Enhancement of Anomalous Boundary Current by High Temperature
EEnhancement of Anomalous Current by High Temperature
Ruiping Guo, Rong-Xin Miao ∗ School of Physics and Astronomy, Sun Yat-Sen University, 2 Daxue Road, Zhuhai 519082,China
Abstract
Recently it is found that Weyl anomaly leads to novel anomalous currents in thespacetime with a boundary. However, the anomalous current is suppressed by the mass ofcharge carriers and the distance to the boundary, which makes it difficult to be measured.In this paper, we explore the possible mechanisms for the enhancement of anomalous cur-rents. Interestingly, we find that the anomalous current can be significantly enhanced bythe high temperature, which makes easier the experimental detection. For free theories,the anomalous current is proportional to the temperature in the high temperature limit.In general, the absolute value of the current of Neumann boundary condition first de-creases and then increases with the temperature, while the current of Dirichlet boundarycondition always increases with the temperature. ∗ Email: [email protected] a r X i v : . [ h e p - t h ] F e b ontents Weyl anomaly measures the quantum violation of scaling symmetry of a theory, and has awide range of applications in black-hole physics, cosmology and condensed matter physics [1].It is interesting that Weyl anomaly is well-defined for not only conformal field theories but alsothe general quantum field theories [1, 2, 3]. Recently, it is found that, due to Weyl anomaly, anexternal electromagnetic field can induce novel anomalous currents in a conformally flat space[4, 5] and a spacetime with boundaries [6, 7]. It is similar to the anomaly-induced transport[8] such as chiral magnetic effect (CME) [9, 10, 11, 12, 13] and chiral vortical effect (CVE)[14, 15, 16, 17, 18, 19, 20]. See also [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]for related works.In this paper, we focus on the anomalous current in the spacetime with a boundary [6, 7].It takes the following universal form in four dimensions < J µ > = − βF µν n ν x + ..., x ∼ , (1)where β is the beta function, F µν are the field strength, x is the proper distance to theboundary, n µ are the normal vectors and ... denotes higher order terms in O ( x ). It shouldbe mentioned that there are boundary contributions to the current, which cancels the bulk“divergence” of (1) and makes finite the total current [6]. Note that (1) applies to the regionnear the boundary. The anomalous current in the full space is studied in [32], where it is1igure 1: The magnetizing current induced by an external magnetic field near a boundary.In the left figure, there is no magnetic field and the average current is zero. In the rightfigure, we turn on an external magnetic field, which induces a non-zero magnetizing current.The higher the temperature is, the stronger the magnetizing current is.found that the mass and the distance to the boundary heavily suppress the currents. See also[25].In this paper, we explore the mechanism for the enhancement of anomalous currents,which is important for the experimental measurement. We find that the high temperaturecan greatly enhance the anomalous currents. For free complex scalars, the anomalous currentis proportional to the temperature in the high temperature limit. As a result, for any givenmass and distance, one can always produce a detectable anomalous current by increasingthe temperature. At first glance, it is surprising because usually the high temperature tendsto destroy the fragile quantum effects. The typical example is the superconductivity, whichusually