Magneto-vortical effect in strongly coupled plasma

Based on a holographic model incorporating both the chiral anomaly and the gravitational anomaly, we study the effect of magneto-vortical coupling on the transport properties of a strongly coupled plasma. The focus of present work is on the generation of a vector charge density and an axial current, as response to vorticity in a magnetized plasma. The transport coefficients parameterizing the vector charge density and axial current are calculated both analytically (in the weak magnetic field limit) and also numerically (for general values of the magnetic field). We find the generation of vector charge receives both non-anomalous and anomalous contributions, with the non-anomalous contribution dominating in the limit of a strong magnetic field and the anomalous contribution sensitive to both chiral anomaly and gravitational anomaly. On the contrary, we find the axial current is induced entirely due to the gravitational anomaly, thus we interpret the axial current generation as chiral vortical effect. The corresponding chiral vortical conductivity is found to be suppressed by the magnetic field. By the Onsager relation, these transport coefficients are responsible for the generation of a thermal current due to a transverse electric field or a transverse axial magnetic field, which we call the thermal Hall effect and the thermal axial magnetic effect, respectively.


Introduction
The effect of magnetic field and vorticity on QCD matter has attracted much attention over the past few years. At very high temperature, when quarks become asymptotically free, a charged-neutral QCD matter can be either magnetized in the magnetic field or polarized in the vorticity field. Close to the chiral phase transition, when the interaction among quarks becomes strong, more interesting phenomena such as inverse a e-mail: yybu@hit.edu.cn b e-mail: linshu8@mail.sysu.edu.cn (corresponding author) magnetic catalysis [1][2][3][4] and vector meson condensation [5,6] can emerge. Similarly, the vorticity field may suppress the chiral condensation [7,8].
Recently, the interplay of a strong magnetic field and a vorticity was found to lead to new transport phenomena such as dynamical generation of a vector charge [17]; see also [18][19][20][21]. Under the lowest Landau level (LLL) approximation, Hattori and Yin found the generation of a vector charge from spin-vorticity coupling as [17] 1 with C A = 1 2π 2 the chiral anomaly coefficient. In fact, such a contribution should be viewed as a large B, free limit of a QED plasma. More generally, one would expect from the viewpoint of polarizable matter [22] that Moreover, if we could associate an effective chemical potential for the generated vector charge, this can further give rise to the generation of an axial current by the chiral anomaly and gravitational anomaly. Note that the vector charge susceptibility is χ = C A |q f |B in the LLL approximation, and thus (1) corresponds to the effective chemical potential μ eff = sgn(q f ) 2 ·B withB = B/|B|. The vector charge imbalance would result in an axial current through the chiral separation effect [15,16] Again, this is a large B, free limit of a QED plasma. More generally, one would expect an extra contribution from the gravitational anomaly, which always induces a temperaturedependent contribution to the axial current, even in the absence of the chiral imbalance [14,23,24]. Therefore, we expect the more general axial current It is worth noting that the physical picture behind (1) and (3) is the spectral flow: a shift in the background vector gauge field leads to an opposite energy shift for right-and left-handed fermions, generating a net axial charge. In order for the spectral flow picture to generate vector charge, we would need an axial gauge field, whose couplings to rightand left-handed fermions differ in sign, thus leading to the same energy shift for them. In the analysis of Hattori and Yin [17], the role of the axial gauge field is played by the vorticity. Indeed, in free theory, we have S = J 5 so that we can identify A 5 with by comparing the coupling · S with J 5 · A 5 . However, in an interacting theory, the "equivalence" of A 5 with is far from obvious. First of all, even in a free theory the presence of an axial gauge field as a source poses an ambiguity in the definition of currents: consistent current and covariant current could differ by terms proportional to the axial gauge field [25]. A similar ambiguity does not exist in the case with the vorticity as a source. Secondly, in an interacting theory, the vorticity couples to the angular momentum as a whole. The separation of the spin from the total angular momentum is often ambiguous. Therefore, it is desirable to go away from the free theory limit to test the robustness of the mechanism. In this paper, we go to the opposite limit, where the theory is strongly coupled. Specifically, we will study the response of a strongly coupled magnetized plasma to the vorticity field by a holographic model.
