Anomalous transport from holography: Part II

# Anomalous transport from holography: Part II

Yanyan Bu yybu@post.bgu.ac.il Department of Physics, Ben-Gurion University of the Negev,
Beer-Sheva 84105, Israel
Michael Lublinsky lublinm@bgu.ac.il Amir Sharon sharon.amir24@gmail.com Department of Physics, Ben-Gurion University of the Negev,
Beer-Sheva 84105, Israel
###### Abstract

This is a second study of chiral anomaly induced transport within a holographic model consisting of anomalous Maxwell theory in Schwarzschild- 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 -correction to the diffusion constant. We also find non-linear in 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.

## 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. Derivative expansion in fluid-dynamic variables such as 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 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 long history, 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 to make the fluid dynamical equations causal. Refs. [3, 4, 5, 6] introduced retardation effects for irreversible currents, which, via 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 only111In fact there are infinitely many such variables (see Ref. [7] for a discussion)., which emerges after most of the degrees of freedom of the underlying microscopic theory are integrated out.

The most general parity-even linear in external fields and charge density off-shell constitutive relation for a vector current has the following form

 Jt=ρ,            →J=−D→∇ρ+σe→E+σm→∇×→B, (1)

where is a vector charge density and the diffusion , electric/magnetic conductivities are functionals of space-time derivatives. In terms of hydrodynamic expansion, the constitutive relation (1) provides all order resummation of gradients of the fluid-dynamic variables (the charge density ) and external fields ( and ). In momentum space and are functions of frequency and momentum squared (assuming isotropic medium), which we refer to as transport coefficient functions (TCFs). Via inverse Fourier transform, TCFs appear as memory functions in the constitutive relation [8].

For a holographic charged plasma dual to Maxwell theory in Schwarzschild- TCFs were studied in depth in [9]. The derivative resummation in the constitutive relation was implemented via the technique of [10, 11, 7, 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 fluid222One 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 non-linear 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.. 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.

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 currents are coupled to external electromagnetic fields, the triangle anomaly renders the axial current into non-conserved,

 ∂μJμ=0,                  ∂μJμ5=12κ→E⋅→B, (2)

where / are vector/axial currents, and is an anomaly coefficient. For gauge theory with a massless Dirac fermion in fundamental representation, , and is an electric charge which below will be set to unit.

Presence of triangle anomalies requires modification of usual constitutive relations for the currents. An example of such modification is the chiral magnetic effect (CME) [15, 16, 17, 18, 19]333See also [20, 21, 22] for earlier related works., 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 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 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) [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 references therein for comprehensive reviews on the subject of anomalous transports.

In [68] we went beyond [9] focusing on transport properties induced by the chiral anomaly. The holographic model was modified to be anomalous Maxwell theory in Schwarzschild-. 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 the dual gravity, the probe limit ignores backreaction of the gauge dynamics on the geometry. The holographic model under study consists of two Maxwell fields in the Schwarzschild- 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 [69].

Before diving into the details presented in the following sections, we summarise our main results. The paper is split into two largely independent parts. In the first part, we consider a setup in which a static but spatially-varying magnetic field is the only external field that is turned on. Then the constitutive relations for the vector/axial currents are

 Jt=ρ,           Ji=−12∂iρ+12κμ5Bi–––––––––−Gi(x=∞), (3)
 Jt5=ρ5,          Ji5=−12∂iρ5+12κμBi––––––––−Hi(x=∞), (4)

where are vector/axial charge densities, the underlined terms in are the chiral magnetic/separation effects. contain derivatives of and are defined in section 3. It is important to stress that in contrast to the above discussion of linearized hydro, (3,4) are exact, without any approximations for . The nonlinearity of the CME/CSE in external magnetic field is completely accounted for by the chemical potentials . The non-derivative part of (3) is consistent with the “non-renormalisability” of CME [70, 71, 48, 49]. However, as will be clear from (48,49), the derivative corrections in introduce new effects, which do modify the original CME. Particularly, the currents along the direction of get affected.

When vary slowly from point to point, can be calculated order-by-order within boundary derivative expansion. Let us introduce a scaling parameter :

 ∂μ=(∂t,∂i)⟶(λ∂t,λ∂i). (5)

Then, derivative counting goes by powers of . Up to second order in derivative expansion, we calculated and chemical potentials without any further assumptions. Given that these results are rather lengthy, we postpone to present them in section 3, see (48,49,3). At third order , for we calculated only terms that are linear in , see (51,52) for a complete listing. Among these third order terms, the diffusion constant (i.e., the DC limit of the diffusion function ) gets a negative -dependent correction

 D0=12−18(2log2−1)κ2B2. (6)

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 [72].