appears at low temperatures. Another example is the quantum entanglement, whichcan be easily destroyed by the environment at high temperature due to the decoherence.Notice that the anomalous current originates from the vacuum magnetization [6]. Fromthe viewpoint of magnetization, it is natural that the anomalous current can be enhanced bythe high temperature. That is because, usually, the higher the temperature is, the strongerthe magnetizing current is. Take the classical magnetization as an example. Suppose thatthere is a box of free electron gas at a finite temperature. At the beginning, the averagecurrent is zero due to the disorderly thermal motions of electrons. Now turn on a backgroundmagnetic field. Although the external magnetic field does not do any work, it selects a specialmovement direction of electrons under the influence of the boundary. In other words, themagnetic field induces a magnetizing current near the boundary. The electron gas at highertemperature has a larger average kinetic energy and results in a stronger magnetizing current.Thus it is not surprising that the anomalous current is enhanced by the high temperature.See Fig. 1 for the physical picture of magnetizing currents.2et us summarize the properties of anomalous currents at a finite temperature below. Forsimplicity, we focus on free complex scalars. . The anomalous current is proportional to the temperature in the high temperaturelimit. Remarkably, the coefficient is just the anomalous current in lower dimensions at zerotemperature. See (43). . The anomalous current of Dirichlet boundary condition (DBC) always increases withtemperature, while the absolute value of the current of Neumann boundary condition (NBC)first decreases and then increases with temperature. See Fig. 3 and Fig. 4. . Although the large mass always suppresses anomalous currents, the small mass couldenhance the anomalous current for NBC. See Fig. 6 (right).The paper is organized as follows. In section 2, by applying the heat-kernel method[37, 38], we study the anomalous current for free complex scalars at finite temperature upto the linear order of magnetic fields. We find that it is proportional to the temperature inthe high temperature limit. In section 3, we obtain a non-perturbative formal expression ofthe anomalous current, which can be evaluated numerically. Again, the anomalous current isenhanced by the high temperature. Finally, we conclude with some discussions in section 4. In this section, by applying the heat kernel method [37, 38], we study the anomalous currentof complex scalars at a finite temperature. It is found that, in the high temperature limit, theanomalous current increases linearly with temperature. As for the case of low temperature,the anomalous current increases with temperature for Dirichlet boundary condition (DBC),while decreases with temperature for Neumann boundary condition (NBC).For simplicity, we focus on a flat half space x ≥ B parallelto the boundary. We have coordinates x µ = ( τ, x, y a ) = ( τ, x, y , ..., y d − ), background vectorfield A µ = (0 , , Bx, , ...,