The rest of the paper is organized as follows: In Sect. 2, we present the setup of the holographic model. In Sect. 3, we turn on a metric perturbation as a proxy for the vorticity in the magnetized plasma. We will study the response of the vector charge density and axial current to the vorticity. In Sect. 4, we will present both analytic results in the small B regime and numerical results for general B. In Sect. 5, we use the Onsager relation to obtain the thermal Hall effect and the thermal axial magnetic effect. We conclude and discuss implications of our results in Sect. 6. Details of the computations are collected in Appendices A, B and C.
2 Holographic setup: magnetic brane in Ad S 5

Gravity action and dictionary
We extend the holographic model initially considered in [26,27] by including both vector and axial gauge fields. The full action is where F V = dV and F a = d A. The last line of (5) corresponds to boundary terms defined on the hypersurface of constant r . The notation γ denotes the determinant of the induced metric γ μν on : We also need the out-pointing unit normal vector of the surface : Moreover, K = γ μν K μν whereas K μν is the extrinsic curvature tensor The Levi-Civita tensor is M N P Q R = (M N P Q R)/ √ −g whereas (M N P Q R) is the Levi-Civita symbol under the convention (rt x yz) = +1. The purely gauge Chern-Simons action (α-terms) mimics the chiral anomaly, while the mixed gauge-gravitational Chern-Simons term (λ-term) is to model the gravitational anomaly of the boundary field theory.
As explained in [27], in order to get a correct form of the gravitational anomaly (i.e. guarantee the gauge variation of the bulk action to be a total derivative), one needs to add the term where∇ is compatible with the induced metric γ AB . The counter-term action is where C t cancels the logarithmic divergences [28,29] Note that C t non-vanishes only when non-trivial sources (either external gauge fields or non-flat boundary metric) are turned on for the boundary theory. In addition, in C t we employ the minimal subtraction scheme so that it will not generate finite contribution to the boundary currents and stress tensor. According to the holographic dictionary, the expectation values of the stress tensor and currents of the boundary theory are defined as Explicitly, the vector current is (from now on, we set 2κ 2 = 1 for convenience) However, the axial current and stress tensor are somewhat subtle and complicated: where T C μν arises from the functional derivative of C t , J μ CSK is due to the added action S CSK , and T Gra μν comes from the gravitational Chern-Simons term. The expressions for all of them are [27,30] where T C 1 μν vanishes for a flat boundary. T μν Gra was first derived in [30] based on the ADM decomposition approach. Above, we stick to the consistent current formalism. Indeed, in the absence of a background for the axial gauge field, there will be no difference between the consistent current and covariant current [25]. The authors of [26] presented a thorough analysis for the holographic renormalization of the model, but did not get the term S CSK . Additionally, the authors of [26] addressed the fact that the gravitational Chern-Simons term will make a contribution to the boundary stress tensor. See also [30,31] for more recently updated formulas for stress tensor and axial current of the boundary theory. The holographic model does correctly describe the chiral/gravitational anomalies for the boundary field theory [27]: where a hat means that the corresponding quantity is defined on the boundary. Under the variation one obtains the Einstein equation and anomalous Maxwell equations The bulk stress tensor T bulk M N could be split into two parts, where Alternatively, the Einstein equation could be rewritten as wherẽ

Neutral magnetic brane background
To proceed, we consider the background solution of the holographic model (5). For simplicity, we focus on a neutral magnetized plasma. To this end, we turn on a constant magnetic field along the z-direction, which obviously breaks the SO(3) rotational symmetry to SO(2) ⊥ on the x y-plane. As a result, the background metric takes the form Note that in writing down (28), the ingoing Eddington-Finkelstein coordinate has been employed in order to avoid a coordinate singularity. The background metric (28) has an event horizon at r = r h so that while W T , W L are regular at r = r h . The Hawking temperature, identified as the temperature of the dual gauge theory, is Generically, both W T (r ) and W L (r ) will depend on r h nontrivially.
