# Anomalous transport from holography: part II

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## Abstract

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 the 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 modifications to the chiral magnetic wave nonlinear in *B*. 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 nonlinear axial current (CSE) is found in the first case. The dependence on magnetic field and frequency of linear transport coefficient functions is explored in the second.

## 1 Introduction and summary

Fluid dynamics [1, 2] is an effective long-wavelength description of most classical or quantum many-body systems at nonzero temperature. It is defined in terms of constitutive relations, which relate thermal expectation values of conserved currents to thermodynamical variables and external fields. The derivative expansion in fluid-dynamic variables such as the velocity or charge densities accounts for deviations from thermal equilibrium. At each order, the derivative expansion is fixed by thermodynamic considerations and symmetries, up to a finite number of transport coefficients, such as the viscosity, diffusion constant and conductivity. The latter are not calculable from hydrodynamics itself, but have to be determined from underlying microscopic theory or experimentally.

Although fluid dynamics has a long history, the theoretical foundations of relativistic viscous hydrodynamics are not yet fully established. The Navier–Stokes hydrodynamics leads to violation of causality: the set of fluid dynamical equations makes it possible to propagate signals faster than light. To overcome this problem, simulations of relativistic hydrodynamics are usually based on phenomenological prescriptions of [3, 4, 5, 6], which admix viscous effects from second order derivatives, so as to make the fluid dynamical equations causal. References [3, 4, 5, 6] introduced retardation effects for irreversible currents, which, via the equations of motion, become additional degrees of freedom. In other words, one needs to include higher order gradient terms in the derivative expansion in order to obtain a causal formulation. In general, causality is violated if the derivative expansion is truncated at any *fixed* order. It is supposed to be restored when all-order gradient terms are included, which we refer to as *all-order resummed* hydrodynamics. Resummed hydrodynamics is UV complete in a sense that it has a well-defined large frequency/momenta limit. Yet it is an effective theory of hydrodynamic variables only,^{1} which emerges after most of the degrees of freedom of the underlying microscopic theory are integrated out.

For a holographic charged plasma dual to *U*(1) Maxwell theory in Schwarzschild–AdS\(_5\) TCFs were studied in depth in [9]. The derivative resummation in the constitutive relation was implemented via the technique of [7, 10, 11, 12], which was originally invented to resum all-order velocity gradients (linear in the velocity amplitude) in the energy-momentum tensor of a holographic conformal fluid.^{2} It is important to stress that this linearisation procedure is a mathematically well-controlled approximation: the perturbative expansion corresponds to a formal expansion in the amplitudes of fluid-dynamic variables and external fields, without any additional assumptions. In this respect, the implemented approximation is identical to that of the linear response theory based on two-point correlators.

Our technique follows closely the original idea of [14], which relates fluid’s constitutive relations for the boundary theory to solving equations of motion in the bulk. However, an important new element of our formalism is that it is not based on current conservation (i.e., “off-shell” formalism), which makes it essentially different from the “on-shell” formalism of [14]. Constitutive relations and TCFs can be uniquely determined from dynamical components of the bulk equations only, while the constraint component in the bulk is equivalent to continuity equation on the boundary.

*U*(1) currents are coupled to external electromagnetic fields, the triangle anomaly renders the axial current non-conserved,

*e*is an electric charge which below will be set to unit.

The presence of triangle anomalies requires a modification of the usual constitutive relations for the currents. An example of such a modification is the chiral magnetic effect (CME) [15, 16, 17, 18, 19],^{3} that is, the induction of an electric current along the applied magnetic field. CME relies on chiral imbalance, which is usually parameterised by an axial chemical potential. Studies of CME can be found in e.g. [23, 24, 25, 26, 27, 28, 29] based on perturbation theory, in e.g. [30, 31, 32, 33, 34, 35] within lattice simulations, and in e.g. [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50] for the strongly coupled regime based on the AdS/CFT correspondence [51, 52, 53].

The chiral separation effect (CSE) [54, 55] is another interesting phenomenon induced by the anomalies. It is reflected in a separation of the chiral charges along an external magnetic field at finite density of vector charges. Chiral charges can also be separated along external electric field, when both vector and axial charge densities are nonzero, the so-called chiral electric separation effect (CESE) [56, 57].

In heavy ion collisions, experimentally observable effects induced by the anomalies were discussed in [58, 59, 60, 61, 62]. We refer the reader to [63, 64, 65, 66, 67] and the references therein for comprehensive reviews of the subject of anomalous transports.