With the third order results for and , we also computed the dispersion relation for a free mode that can propagate in the medium:

 ω=[∓1+36(1−2log2)κ2B2]6κ→q⋅→B−[12+18(1−2log2)κ2B2]iq2−i8q4log2+⋯, (7)

where means a constant magnetic field. The first term in (7) represents the chiral magnetic wave (CMW) [70]. Interestingly, we see nonlinear in corrections to both the speed of CMW and its decay rate. Note that we also expect emergence in (7) of the following terms , , , and . 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 systems444We thank Dmitri Kharzeev for proposing us this study.. CME emerges from a nonzero axial chemical potential , which is usually assumed to have some background profile. It is, however, possible to induce (and thus ) dynamically through 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 and a time-dependent but spatially-homogeneous electric field . For simplicity the charge densities will be assumed to be spatially-homogeneous too555In principle it is not excluded that the charge densities could be spatially-inhomogeneous. Yet such spatial inhomogeneity would render the derivative resummation highly complicated.. From (2), could be set to zero. The constitutive relations for the vector/axial currents are

 Jt=0,             Ji=Ei+∂tEi+12κμ5Bi−12κϵijkAj(1)Ek+¯¯¯¯Gi(x=∞), (8)
 Jt5=ρ5,               Ji5=12κμBi−12κϵijkVj(1)Ek+¯Hi(x=∞), (9)

where , , and depend on , and nonlinearly and will be computed below. Our study is further split into two parts. In section 4.1, , , and will be evaluated perturbatively within the gradient expansion (5). These perturbative results can be found in (65-68). In section 4.2, we will consider another approximation—linearisation of the constitutive relations in the external electric field.

In the linearised regime, we assume the following scaling for , and

 ρ5∼O(ϵ),         →E(t)∼O(ϵ),         →B∼O(ϵ0), (10)

which will be referred to as amplitude expansion. To linear order in , the vector/axial currents are

 Jt=0,     →J=σe→E+τ1κρ5→B+τ2κ2(→E⋅→B)→B;          Jt5=ρ5,     →J5=0, (11)

where is a limit of the electric conductivity introduced in (1), while are new TCFs. As with other TCFs, they are functionals of time derivative operator and become functions of frequency in Fourier space,

 σe[∂t]⟶σe(ω),               τ1,2[∂t]⟶τ1,2(ω). (12)

At the linear level (in external fields and hydro variables), the transport coefficient functions in [9, 68] were proved to be frame independent. Along this line of proof, we expect that are also independent of hydro frame choice. Imposing the continuity equation (2), the electric current is put on-shell,

 Ji=σijEj,     σij=σeσT(δij−BiBjB2)+[σe−(12iωτ1−τ2)κ2B2]σLBiBjB2, (13)

where the transverse conductivity is not affected by the magnetic field in contrast to the longitudinal conductivity which gets corrected by the magnetic field via the chiral anomaly. In section 4.2 the TCFs will be first analytically evaluated in the hydro limit and then numerically for arbitrary frequency.

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 in DC limit, while [74] calculated its -dependence. Back-reaction effects on were considered in [76]. Ref. [77] performed similar study focusing 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] focused on a weak electric field, in which the axial current vanishes. So, 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 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 [10, 11, 7, 12, 9], only off-shell construction reveals transport properties of the system in full. Particularly, there are three independent TCFs ( and ) 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 and fields. This is in contrast to [74, 77], which limited their study to the case of parallel fields only, primarily focusing on the longitudinal electric conductivity . By varying the relative angle between and fields, one can separate the anomaly induced effects (parametrised by and ) from the ones that are not related to the anomaly ().

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 (10) and calculate the dependence of AC conductivity on magnetic field. The last section 5 presents the conclusions. Two Appendices supplement computations of sections 3 and 4.

## 2 The holographic model: U(1)V×U(1)A

The holographic model is the theory in the Schwarzschild-. The chiral anomaly of the boundary field theory is modelled via the gauge Chern-Simons terms in the bulk action

 S=∫d5x√−gL+Sc.t., (14)

where

 L=−14(FV)MN(FV)MN−14(Fa)MN(Fa)MN+κϵMNPQR2√−g×[3AM(FV)NP(FV)QR+AM(Fa)NP(Fa)QR], (15)

and the counter-term action is

 Sc.t.=14logr∫d4x√−γ[(FV)μν(FV)μν+(Fa)μν(Fa)μν]. (16)

The field strengths and are defined as

 (FV)MN=∂MVN−∂NVM,              (Fa)MN=∂MAN−∂NAM. (17)

is the Levi-Civita symbol with the convention , and the Levi-Civita tensor is . Our choice for (16) 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

 ds2=gMNdxMdxN=2dtdr−r2f(r)dt2+r2δijdxidxj, (18)

where , so that the Hawking temperature (identified as temperature of the boundary theory) is normalised to . On the constant hypersurface , the induced metric is

 ds2|Σ=γμνdxμdxν=−r2f(r)dt2+r2δijdxidxj. (19)