It is a simple exercise to check that, given the above ansatz (27) and (28), both the gauge and the gravitational Chern-Simons terms do not affect the bulk equations of motion. Therefore, the background geometry is simply the "magnetic brane" solution initially studied in [32].
The ordinary differential equations (ODEs) for the metric functions in (28) are where the prime denotes a derivative with respect to r . Equations (33) and (34) look different from those of [32]. However, suitable combinations of the above equations give rise to the results of [32]: Obviously, not all the equations in (31)-(34) are independent: we will take (31) as the constraint and solve the remaining ones to determine the metric functions f (r ), W T (r ), W L (r ).
In order to fully determine f (r ), W T (r ), W L (r ), we have to impose two boundary conditions for each of them. For f (r ), we impose However, it is found that the second boundary condition (underlined above) is automatically satisfied by the bulk EOMs. This requires one to impose another condition for f (r ), which is explained in Appendix 1. For W T (r ) and W L (r ), the boundary conditions are where the last two equations are read off from (33) and (34) by requiring regularity of W T (r ), W L (r ) at the horizon r = r h . We solve the bulk EOMs (31)-(34) analytically when the magnetic field is weak (i.e., B/T 2 1) and numerically when B is general. The calculational details as well as the main results are deferred to Appendix A.

Bulk perturbations
In this section, we study the linear response of the magnetized plasma to a fluid vorticity. A weak fluid vorticity would be mimicked by a gravito-magnetic field [27,31]. More precisely, one perturbs the boundary Minkowski spacetime (where the fluid flows) as Then the vorticity is generated at linear order in h ti as with the unperturbed fluid velocity u μ = (1, 0, 0, 0). Thus, the curl of h ti could be thought of as a fluid vorticity. We take which gives rise to a stationary vorticity along the z-direction, i.e. parallel to the magnetic field. Then we can obtain the Kubo formulas for the transport coefficients defined in (2) and (4) as These will be used in holographic calculations.
To turn on a gravito-magnetic field in the bulk, it is convenient to use the Poincaré coordinate system so that the bulk metric takes a diagonal form, where we still denote the time of the bulk theory by t. On top of the background (44) and (27), it is consistent to turn on the following fluctuation modes: while setting all the rest corrections to zero. Here, we have assumed (t, x)-dependent fluctuations for a reason to be discussed in the next subsection and we consider a plane wave ansatz: In what follows we record the bulk equations of motion for the fluctuation modes. First, we consider the constraint equations. The constraint E r y = 0 is The constraint E V r = 0 gives Next, we turn to the dynamical components of the bulk EOMs. The Einstein equation E ty = 0 reads The Einstein equation E xy = 0 is

Adiabatic limit and boundary conditions
From the Kubo formulas (43), it seems as if we could set ω = 0 from the beginning. However, the boundary condition at the horizon cannot be uniquely determined in this case. This ambiguity is related to the ambiguity in the Kubo formula itself. It can be evaluated in any equilibrium state, be it charged or neutral. For our purpose, it should be evaluated in the unperturbed neutral plasma state. The boundary condition to use should correspond to the neutral state. In practice, we specify the state as follows: the state is realized by turning on the vorticity field adiabatically to the original neutral magnetized plasma. We will seek solutions to (47) through (53) in the adiabatic limit ω → 0. To this end, we expand the bulk perturbations in powers of ω: with X = δg xy , δV x , δg ty , δV t and δ A z . In fact, we only need the leading order solution X (0) , for which we suppress the superscript (0). The fields decouple into two sets {δg xy , δV x } and {δg ty , δV t , δA z }. The set {δg ty , δV t , δA z } satisfies the following equations: The boundary conditions on the horizon need to be derived by matching with the horizon solutions in the limit ω → 0. We elaborate on the derivation in Appendix B. The resultant boundary conditions on the horizon are given by The free parameters for the three fields can be chosen as horizon derivatives of δg ty , δV t and horizon value of δ A z . The three parameters on the horizon can be mapped to boundary values of the three fields. We can further simplify Eqs. (55)-(57) by considering the limit q → 0. Note that δg ty has an opposite parity to those of δV t and δ A z , and δg ty is the only field sourced on the AdS boundary. Therefore, the AdS boundary conditions are We expect the scaling behaviors δg ty ∼ O(q 0 ), and δV t , δA z ∼ O(q). Defining δV t = iqδṼ t and δ A z = iqδÃ z , we can further simplify (55) through (57) by keeping the leading terms in the q-expansion

The correlators J t T t y and J z 5 T t y
In this section, we calculate the generation of J t and J z 5 as linear responses to the external source h ty . Note that h ty couples to T ty in the boundary theory. We can express the responses in terms of retarded correlators J t T ty and J z 5 T ty as with the retarded correlators defined as Equation (61) relates the above retarded correlators in equilibrium to the responses of J t and J z 5 to h ty , which are exactly the quantities we will calculate. So, instead of directly computing the retarded correlators (62), we will read off them by computing the one-point functions of J t and J z 5 (with the only source h ty turned on).