In [68] we went beyond [9] focussing on transport properties induced by the chiral anomaly. The holographic model was modified to be anomalous \(U(1)_V\times U(1)_A\) Maxwell theory in the Schwarzschild–AdS\(_5\) case. Under various approximations, off-shell constitutive relations were derived for vector/axial currents. In a weak external field approximation, all-order derivatives in the vector/axial currents were resummed into six momenta-dependent TCFs: the diffusion, the electric/magnetic conductivity, and three anomaly-induced TCFs. The latter generalise the chiral magnetic/separation effects. Beyond weak external field approximation, nonlinear transports were also revealed when constant background external fields are present. Particularly, the chiral magnetic effect, including all-order nonlinearity in magnetic field, was proven to be exact when all external fields except for a constant magnetic field are turned off. Nonlinear corrections to the currents’ constitutive relations due to electric and axial external fields were computed.

In the present work we continue the study of anomaly-induced transports within the holographic model of [68]. No axial external fields will be turned on in this work. As in [9, 68] we work in the probe limit so that the currents and energy-momentum tensor decouple. In dual gravity, the probe limit ignores the back-reaction of the gauge dynamics on the geometry. The holographic model under study consists of two Maxwell fields in the Schwarzschild–AdS\(_5\) black brane geometry. The chiral anomaly is holographically realised via the gauge Chern–Simons actions for both Maxwell fields. Such a holographic setup can be realised via a top–down brane construction of \(D4/D8/\overline{D8}\) [69].

*B*-dependent correction

^{4}CME emerges from a nonzero axial chemical potential \(\mu _{_5}\), which is usually assumed to have some background profile. It is, however, possible to induce \(\rho _{_5}\) (and thus \(\mu _{_5}\)) dynamically through the interplay between the electric and magnetic fields, as is clear from the continuity equation (2). Specifically, we are ready to consider a constant magnetic field \(\vec {\mathbf {B}}\) and a time-dependent but spatially homogeneous electric field \(\vec {E}(t)\). For simplicity the charge densities \(\rho , \rho _{_5}\) will be assumed to be spatially homogeneous too

^{5}. From (2), \(\rho \) could be set zero. The constitutive relations for the vector/axial currents are

While there is some overlap between our results and the literature, differences between the present study and those of [73, 74, 75, 76] must be clarified. Utilising the weak electric field approximation (10), [73] analytically evaluated the magnetic field dependence of the longitudinal conductivity \(\sigma _{\text {L}}\) in DC limit, while [74] calculated its \(\omega \)-dependence. Back-reaction effects on \(\sigma _{\text {L}}\) were considered in [76]. References [77] performed a similar study, focussing on time evolution of the induced vector current, given some specially chosen initial profile for the electric field. All the studies [73, 74, 75, 76] focussed on a weak electric field, in which the axial current vanishes. Therefore, our nonlinear results and particularly the axial charge separation current (66) appears as new. As for the linearised setup (10), [73, 74, 75, 76] imposed the continuity equation and replaced the axial charge density \(\rho _{_5}\) 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 [7, 9, 10, 11, 12], only off-shell construction reveals transport properties of the system in full. Particularly, there are three independent TCFs (\(\sigma _e\) and \(\tau _{1,2}\)) in the constitutive relation (11), all of which we are able to determine separately, compared to only two independent conductivities in (13).

Another difference worth mentioning is that we explicitly trace all the effects in the induced current that arise from the relative angle between \(\vec {E}(t)\) and \(\vec {\mathbf {B}}\) fields. This is in contrast to [74, 77], which limited their study to the case of parallel fields only, primarily focussing on the longitudinal electric conductivity \(\sigma _{\text {L}}\). By varying the relative angle between \(\vec {E}(t)\) and \(\vec {\mathbf {B}}\) fields, one can separate the anomaly-induced effects (parametrised by \(\tau _1\) and \(\tau _2\)) from the ones that are not related to the anomaly (\(\sigma _e\)).

The paper is structured as follows. In Sect. 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 Sect. 4, CME/CSE in the presence of constant magnetic and time-varying electric fields are analysed. This study is further split into two subsections. The exploration of nonlinear phenomena in the induced vector/axial currents is done in Sect. 4.1. In Sect. 4.2 we focus on the linearised regime (10) and calculate the dependence of AC conductivity on magnetic field. Section 5 presents the conclusions. Two appendices supplement computations of Sects. 3 and 4.

## 2 The holographic model: \(U(1)_V\times U(1)_A\)

*finite*contribution to the boundary currents.

*r*hypersurface \(\Sigma \), the induced metric \(\gamma _{\mu \nu }\) is

*V*and

*A*fields are as follows.