0) and the metric g µν = δ µν = diag(1 , , ...., τ (cid:39) τ + β isthe Euclidean time, β = 1 /T is the inverse temperature and x denotes the distance to theboundary. The heat kernel of complex scalars satisfies the equation of motion (EOM) ∂ t K ( t, x µ , x (cid:48) µ ) − δ µν ( ∂ µ + A µ )( ∂ ν + A ν ) K ( t, x µ , x (cid:48) µ ) = 0 (2)3ogether with the following boundary conditions (BC)lim t → K ( t, x µ , x (cid:48) µ ) = δ d ( x µ − x (cid:48) µ ) , (3) K ( t, τ, τ (cid:48) ) = K ( t, τ + β, τ (cid:48) ) = K ( t, τ, τ (cid:48) + β ) , (4)for t and τ , respectively. Besides, one further imposes either DBC K ( t, x µ , x (cid:48) µ ) | x =0 = 0 , (5)or NBC ∂ x K ( t, x µ , x (cid:48) µ ) | x =0 = 0 , (6)on the boundary x = 0.From the heat kernel, we can obtain the Green function G ( x µ , x (cid:48) µ ) = (cid:90) ∞ dtK ( t, x µ , x (cid:48) µ ) , (7)and then derive the expectation value of the current byˆ J µ = lim x (cid:48) → x (cid:104) ( ∂ x µ + A x µ ) − ( ∂ x (cid:48) µ − A x (cid:48) µ ) (cid:105) G ( x µ , x (cid:48) µ ) . (8)In general ˆ J µ is divergent, which can be renormalized by subtracting the value it would havein the space without boundary, J µ = ˆ J µ − ˆ J µ . (9)In general, it is difficult to solve the heat kernel in the spacetime with boundaries, evenfor free theories. For simplicity, we focus on the perturbation solution in the linear order ofthe magnetic field O ( B ). Following [38], we obtain the heat kernel K = ∞ (cid:88) m = −∞ πt ) d − β exp (cid:32) − π m tβ + 2 iπm ( τ − τ (cid:48) ) β − d − (cid:88) a =2 ( y a − y (cid:48) a ) (cid:33) (cid:0) K + K bdy (cid:1) , (10)where K is the heat kernel in a 2d free space K = B π sin( Bt ) exp (cid:18) − B Bt ) (cid:0) ( x − x (cid:48) ) + ( y − y (cid:48) ) (cid:1) + B (cid:0) x (cid:48) + x (cid:1) (cid:0) y (cid:48) − y (cid:1)(cid:19) , (11)and K bdy denotes the correction due to the boundary K bdy = χB π sin( Bt ) exp (cid:18) − B Bt ) (cid:0) ( x + x (cid:48) ) + ( y − y (cid:48) ) (cid:1) + B (cid:0) x (cid:48) + x + f BC (cid:1) (cid:0) y (cid:48) − y (cid:1)(cid:19) . (12)4ere χ = − χ = 1) for DBC (NBC) and f BC is given by f BC = −√ πxx (cid:48) e ( x (cid:48) + x ) t erfc (cid:16) x (cid:48) + x √ t (cid:17) √ t + O ( B ) , DBC , − ( x (cid:48) + x ) + √ πe ( x (cid:48) + x ) t ( t + x +( x (cid:48) ) ) erfc (cid:16) x (cid:48) + x √ t (cid:17) √ t + O (cid:0) B (cid:1) , NBC , (13)where erfc( x ) is the complementary error function. One can check that the heat kernel (10)satisfies EOM (2) and BCs (3,4,5,6) at the linear order of O ( B ). Now we are ready to calculate the anomalous current at finite temperature. From (7,8,9,10),we derive J y = ∞ (cid:88) m = −∞ (cid:90) ∞ dt πB (4 πt ) d/ β e − π m tβ − x erfc (cid:16) x √ t (cid:17) + O ( B ) , DBC , √ txe − x t √ π − (cid:0) t + x (cid:1) erfc (cid:16) x √ t (cid:17) + O (cid:0) B (cid:1) , NBC . (14)Before we try to perform the above complicated sum and integral, let us first consider someinteresting limits, the high temperature limit and low temperature limit. In the high tem-perature limit β →
0, only the term with m = 0 contributes to the sum,lim T →∞ ∞ (cid:88) m = −∞ e − π m tβ = 1 . (15)Substituting (15) into (14) and performing the integral along t, we getlim T →∞ J y = − − d π − d Γ ( d − ) ( d − x d − B T + O ( B ) , DBC , − d (( d − d +8) π − d Γ ( d − ) ( d − d − x d − B T + O (cid:0) B (cid:1) , NBC . (16)It is remarkable that the anomalous current is proportional to the temperature in the hightemperature limit. This provides an interesting mechanism to enhance the anomalous current,which is usually suppressed by the mass and the distance to the boundary. Note that (16)works for d > d > d = 4, one should perform suitableregularization in order to get finite results. Taking the regularization [32] J dy = lim (cid:15) → J y ( d = 4 + (cid:15) ) + J y ( d = 4 − (cid:15) )2 , (17)we derive the current in four dimensionslim T →∞ J dy = (cid:40) − π B T + O ( B ) , DBC , x )+3+log(16)+2 log( π ) − ψ (0) ( ) π B T + O (cid:0) B (cid:1) , NBC , (18)5here ψ (0) is the PolyGamma function.Let