The bulk EOMs (60) will be solved under the boundary conditions (58) and (59). This section will be further split into two parts: an analytical study when the magnetic field is weak versus a numerical study when the value of the magnetic field is generic. In these two complementary studies, we will utilize the results of the background metric functions summarized in Appendix A.
In the limit ω → 0 and q → 0, the vector charge density and the axial current are (in terms of the bulk fields) Near the AdS boundary, the bulk fluctuations behave as So, the vector charge density and axial current for the boundary theory are Below we solve for J t and J z 5 perturbatively in B and also numerically for generic B.

Weak magnetic field: a perturbative study
When the magnetic field B is weak, the bulk fluctuations are expandable, ty + 2 δg [2] ty + · · · , δṼ t = δṼ [1] t where ∼ B. At the lowest order O( 0 ), first we have Then we have whose solution is At the first order O( 1 ), which is solved by At the second order O( 2 ): Here, we would like to recall that W (2) T is obtained in (114). The solution for δg [2] ty would be δg [2] T (x) where the integration constant C is fixed as T (x) So, The equation for δÃ [2] z is where The solution and the near-boundary expansion for δÃ [2] z are δÃ [2] Up to O(B 2 ), the vector charge density and axial current on the boundary are The transport coefficients ξ and σ are where we have substituted the perturbative expression (116) for r p . The transport coefficient ξ contains both non-anomalous contribution and anomalous contribution proportional to αλ. The non-anomalous contribution is consistent with the prediction of the non-anomalous magnetohydrodynamics (MHD) [33], which for a neutral plasma has 2 with M ≡ ∂F ∂(B· ) being the magneto-vortical susceptibility and 2 p ,B 2 ≡ 2 ∂ p ∂(B 2 ) being the magnetic susceptibility. Here p is the pressure and F is the free energy density [33]. Note that in (81) we ignored terms nonlinear in . Both susceptibilities M and 2 p ,B 2 can be calculated independently. In the weak B field limit, the magnetic susceptibility 2 p ,B 2 can be calculated from the perturbative background we already obtain in Appendix A. While the magneto-vortical susceptibility M vanishes for a neutral plasma by charge conjugation symmetry, M ,μ ≡ ∂ M /∂μ does not. The quantity M ,μ can even be calculated in a charged plasma at B = 0.
In Appendix C, we calculate both susceptibilities with the following results: The negative value of M ,μ is consistent with the fact that spin-vorticity coupling lowers/raises energy of particle/antiparticle. Clearly, (81) and (82) are in perfect agreement with (79).