The currents (24) are defined independently of the constraint equations (21). Throughout this work, the radial gauge \(V_{r}=A_{r}=0\) will be assumed. Consequently, in order to completely determine the boundary currents (25) it is sufficient to solve the dynamical equations (20) for the bulk gauge fields \(V_\mu ,A_\mu \) only, leaving the constraints aside. The constraint equations (21) give rise to the continuity equations (2). In this way, the currents’ constitutive relations to be derived below are off shell.

## 3 CME/CSE with time-independent inhomogeneous magnetic field

*r*estimates for \(\mathbb {V}_t\) and \(\mathbb {A}_t\), the frame convention (35) was used to fix the coefficients of \(1/r^2\) in the near-boundary expansion for \(V_t,A_t\) (thus those of \(\mathbb {V}_t\) and \(\mathbb {A}_t\)). The dynamical equations (37)–(40) get simplified,

*r*-coordinate, which can be solved via direct integration over

*r*. The results for \(\mathbb {V}_\mu ^{[n]}\) and \(\mathbb {A}^{[n]}\) up to \(n=2\) can be found in Appendix A; see (100)–(106).

## 4 CME/CSE with constant magnetic and time-dependent electric fields

^{6}The continuity equation (2) degenerates to

*r*and

*t*only. As a result, the dynamical equations (37)–(40) are reduced to

### 4.1 Nonlinear phenomena: general analysis and derivative expansion

The objective of this subsection is to show that beyond linearised limit (69) the setup (60) also induces a non-vanishing axial current \(\vec {J}_5\), which has been omitted in the literature. To this end, as in Sect. 3, we first give a fully nonlinear analysis for the dynamical equations (61)–(64), followed by perturbative calculations for \(\mathbb {V}_\mu , \mathbb {A}_\mu \) within the derivative expansion (5). All calculational details are addressed in Appendix B.

As in Sect. 3 the formal analysis are based on rewriting the dynamical equations (61)–(64) into integral form, from which one could deduce the near-boundary asymptotic behaviours for \(\mathbb {V}_\mu ,\mathbb {A}_\mu \). The results can be found in (112)–(115). Plugged them into (29), the near-boundary behaviour for \(\mathbb {V}_\mu , \mathbb {A}_\mu \) presented in (112)–(115) is translated into boundary currents (8) and (9). Generically, the quantities \(\mathbb {V}_i(1)\), \(\mathbb {A}_i(1)\), \(\overline{G}_i(x=\infty )\) and \(\overline{H}_i(x=\infty )\) in (8) and (9) cannot be computed analytically. However, as in Sect. 3, the formal analysis determines the generic forms for \(J^\mu /J_5^\mu \).

^{7}

### 4.2 Linear in \(\vec {E}\) phenomena

*r*once, we get

The differential equation (76) is *linear* in the correction \(\mathbb {V}_i\). Therefore, (76) can be solved via the technique developed in [7, 10, 11, 12]. The bulk equations reduce to linear inhomogeneous partial differential equations while the inhomogeneous terms are built from boundary derivatives of the fluid-dynamic variables and external fields. The equations then can be exactly solved using the Green function formalism: the bulk fields are decomposed in terms of all possible basic vector structures constructed from the fluid-dynamic variables and external fields. These decomposition coefficients (components of the inverse Green function) are functions of the holographic radial coordinate and functionals of the boundary derivative operators. The functional dependence of the decomposition coefficients on the boundary derivative operation encodes all-order linear derivatives in the constitutive relations. Transformed into momentum space, the bulk equations give rise to ordinary differential equations for those decomposition coefficients, which are RG-like equations in AdS space. Solving the RG-like equations completely determines the fluid’s constitutive relations and all transport coefficients. Below we implement these steps.

^{8}

For arbitrary \(\omega \), we resort to numerical methods and solve ODEs (80)–(82) for representative values of \(\kappa {\mathbf {B}}\). The numerical procedure is identical to that of [68] and for all the numerical details we refer the reader to this publication. In Fig. 2 we show the \(\omega \)-dependence for \(\tau _1\) and \(\tau _2\) for sample choices of \(\kappa {\mathbf {B}}\). In Fig. 3 we plot the normalised TCFs \(\tau _1/\tau _1^0\) and \(\tau _2/\tau _2^0\). Overall, \(\tau _1\) and \(\tau _2\) display quite similar dependences on the frequency \(\omega \). After some oscillations, both \(\tau _1\) and \(\tau _2\) approach zero asymptotically.