us go on to discuss the low temperature limit, where the discrete summation can bereplaced by a continuous integrationlim T → ∞ (cid:88) m = −∞ β e − π m tβ = (cid:90) ∞−∞ e − π z t dz = 12 √ πt , (19)where z = m/β . Substituting (19) into (14), we derivelim T → J y = − − d π − d Γ ( d ) ( d − x d − B + O ( B ) , DBC , − d ( d − d +2 ) π − d Γ ( d − ) ( d − d − x d − B + O (cid:0) B (cid:1) , NBC , (20)which agree with the results of [32, 38]. This can be regarded as a check for our calculations.There is another method to study the current in the high and low temperature limits. Byapplying the transformation ∞ (cid:88) m = −∞ e − π m tβ β = ∞ (cid:88) m = −∞ e − β m t √ π √ t , (21)we can rewrite the current (14) into the following form J y = ∞ (cid:88) m = −∞ (cid:90) ∞ dt πB (4 πt ) ( d +1) / e − β m t − x erfc (cid:16) x √ t (cid:17) + O ( B ) , DBC , √ txe − x t √ π − (cid:0) t + x (cid:1) erfc (cid:16) x √ t (cid:17) + O (cid:0) B (cid:1) , NBC . (22)Note that the transformation (21) maps the high (low) temperature to the low (high) tem-perature. Now the term with m = 0 dominates the sum in the low temperature limitlim T → ∞ (cid:88) m = −∞ e − β m t √ πt = 12 √ πt , (23)which exactly agrees with (19,21). Substituting (23) into (22), we re-derive (20). As in thehigh temperature limit, the sum can be replaced by the following integrallim T →∞ ∞ (cid:88) m = −∞ e − β m t √ π √ t = (cid:90) ∞−∞ e − ¯ z t √ π √ tβ d ¯ z = 1 β , (24)which agrees with (15,21). From (22,24), we reproduce the anomalous current (16) in hightemperature limit. Now we have obtained the currents in the low and high temperature limitsby using two methods. This is a double check for our calculations.6igure 2: Anomalous currents of complex scalars for DBC (left) and NBC (right) in fourdimensions. The current of NBC is much larger than the current of DBC. Here we have set B = x = 1.Let us go on to consider the general temperature. Summing (22), we get J y = (cid:90) ∞ dt πBϑ (cid:18) , e − β t (cid:19) (4 πt ) ( d +1) / − x erfc (cid:16) x √ t (cid:17) + O ( B ) , DBC , √ txe − x t √ π − (cid:0) t + x (cid:1) erfc (cid:16) x √ t (cid:17) + O (cid:0) B (cid:1) , NBC , (25)where ϑ is the Elliptic theta function. Although it is difficult to work out the exact expressionof (25), it can be evaluated numerically. See Fig.2, Fig.3 and Fig.4 for examples. Withoutloss of generality, we set B = x = 1 for all the figures of this paper. We find that, in thehigh temperature limit, the currents increase linearly with temperature for both DBC andNBC in general dimensions. It is interesting that the currents of DBC and NBC approachthe same high-temperature limit in five dimensions. It is also interesting that, in dimensionshigher than four, the absolute values of the currents of NBC first decrease and then increasewith temperature, while the currents of DBC always increase with temperature. We focus on massless scalars in the above discussions, where exact expressions of the currentscan be derived in the high and low temperature limits. In this subsection, let us studythe mass effect, which can be taken into account by adding e − M t to the heat kernel (10).Following the approach of sect. 2.2, we obtain the anomalous currents for massive scalars as J y = (cid:90) ∞ dt πBϑ (cid:18) , e − β t (cid:19) e − M t (4 πt ) ( d +1) / − x erfc (cid:16) x √ t (cid:17) + O ( B ) , DBC , √ txe − x t √ π − (cid:0) t + x (cid:1) erfc (cid:16) x √ t (cid:17) + O (cid:0) B (cid:1) , NBC , (26)7igure 3: Anomalous currents of complex scalars for DBC (orange line) and NBC (blue line)in five dimensions. It is