The dependence log r h may look odd at the first sight. To restore units, we should use the replacement log r h → log(r h L). In fact, this transport coefficient is scheme dependent. The appearance of the AdS radius L comes from the fact that we use 1/L as our renormalization scale [34]. Other physically significant renormalization scales could be used, which could alter this term [34]. It is also interesting to note that the scheme dependence is related conformal anomaly. In fact, a different scheme would correspond to adding a finite counter-term as Such a counter-term would give the following contribution to the vector current: In the presence of h ty (x), we can easily obtain J t ∼ a B · . Therefore the combination M ,μ − 2 p ,B 2 can be shifted by a constant. Note that the scheme dependence of the vector charge density is absent in a free theory. The anomalous contribution is proportional to λα B. The EOM (60) suggests the following chain of responses: δ A z ∼ O(λ) is induced in response to vorticity and then backreaction of δ A z to δV t gives J t ∼ O(λα B). This corresponds to the backreaction of J z 5 generated by CVE to J t on the field theory side. A possible α 2 B 2 -term would emerge at the next order O(B 2 ). In this case, (60) suggests the following chain of responses: a non-anomalous contribution to J t is generated by magneto-vortical coupling. Then J z 5 is induced by CSE. The backreaction of J z 5 to J t would give the O(α 2 B 2 ) contribution. However, the above reasoning is not quite accurate. As we show below, in fact CSE is not generated in the presence of J t . Nevertheless the bulk profile of δ A z does backreact to δV t to give α 2 B 2 correction to J t .
In contrast to ξ , the transport coefficient σ is scheme independent. We can add an analogous counter-term as (83). It would not contribute to J z 5 in the absence of a background axial gauge field. The structure of σ is relatively simple. Aside from the T 2 -correction to the CVE (i.e., the first piece in σ ), σ encodes the correction to chiral vortical coefficient from B. However, there is no contribution proportional to α.
In other words, no CSE is seen despite the generation of J t . This is in contrast to the naive expectation from CSE with χ being the vector charge susceptibility. In fact, from the holographic model, the absence of CSE holds more generally: if we integrate (57) from the horizon to an arbitrary r , we obtain (with λ = 0) where the horizon boundary conditions (58) have been utilized to fix the integration constant. Taking r → ∞ and notingṼ t (r → ∞) = 0, we have Following [35], we should identify the difference of δV t on the boundary and on the horizon as the vector chemical potential It then follows naturally from (86) that J z 5 = 8α Bμ. In our case, we have μ = 0 but J t = 0.
To understand the physical difference between J t and μ, we note μ (−μ) is the extra energy cost to create one unit of particle (anti-particle), but J t depends on actual distribution of particles and anti-particles. In our case μ = 0 implies that it costs the same energy to create both particle and antiparticle. Indeed, we can view δV t as a zero mode as it vanishes both on the horizon and on the boundary, which supports the picture of vanishing energy cost for creating a particle. Since the state is obtained by the adiabatic limit, it means J t is generated dynamically 3 . The CSE seems to be only sensitive to the energy difference μ, not the charge density. The absence of CSE can also been understood from the scheme independence J z 5 : unlike J t , J z 5 is unaffected by the choice of scheme from (63). This makes natural for J z 5 to depend on μ, rather than on J t .

Generic magnetic field: a numerical study
For generic values of B, we will solve the fluctuation EOMs (60) through the shooting technique. First, we find the nearhorizon solution: where, thanks to the horizon condition (58), only δg 1 ty , δṼ 1 t and δÃ 0 z are undetermined. Then we will choose a reasonable value of δg 1 ty (this corresponds to turning on a specific source h ty ) and fine-tune δṼ 1 t , δÃ 0 z until δṼ t (r = ∞) = δÃ z (r = ∞) = 0 are satisfied. From the numerical solution, we can read off the expectation values of J t and J z 5 , as responses to the source h ty only. In practical numerics, we set the horizon data δg 1 ty = −1. However, there is one problem in the procedure mentioned above. Since we intend to solve the background EOMs (32)-(34) using the initial conditions (122) and (124), we should be careful in solving the fluctuation EOMs (60). More precisely, the correct solutions are where the starred functions δg * ty , δV * t and δ A * z are solved from (60) using the "incorrect" numerical background metric functions, as discussed in Appendix A. Adapted to the tilde variables, we have Here, for the sake of numerical calculation, we have further re-scaled the term δṼ t of (60) by a factor of B.
