 5 ). Note σχ does not depend on the vector/axial chemical potentials at all, as can be seen from the relevant ODEs (A2), (A6), (A7). In Fig. 2 we show the 3D plot of σχ . The plots in Fig. 3 are 2D slices of Fig. 2 when either ω = 0 or q = 0. While σχ is different from the chiral magnetic conductivity σ χ of [1], it has roughly the same dependence on frequency/momentum as σ χ as is clear from these plots. Namely, σχ shows a relatively weak dependence on q 2 while its dependence on ω is more profound: damped oscillations towards asymptotic regime around ω 5 where σχ vanishes essentially. As will be clear later, this damped oscillating behavior is also observed in all other TCFs. This phenomenon can be related to quasi-normal modes in the presence of background fields, but here we are not pursuing this connection any further. When q = 0 we computed the inverse Fourier transform of σχ , that is the memory functioñ σχ (t) of (13), as displayed in Fig. 1.
Next we consider TCFs D H ,D H , σ aχ H andσ aχ H multiplying second order derivative structures. These second Thus, we will mainly focus on D H . σ aχ H is the anomalous chiral Hall TCF andσ aχ H is its axial analogue. Since V 4 = q 2 V 5 andV 4 = q 2V 5 (see (A19)), from the ODE (A13) it is obvious thatσ aχ H has an overall q 2 factor, so we will plotσ aχ H /q 2 in order to see non-trivial behavior.

CMW dispersion relation to all orders: non-dissipative modes
The TCF σχ enters the dispersion relation of CMW: The dispersion relation (71) is exact to all orders in q 2 , provided κB 1. General solutions of this equation are complex and cannot be studied with our present results. This is because σχ (ω, q 2 ) and D(ω, q 2 ) have been computed for real values of ω only. We believe that beyond the hydrodynamic limit, Eq. (11) has infinitely many gapped modes. Exploring this point in general would require going into complex ω plane for the TCFs, which is beyond the scope of the present work. Yet, quite intriguingly, there is a set of purely real non-dissipative solutions to (71). In order to find these solutions we have devised the following procedure.
First, the equation is split into real and imaginary parts (assuming q parallel to B): For a fixed value of κB, say κB = 0.33, the functions φ I and φ R are shown in Fig. 21 (left) as contour plots in (ω, q 2 ) space (the function D(ω, q 2 ) is taken from [13] ). The dashed (blue) and solid (red) curves stand for φ I and φ R respectively. The numbers indicated on the curves correspond to the values of these functions along the curves. Our interest is when both functions vanish simultaneously, that is a crossing point of φ I = 0 and φ R = 0 curves. Such crossing is clearly seen in the region ω < 0.5 and q 2 < 0.5. We denote this point by  (ω B , q B ). This is a discrete density wave mode propagating in the medium without any dissipation. The procedure could be repeated for other values of κB. The result is a one dimensional curve in a 3d parameter space depicted in Fig. 21 (right). A few comments are in order. First, there is a minimal value of κB 0.33 for which there exists such a solution. Second, in fact there are multiple solutions corresponding to several disconnected branches in Fig. 21, which we do not display.

Conclusion
In this work, we have continued exploration of nonlinear chiral anomaly-induced transport phenomena based on a holo-graphic model with two U (1) fields interacting via gauge Chern-Simons terms. For a finite temperature system, we constructed off-shell constitutive relations for the vector and axial currents.
The constitutive relations contain nine terms which are linear simultaneously in the charge density fluctuations and constant background external fields. The nine terms summarised in (7), (8) correspond to all order resummation of gradients of the charge density fluctuations parameterised by TCFs, first computed analytically in the hydrodynamic limit (Sect. 4.2) and then numerically for large frequency/momentum (Sect. 4.3). A common feature of all TCFs in (7), (8) is that they depend weakly on spatial momentum but display pronounced dependence on frequency in the form of damped oscillations vanishing asymptotically at ω 5.
Most of our results are presented in summary Sect. 3. Among new results worth highlighting is the CME memory function computationσχ (t − t ). The memory function is found to differ dramatically from a delta-function form of instantaneous response. In fact,σχ (t − t ) vanishes at t = t and the CME response gets built only after a finite amount of time of order temperature.
Another result we find of interest is related to CMW dispersion relation, which for the first time was considered to all orders in momentum q. Beyond the perturbative hydrodynamic limit, we found a continuum set of discrete density wave modes, which can propagate in the medium without any dissipation. While the original CMW dissipates and that could be one of the problems for its detection, the new modes that we discover should be long lived and have some poten-tial experimental signature. 2 It is important to remember that our calculation of the CMW dispersion relation is done for a weak magnetic field only. One can obviously question the validity of the results beyond this approximation. Both TCFs σχ and D that enter the CMW dispersion relation are functions of E and B. In our previous work [58], we initiated this study, still in perturbative in E and B regions, but a full non-perturbative analysis will be reported elsewhere [70].
We have found a wealth of non-linear phenomena all induced entirely by the chiral anomaly. An important next step in deriving a full chiral MHD would be to abandon the probe limit adopted in this paper and include the dynamics of a neutral flow as well. This will bring into the picture additional effects such as thermoelectric conductivities, normal Hall current, the chiral vortical effect [71,72], and some nonlinear effects discussed in [47]. We plan to address these in the future. sub-sector (i): sub-sector (ii): The remaining decomposition coefficients satisfy the same ODEs as above. More specifically, the sub-sector 9 , V 9 } satisfies the same equations as the sub-sector (i): 10 , V 10 ,V 11 , V 11 ,V 12 , V 12 } obeys the same equations as sub-sector (ii): In what follows, we explore some "mirror symmetry relations" among these decomposition coefficients, which are useful in simplifying the expressions for currents' constitutive relations at the order O( 1 α 1 ). First, notice that since these two sub-sectors satisfy identical system of ODEs and have the same boundary conditions. Following this reasoning, The "equal sign" in (A17), (A18) should be understood in the specific order as shown therein. Certain relations can be established among the decomposition coefficients in (49)- (52). It follows from the Landau frame convention (62) and boundary conditions (55), (56) that Now lets explore the mirror symmetry for the decomposition coefficients under exchangeρ ↔ρ 5 . The decomposition coefficients in the sub-sector {S 1 ,S 1 , V 1 ,V 1 , V 2 ,V 2 , V 3 ,V 3 } are found symmetric with respect toρ,ρ 5 : S 1 (ρ,ρ 5 ) = S 1 (ρ 5 ,ρ),S 1 (ρ,ρ 5 ) =S 1 (ρ 5 ,ρ).
The symmetry relations (A20), (A21) guide the choice of values for κρ and κρ 5 in numerical procedure for the ODEs. Given these relations, the choice κρ κρ 5 can be applied when solving V 1 ,V 1 , V 3 ,V 3 , S 1 , V 2 ,S 1 ,V 2 and V 4 ,V 4 , V 6 ,V 6 without loosing generality. These relations also help to reduce the number of the ODEs to be solved.