remarkable that the currents increase linearly with temperature andapproach the same value for DBC and NBC in the high temperature limit.Figure 4: Anomalous currents of complex scalars for DBC (orange line) and NBC (blue line)in six dimensions. The current of NBC first decreases and then increases with temperature,while the current of DBC always increases with temperature.where M is the scalar mass. It is clear that the current (26) is heavily suppressed by largemass. In the high and low temperature limits, the anomalous current (26) becomeslim T →∞ J y = T (cid:90) ∞ dt πB (4 πt ) d/ e − M t − x erfc (cid:16) x √ t (cid:17) + O ( B ) , DBC , √ txe − x t √ π − (cid:0) t + x (cid:1) erfc (cid:16) x √ t (cid:17) + O (cid:0) B (cid:1) , NBC , (27)andlim T → J y = (cid:90) ∞ dt πB (4 πt ) ( d +1) / e − M t − x erfc (cid:16) x √ t (cid:17) + O ( B ) , DBC , √ txe − x t √ π − (cid:0) t + x (cid:1) erfc (cid:16) x √ t (cid:17) + O (cid:0) B (cid:1) , NBC . (28)It is remarkable that, similar to the massless case, the anomalous currents of massive scalarsalso increase linearly with temperature in the high temperature limit. This means that, for a8igure 5: Mass effects of currents for DBC (left) and NBC (right) in five dimensions. It isinteresting that the mass can change the sign of currents of NBC.Figure 6: Mass effects of currents for DBC (left) and NBC (right) in six dimensions. It isremarkable that a suitable small mass can enhance the currents of NBC.given charge carrier with fixed mass, by increasing the temperature, one can always producea detectable anomalous current in laboratory.To end this section, let us draw some figures to illustrate the mass effect of anomalouscurrents. See Figs. 5,6,7. In general, the large mass suppresses but does not change the high-temperature behaviors of the currents. In other words, the currents are always enhanced byhigh temperatures. It is remarkable that, as is shown in figure 4, the current of NBC canalso be enhanced by increasing the mass slightly. In the above section, we focus on the currents at the linear order of the magnetic field O ( B ).In this section, we generalize our discussions to non-perturbative currents. We follow themethod of [32], where the anomalous current at zero temperature is investigated.9igure 7: Mass effects of currents for DBC (left) and NBC (right) in seven dimensions. Themass suppresses the currents of DBC and NBC.Green’s function obeys EOM[ − D µ D µ + M ] G ( x, x (cid:48) ) = δ ( d ) ( x, x (cid:48) ) , (29)and DBC G ( x µ , x (cid:48) µ ) | x =0 = 0 , (30)or NBC ∂ x G ( x µ , x (cid:48) µ ) | x =0 = 0 , (31)on the boundary x = 0. Performing Fourier transform with the correct period τ (cid:39) τ + β , G = 1 β ∞ (cid:88) m = −∞ (cid:90) dk d − (cid:107) (2 π ) d − ˜ G ( k, m ) e i πmβ ( τ − τ (cid:48) ) e − ik a ( y a − y (cid:48) a ) (32)we can rewrite (29) as[ − ∂ x + ( M + k a + ( 2 πmβ ) ) − Bxk + B x ] ˜ G = δ ( x − x (cid:48) ) , (33)where k = k y . We split Green’s function ˜ G into the one in a free space and the correctiondue to the boundary ˜ G = G free + G bdy , (34)where G free is given by [39, 40] G free = (cid:113) πB Γ( λ k ) D − λ k (cid:0) √ (cid:0) ¯ x − ¯ k (cid:1)(cid:1) D − λ k (cid:0) √ (cid:0) ¯ k − ¯ x (cid:48) (cid:1)(cid:1) , x > x (cid:48) , (cid:113) πB Γ( λ k ) D − λ k (cid:0) √ (cid:0) ¯ k − ¯ x (cid:1)(cid:1) D − λ k (cid:0) √ (cid:0) ¯ x (cid:48) − ¯ k (cid:1)(cid:1) , x < x (cid:48) . (35)10ere D denotes the parabolic cylinder function, λ k = ( B + M + k a + ( πmβ ) − k ) / (2 B ),¯ k = k / √ B and ¯ x = √ Bx . Imposing BCs (30, 31), we solve the