For convenience, we set r h = 1 in our numerical calculations. So, the dimensionful quantities (J t , J z 5 etc.) to be plotted in Sects  When the chiral anomaly and gravitational anomaly are turned off (i.e., α = λ = 0), the transport properties of the magnetized plasma get non-anomalous contributions from the medium only. The medium effects are not covered by the study of [17] since the calculations therein are essentially based on the vacuum state. In this situation, as we discussed in the previous section we only see a dynamically generated vector charge density J t , whereas the correlator J z 5 T ty /(iq) (and thus J z 5 ) vanishes identically as seen from (87). In Fig. 1, we show the correlator J t T ty /(iq) and the transport coefficient ξ as a function of B/T 2 . For the purpose of probing the strong magnetic field limit, we have improved our numerical calculations and generate plots up to B/T 2 ∼ 3000.
From the left panel of Fig. 1, it seemingly implies a quasilinear growth for J t T ty /(iq) as B/T 2 is increased, which as we will show is inaccurate. The right panel of Fig. 1 reveals more information: −ξ approaches 1 from above. In Fig. 2, we fit our numerical result for ξ in the strong magnetic field limit by the following function:  It is tempting to conclude that −ξ → 1 asymptotically. The correction in (92) can be understood as the v 2 t term in the general expression (65) by noting that r h = 1 in our numerical results.

Anomalous effects: λ = 0
We now turn attention to the anomalous contributions to the transport properties of the magnetized plasma. While the two anomaly coefficients, α, λ, are fixed for a specific QFT on the boundary, we here take a phenomenological viewpoint and think of α, λ as free parameters. First of all, taking λ = 0 will kill J z 5 completely, as seen from the bulk EOMs (60). Thus, our representative choices for the anomaly coefficients are We begin with the fate of the CVE conductivity σ . First, the last equation of (60) could be formally integrated from the horizon to the AdS boundary, yielding where μ is defined in (88). Here, we stress that the CVE conductivity σ depends on λ linearly and is independent of α.
In the left panel of Fig. 3, we plot the CVE conductivity σ as a function of B/T 2 , taking all choices for α, λ from (93). From the plot, we obviously see perfect overlapping of different curves, confirming our claim that σ linearly depends on λ only. Intriguingly, the magneto-vortical coupling effect tends to suppress the CVE conductivity and eventually renders it vanishing at large magnetic field. Asymptotically, σ ∼ B −1 as demonstrated in the right panel of Fig. 3. Similar suppression effects due to the quark mass [38,39] and spacetime curvature [40] are seen. Our findings are in contrast to the proposal of [17] that the magneto-vortical coupling (through the chiral anomaly) generates a term to σ linear in B when B/T 2 becomes very large. Next, we consider anomalous contributions to the generation of the vector charge density J t and the transport coefficient ξ . Given that J t always gets a non-anomalous contribution, we find it more transparent to consider In accord with the different choices for α, λ as made in (93), we show anomalous corrections δ J t T ty /(iq) and δξ in  From Fig. 4, we read that the anomalous contribution to the vector charge δ J t has opposite sign to their non-anomalous counterpart for a weak magnetic field. As the magnetic field becomes stronger, δ J t changes sign and continues to grow mildly at large B/T 2 . More precisely, the numerical results at large B imply the following asymptotic behaviors for the anomalous corrections: which are clearly confirmed by the plots of Fig. 6. It is worth noting that from (92) and (96) the large B limit of ξ is dominated by the non-anomalous medium contribution.