Equations of motion for and fields are

 dynamical~{}~{}equations:   EVμ=EAμ=0, (20)
 constraint~{}~{}equations:   EVr=EAr=0, (21)

where

 EVM≡∇N(FV)NM+3κϵMNPQR√−g(Fa)NP(FV)QR, (22)
 EAM≡∇N(Fa)NM+3κϵMNPQR2√−g[(FV)NP(FV)QR+(Fa)NP(Fa)QR]. (23)

The boundary currents are defined as

 Jμ≡limr→∞δSδVμ,             Jμ5≡limr→∞δSδAμ, (24)

which, in terms of the bulk fields, are

 Jμ=limr→∞√−γ{(FV)μMnM+6κϵMμNQR√−gnMAN(FV)QR−˜∇ν(FV)νμlogr},Jμ5=limr→∞√−γ{(Fa)μMnM+2κϵMμNQR√−gnMAN(Fa)QR−˜∇ν(Fa)νμlogr}, (25)

where is the outpointing unit normal vector on the slice , and is compatible with the induced metric .

The currents (24) are defined independently of the constraint equations (21). Throughout this work, the radial gauge 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 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.

It is useful to reexpress the currents (25) in terms of the coefficients of near boundary asymptotic expansion of the bulk gauge fields. Near ,

 Vμ=Vμ+V(1)μr+V(2)μr2−2VLμr2logr+O(logrr3),        Aμ=A(2)μr2+O(logrr3), (26)

where

 V(1)μ=FVtμ,           4VLμ=∂νFVμν. (27)

In (26) the constant term for is set to zero given that axial external fields are turned off in our present study. The holographic dictionary implies that is a gauge potential of external electromagnetic fields and ,

 Ei=FVit=∂iVt−∂tVi,    Bi=12ϵijkFVjk=ϵijk∂jVk. (28)

When obtaining (26,27), only the dynamical equations (20) were utilised. The near-boundary data and have to be determined by completely solving (20) from the horizon to the boundary. The currents (25) become

 Jμ=ημν(2V(2)ν+2VLν+ησt∂σFVtν),         Jμ5=ημν2A(2)ν. (29)

The remainder of this section is to outline the strategy for deriving the constitutive relations for and . 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 (20) assuming some charge densities and external fields, but without specifying them explicitly.

Following [9] we start with the most general static and homogeneous profiles for the bulk gauge fields which solve the dynamical equations (20),

 Vμ=Vμ−ρ2r2δμt,            Aμ=−ρ52r2δμt, (30)

where are all constants for the moment. Regularity requirement at fixes one integration constant for each and . As explained below (27), the constant in is set to zero. Through (29), the boundary currents are

 Jt=ρ,   Ji=0;           Jt5=ρ5,   Ji5=0. (31)

Hence, and are identified as the vector/axial charge densities.

Next, following the idea of fluid/gravity correspondence [14], we promote into arbitrary functions of the boundary coordinates

 Vμ→Vμ(xα),           ρ→ρ(xα),           ρ5→ρ5(xα). (32)

Then, (30) ceases to be a solution of the dynamical equations (20). To have them satisfied, suitable corrections in and have to be introduced:

 Vμ(r,xα)=Vμ(xα)−ρ(xα)2r2δμt+Vμ(r,xα),  Aμ(r,xα)=−ρ5(xα)2r2δμt+Aμ(r,xα), (33)

where will be determined from solving (20). Appropriate boundary conditions have to be specified. First, and have to be regular over the whole integration interval of . Second, at the conformal boundary , we require

 Vμ→0,      Aμ→0       as      r→∞, (34)

which amounts to fixing external gauge potentials to be and zero (for the axial fields). Additional integration constants will be fixed by the Landau frame convention for the currents,

 Jt=ρ(xα),           Jt5=ρ5(xα). (35)

The Landau frame choice can be identified as a residual gauge fixing for the bulk fields.

The vector/axial chemical potentials are defined as

 μ=Vt(r=∞)−Vt(r=1)=12ρ−Vt(r=1),μ5=At(r=∞)−At(r=1)=12ρ5−At(r=1). (36)

Generically, are nonlinear functionals of densities and external fields.