Approach to the asymptotic regime, however, depends on strength of the magnetic field. When \(\kappa {\mathbf {B}}\) is increased, the asymptotic behaviour is delayed towards larger \(\omega \). What is more intriguing is that increasing \(\kappa {\mathbf {B}}\) renders \(\tau _1\) and \(\tau _2\) to develop a resonance-like enhancement at finite \(\omega \). This could be an interesting experimentally observable feature. For very strong magnetic fields \(\kappa {\mathbf {B}}\rightarrow \infty \), the chiral anomaly-induced effects would be pushed to the UV, corresponding to early time effects, such as in [77].

In Fig. 4 we show two-point correlators \(G^{\text {T,L}}\) for different choices of \(\kappa {\mathbf {B}}\). However, it is difficult to appreciate the anomaly-induced effects from Fig. 4 because in the correlators they get mixed with non-anomalous ones. To illuminate \(\kappa {\mathbf {B}}\)-correction to \(G^{\text {L}}\), in Fig. 5 we plot the difference \(\delta G^{\text {L}}=G^{\text {L}}-G^{\text {T}}\). From these plots, the effect of the chiral anomaly on the induced vector current is seen more clearly. We again notice a remarkable relative enhancement at intermediate values of \(\omega \).

## 5 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 the CSE get corrected by derivative terms; see (3) and (4). Within the derivative expansion, we analytically calculated corrections up to third order in the expansion; see (48) and (49) and (51) and (52). Apart from the derivative corrections to CME and CSE, the diffusion constant \(\mathcal {D}_0\) was found to receive a negative anomaly-induced correction; see (6). The dispersion relation of the chiral magnetic wave was also found to be modified; see (7).

In the second part of our study, we focussed on the case of time-varying electric and constant magnetic fields without any externally enforced axial charge asymmetry, though the \(\vec {E}(t)\cdot \vec {\mathbf {B}}\) term in the continuity equation (59) generates the axial charge density \(\rho _{_5}\) (and thus \(\mu _{_5}\)) dynamically. For such a configuration of the external fields, we first analysed the most general constitutive relations for the vector/axial currents; see (8) and (9). Then, within the derivative expansion, we explicitly calculated the currents up to third order at nonlinear level; see (65) and (66). When put on shell, the axial current \(\vec {J}_5\) is fully nonlinear in the external electric field.

Employing another approximation, we linearised the constitutive relations assuming the electric field is weak (10). 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 \(\sigma _e\), and two chiral anomaly-induced conductivities \(\tau _1,\tau _2\); see (11). In the DC limit, we analytically computed these conductivities; see (88) and (89). Then, for generic \(\omega \), the numerical plots were presented in Sect. 4.2. Based on these studies, we notice that the anomaly-induced effects get enhanced at some finite frequency \(\omega \), 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.

## Footnotes

- 1.
In fact there are infinitely many such variables (see Ref. [7] for a discussion).

- 2.
One might be concerned that the hydrodynamic derivative expansion forms an asymptotic series with zero radius of convergence [13]. However, contrary to our linearised study, this conclusion applies to nonlinear hydrodynamics in which the number of terms grows factorially with the number of gradients. What is more important is that our approach does not rely on explicit resummation of the gradient series and thus is safe from any convergence related uncertainties.

- 3.
- 4.
We thank Dmitri Kharzeev for proposing us this study.

- 5.
In principle it is not excluded that the charge densities \(\rho ,\rho _{_5}\) could be spatially inhomogeneous. Yet such spatial inhomogeneity would render the derivative resummation highly complicated.

- 6.
While from the continuity equation (2) the charge densities can still have a nontrivial spatial-dependence, we found that such spatial inhomogeneity of the charge densities would make the gradient resummation out of control.

- 7.
While we suspect that the chemical potential \(\mu \) is zero to all orders in the gradient expansion, we have not been able to prove this.

- 8.
In the decomposition for \(\mathbb {V}_i\), one could have included a term \(C_4 \vec {E}\times \vec {\mathbf {B}}\). However, the coefficient \(C_4\) would satisfy a homogeneous ODE. Under the same arguments leading to \(\mathbb {A}_i=0\), \(C_4\) has to be zero too.

## Notes

### Acknowledgements

We would like to thank Dmitri E. Kharzeev, Alex Kovner, Andrey Sadofyev, Derek Teaney, and Ho-Ung Yee for useful discussions related to this work. YB would like to thank KITPC (Beijing) for financial support and hospitality, Physics Department of the University of Connecticut for hospitality where part of this work was done. This work was supported by the ISRAELI SCIENCE FOUNDATION Grant #1635/16, BSF Grant #012124, the People Program (Marie Curie Actions) of the European Union’s Seventh Framework under REA Grant agreement #318921; and the Council for Higher Education of Israel under the PBC Program of Fellowships for Outstanding Post-doctoral Researchers from China and India (2015–2016).

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