Thermal Hall effect and thermal axial magnetic effect
In the previous section, we have obtained the following transport coefficients: which can be viewed as responses of J t and J z 5 to h ty (cf. (61)): where we have substituted the expression for in (41). The correlators in (98) give the following transposed correlators by the Onsager relation: The subscript indicates the expectation values are taken with −B. We can then obtain the response of T ty to both A t and A 5 z using (99) with the signs of both q and B flipped where we have used E x (q) = −iq A t (q) and B 5y (q) = −iq A 5 z (q). In a neutral plasma, (100) corresponds to the generation of thermal current by transverse electric field and transverse axial magnetic field, which we coin the thermal Hall effect and the thermal axial magnetic effect, respectively. Below we will derive (100) more rigorously using the time-reversal symmetry. We start with the following correlators for the responses to h ty : with the correlators in (101) being the limit ω → 0 of the following retarded correlator: By time-reversal symmetry, we can obtain the transposed correlators by [41] with γ a = ±1 corresponding to eigenvalues of operator O a under time-reversal. Note that B flips sign under timereversal, which we indicate in the subscript. For the operators of our interest T ty , J t , J z 5 , we have γ = −, +, +, respectively. Therefore, (101) and (103) give the following transposed correlators (with the direction of B reversed and ω = 0): Note that J t (q) and J z 5 (q) couple to sources V t (−q) and A z (−q), respectively. We can rewrite (104) in a more intuitive way: The thermal Hall effect contains both non-anomalous and anomalous contributions, where the non-anomalous contribution can be understood from non-anomalous MHD [33]. Naively, turning on E x necessarily induces a steady flow along v y due to the Lorentz force acting on positive and negative charge carriers. However, this is not true for a stationary state. The stationary state can be obtained simply by setting ω = 0 in (47) through (53). In this case, the dynamics of the fields δg ty , δV t , δA z decouple from δV x , δg xy . We can thus consistently set δV x and δg xy to zero, leading to vanishing J x and T xy . This strongly constrains the hydrodynamic analysis. Note that both J x and T xy contain the dissipative terms as follows: The shear contribution in T xy cannot be canceled by other terms. The only possibility for T xy to vanish is to have v y = 0. This implies that an inhomogeneous vector charge density is needed for the stationary state: E x − ∂ x μ = 0. Indeed, this is consistent with the holographic analysis if we identify With v y = 0, the nonanomalous contribution to the thermal current is given by [33,[42][43][44]) The first term can be identified as −E x M by noting 2 p ,B 2 B = M. The second term can be interpreted as −P x B if we identify E x M ,μ as an effective polarization P x . Apart from the non-anomalous contribution, we also obtain an anomalous contribution that requires at least the chiral anomaly to exist. This can be seen from the middle equation in (60). When α = 0, the dynamics of δ A z decouples from that of δV t , leaving only a non-anomalous contribution. On the contrary, the thermal axial magnetic effect contains only an anomalous effect. Its existence relies on the gravitational anomaly.

Conclusion
In this work, based on a holographic model, we considered the effects of the magneto-vortical coupling on the transport properties of a strongly coupled plasma. First of all, even when the chiral and gravitational anomalies are turned off, the coupling of a magnetic field and a weak fluid's vorticity dynamically generate a contribution to the vector charge density J t , which we refer to as a non-anomalous medium contribution. The non-anomalous medium contribution in J t grows linearly in B and the relevant transport ξ approaches a constant in the strong magnetic field limit. However, a similar non-anomalous contribution is not observed for the axial current. Secondly, the magneto-vortical coupling also generates anomalous contributions to the vector charge density J t and axial current J z 5 . Thanks to the absence of a background for the vector chemical potential, the anomalous contribution to J z 5 is completely induced by the gravitational anomaly (i.e., insensitive to the chiral anomaly). In this sense, it would be more natural to interpret the anomalous contribution to J z 5 as a CVE contribution [27], rather than a CSE contribution. The corresponding chiral vortical conductivity receives a correction at finite B. In particular, in the strong magnetic field limit, the magneto-vortical coupling renders the CVE conductivity vanishing asymptotically. This is quantitatively different from the conclusion of [17]. In contrast to that of J z 5 , the anomalous contribution to J t requires the chiral anomaly to exist. The presence of a gravitational anomaly can also affect the generation of J t . Thus, the anomalous contribution to J t would contain more fruitful physics. Particularly, in the strong magnetic field limit, the anomalous part of J t seemingly grows logarithmically as a function of B. This is to be compared with the results of [17] where a term linear in B was generated in J t by the chiral anomaly.