In terms of and , the dynamical equations (20) are

 0=r3∂2rVt+3r2∂rVt+r∂r∂kVk+12κϵijk[∂rAi(∂jVk+∂jVk)+∂rVi∂jAk], (37)
 (38)
 0=r3∂2rAt+3r2∂rAt+r∂r∂kAk+12κϵijk[∂rVi(∂jVk+∂jVk)+∂rAi∂jAk], (39)
 0=(r5−r)∂2rAi+(3r4+1)∂rAi+2r3∂r∂tAi−r3∂r∂iAt+r2(∂tAi−∂iAt)+r(∂2Ai−∂i∂kAk)−12∂iρ5+12κr2ϵijk(∂jVk+∂jVk)(∂rVt+1r3ρ)−12κr2ϵijk∂rVj[(∂tVk−∂kVt)+(∂tVk−∂kVt)+12r2∂kρ]−12κr2ϵijk{∂rAj[(∂tAk−∂kAt)+12r2∂kρ5]−∂jAk(∂rAt+1r3ρ5)}. (40)

In the following sections we will present solutions to the dynamical equations (37-40) under two independent setups discussed in section 1.

## 3 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 . From the general results (26,27),

 Vt,At∼O(logrr3),   Vi∼O(logrr2),   Ai∼O(1r2),   % as  r→∞. (41)

In obtaining large estimates for and , the frame convention (35) was used to fix the coefficients of in near-boundary expansion for (thus those of and ). The dynamical equations (37-40) get simplified,

 0=r3∂2rVt+3r2∂rVt+r∂r∂kVk+12κϵijk[∂rAi(∂jVk+∂jVk)+∂rVi∂jAk], (42)
 0=(r5−r)∂2rVi+(3r4+1)∂rVi+2r3∂r∂tVi−r3∂r∂iVt+r2(∂tVi−∂iVt)+r(∂2Vi−∂i∂kVk)−12∂iρ+r∂kFVki+12κr2ϵijk∂r(At−ρ52r2)(∂jVk+∂jVk)−12κr2ϵijk{∂rVj[(∂tAk−∂kAt)+12r2∂kρ5]−∂jAk∂r(Vt−ρ2r2)}−12κr2ϵijk∂rAj[(∂tVk−∂kVt)+12r2∂kρ], (43)
 0=r3∂2rAt+3r2∂rAt+r∂r∂kAk+12κϵijk[∂rVi(∂jVk+∂jVk)+∂rAi∂jAk], (44)
 (45)

For generic profiles of , and , nonlinearity makes it difficult to solve (42-45). To explore general structure of vector/axial currents, we rewrite the dynamical equations (42-45) into integral forms. In this way, near-boundary asymptotic expansion for and could be extracted from the integral forms of (42-45). For simplicity, we deposit the details into Appendix A. Substituting near-boundary behavior (94-97) into (29) produces the results (3,4). As mentioned below (3,4), are functionals of and are presented in (98,99). The formal analyse establishes the structure of , particularly the “non-renormalisation” of CME and its gradient corrections.

We proceed with hydrodynamic gradient expansion for . This requires us to perturbatively solve the dynamical equations (42-45) within the boundary derivative expansion (5),

 ∂μ=(∂t, ∂i)⟶(λ∂t, λ∂i). (46)

The corrections and are expandable in ,

 Vμ=∞∑n=1λnV[n]μ,            Aμ=∞∑n=1λnA[n]μ. (47)

At each order in , and form a system of ordinary differential equations in -coordinate, which can be solved via direct integration over . The results for and up to can be found in Appendix A, see (100-106).

Substituting the first order solutions (100-102) into (98,99) generates hydrodynamic expansion for up to second order in gradient expansion (throughout this work, the electromagnetic fields are thought of as of first order in derivative counting)

 Gi(x=∞)=−π8∂t∂iρ+(32π+3log2)κ∂tρ5Bi+18(1−2log2)κ2(ρ25+ρ2)×ϵijk∂jBk+18(2−3log2)κ2ϵijk(ρ5∂jρ5Bk+ρ∂jρBk)+O(∂3), (48)
 Hi(x=∞)=−π8∂t∂iρ5+(32π+3log2)κ∂tρBi+36(1−2log2)κ2ρρ5ϵijk∂jBk+18(2−3log2)κ2ϵijk(ρ5∂jρBk+ρ∂jρ5Bk)+O(∂3). (49)

Meanwhile, the second order results (103,104) give rise to the gradient expansion of chemical potentials (36)

 μ=ρ2+116(π−2log2)∂2ρ−34(π−2log2)κBk∂kρ5+18(1−2log2)κ2ρB2+O(∂3), μ5=μ(ρ↔ρ5). (50)

In principle, the second order results (103-106) could be inserted into (98,99), producing derivative expansion for and up to third order. However, at third order , computing becomes quite involved. So, at third order we decided to track only linear in terms. As a result, we are able to identify the first anomalous correction to the diffusion constant due to magnetic field. The final expressions are

 G[3]i(x=∞)=π248∂2t∂iρ+116(π−2log2)∂2∂iρ+12#1κ∂2tρ5Bi−π28κ×[∂2(ρ5Bi)−∂i∂k(ρ5Bk)]+34(π−2log2)κ∂i(Bk∂kρ5)+18(1−2