Our findings summarized above would necessitate the formulating of a consistent/complete anomalous magnetohydrodynamics. This requires one to consistently add novel transport phenomena induced by anomalies into the nonanomalous magnetohydrodynamics [33,42]. While our study treated the magentic field as external, the case with a dynamical electromagnetic field is also interesting. The corresponding anomalous MHD has been initially considered in [45] by assuming a small chiral anomaly coefficient. In fact, holographic models corresponding to a dynamical electromagnetic field have been proposed in [46,47]. Including anomalies in the model would allow us to study anomalous MHD without further assumptions on the anomaly coefficients. We leave this for future work.
Last but not least, the transport phenomena we discussed are dissipationless. It is possible to derive them by including anomalies in the partition function approach [48]. This would allow us to obtain more complete dissipationless transport phenomena. We hope to address this in the future. (NSFC) under the grant No.11705037. SL was supported by the NSFC under the grant Nos. 11675274 and 11735007. SL also thanks Yukawa Institute of Theoretical Physics for hospitality and the workshop "Quantum kinetic theories in magnetic and vortical fields" for providing a stimulating environment during the final stage of this work.

Data Availability Statement
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Appendix A: Details of solving the background metric functions
In this appendix, we collect calculational details of solving the background metric functions.
When the magnetic field is weak (i.e. B/T 2 1), we construct the bulk metric functions perturbatively [49]: where a formal parameter ∼ B/T 2 is introduced to mark the perturbative expansion. f (n) (r ), W (n) T (r ) and W (n) L (r ) are solved from the bulk equations. We are interested in n = 2.
At O( 2 ), the constraint equation (31) yields where the integration constant c 1 should be set to zero due to asymptotic AdS requirement (37). By redefinition of the radial coordinate r , we could also set c 2 = 0. Thus, Then the dynamical equation (32) is solved as Substituting (109) and (110) into the dynamical equations (33) and (34), we obtain Obviously, in order to be consistent with (109), one has to set c 1 f = 0. The integration constant c 2 f is fixed by the location of the horizon, which will be presumably shifted due to the presence of a magnetic field, where r (2) p represents the location of the event horizon at O( 2 ). Then From (33), W T is solved as where in the last equality of the second line we have made use of the fact that the difference between r (2) p and r h is of O(B 2 ). At O( 2 ), the relation between the location of the event horizon and the Hawking temperature becomes which is solved as When the value of B is generic, we have to solve the metric functions numerically. We find it more convenient to make the change of variables followed by Then the dynamical bulk equations (32)-(34) turn into where, since we have set r h = 1 above, B should be understood as B/r 2 h . Near the AdS boundary u = 0, the metric functions U, V, W are expanded as where we have made use of the constraint equation (31). Obviously, the asymptotic boundary conditions only give rise to "two" effective requirements! The regularity requirements will yield another three conditions. Just as in the fixing of c 2 , we can utilize the freedom of redefining the radial coordinate u and set U 1 b = 0. To summarize, the boundary conditions at u = 0 (the AdS boundary) are To find the numeric solutions, one can proceed in two different ways. The first approach would be to directly solve (119) under the boundary conditions (121) and (122). A second approach would be to replace the boundary conditions (121) by the following conditions at the horizon: where Note that the choice of U 1 h will set π T = 1. However, solving (119) under the initial conditions (122) and (124), near the boundary u = 0 the solution will behave as Then the correct solution would be obtained by a further rescaling of the boundary coordinates, Due to the "incorrect" asymptotic boundary behavior (125), we have relabeled the magnetic field by b in (125) and (126). When solving the EOMs (119) under the initial conditions (122) and (124), the same relabeling should be made. Recalling the definition of the magnetic field F V = bdx ∧ dy, the physical magnetic field B (in units of r 2 h ) should be Finally, we would like to point out that the background solution obtained with conditions (122) and (124) does not necessarily satisfy U 1 b = 0 (cf. (120)).
The ω → 0 limit of (129) determines the horizon derivative of δg ty and δV t . For the decoupled EOMs (55) through (57), we can take the horizon derivatives of δg ty and δV t , and horizon value of δ A z as free parameters.
The dynamical components of the vector Maxwell equations ∂ r r f (r )∂ r δV y + 2Q∂ r δg ty + 1 r q x q y δV x − q 2 x δV y = 0.