corrections to Green’sfunction G bdy = − Γ ( λ k ) D − λ k (cid:0) √ k (cid:1) π / √ BD − λ k (cid:0) −√ k (cid:1) D − λ k (cid:16) √ (cid:0) ¯ x − ¯ k (cid:1)(cid:17) D − λ k (cid:16) √ (cid:0) ¯ x (cid:48) − ¯ k (cid:1)(cid:17) (36)for DBC and G bdy = Γ ( λ k ) (cid:0) √ D − λ k (cid:0) √ k (cid:1) − ¯ k D − λ k (cid:0) √ k (cid:1)(cid:1) π / √ B (cid:0) √ D − λ k (cid:0) −√ k (cid:1) + ¯ k D − λ k (cid:0) −√ k (cid:1)(cid:1) D − λ k (cid:16) √ (cid:0) ¯ x − ¯ k (cid:1)(cid:17) D − λ k (cid:16) √ (cid:0) ¯ x (cid:48) − ¯ k (cid:1)(cid:17) (37)for NBC.Now we are ready to derive the anomalous current. Substituting (32,36,37) into (8,9) ,we get the renormalized current J y = − β ∞ (cid:88) m = −∞ (cid:90) ∞−∞ dp d − dk (cid:0) ¯ x − ¯ k (cid:1) Γ ( λ p ) D − λ p (cid:0) √ k (cid:1) d − π d − D − λ p (cid:0) −√ k (cid:1) D − λ p (cid:16) √ (cid:0) ¯ x − ¯ k (cid:1)(cid:17) , (38)for DBC and J y = 1 β ∞ (cid:88) m = −∞ (cid:90) ∞−∞ dp d − dk (¯ x − ¯ k )Γ ( λ p ) (cid:0) √ D − λ p (cid:0) √ k (cid:1) − ¯ k D − λ p (cid:0) √ k (cid:1)(cid:1) d − π d − (cid:0) √ D − λ p (cid:0) −√ k (cid:1) + ¯ k D − λ p (cid:0) −√ k (cid:1)(cid:1) × D − λ p (cid:16) √ (cid:0) ¯ x − ¯ k (cid:1)(cid:17) , (39)for NBC. Recall that λ p = ( B + M + ( πmβ ) + p ) / (2 B ), ¯ k = k / √ B and ¯ x = √ Bx . Inprinciple, the formal expressions (38,39) can be evaluated numerically.In the low temperature limit β → ∞ , the sum can be replaced by the integral1 β ∞ (cid:88) m = −∞ = (cid:90) ∞−∞ dp τ π , (40)where p τ = 2 πm/β . And the currents (38,39) reduce to exactly the ones at zero temperature[32] lim T → J y = − (cid:90) ∞−∞ dp d − dk (cid:0) ¯ x − ¯ k (cid:1) Γ ( λ p ) D − λ p (cid:0) √ k (cid:1) d − π d − D − λ p (cid:0) −√ k (cid:1) D − λ p (cid:16) √ (cid:0) ¯ x − ¯ k (cid:1)(cid:17) , (41)for DBC andlim T → J y = (cid:90) ∞−∞ dp d − dk (¯ x − ¯ k )Γ ( λ p ) (cid:0) √ D − λ p (cid:0) √ k (cid:1) − ¯ k D − λ p (cid:0) √ k (cid:1)(cid:1) d − π d − (cid:0) √ D − λ p (cid:0) −√ k (cid:1) + ¯ k D − λ p (cid:0) −√ k (cid:1)(cid:1) D − λ p (cid:16) √ (cid:0) ¯ x − ¯ k (cid:1)(cid:17) , (42)for NBC, where λ p = ( B + M + p a + p τ ) / (2 B ).Note that only the combination M + ( πmβ ) appears in the currents (38,39). Thus ( πmβ )behaves effectively as a mass. In the high temperature limit, the effective mass M eff = M + ( πmβ ) becomes infinite for m (cid:54) = 0, which would heavily suppress the current . As aresult, only the term with m = 0 is dominated in the high temperature limit. Keeping onlythe zero-m terms of (38,39) and using (41,42), we finally obtain the anomalous current in thehigh temperature limit lim T →∞ J dy = T lim T → J d − y , (43)where J d denotes the current in d dimensions. It is remarkable that the non-perturbativecurrent is also proportional to the temperature in the high temperature limit. It is alsoremarkable that the coefficient is just the current in ( d −
1) dimensions at zero temperature.One can check that the perturbative currents obtained in sect. 2 obey the novel relation (43).
In this paper, we explore the mechanism to enhance the anomalous current caused by abackground magnetic field in the spacetime with a boundary. Usually, the anomalous currentis suppressed by the mass and the distance to the boundary, which are the main experi-mental obstructions. Remarkably, we find that the high temperature can greatly enhancethe anomalous current and make easier the experimental measurement. For free complexscalars, it is found that the anomalous current is proportional to the temperature in the hightemperature limit. Interestingly, the coefficient is just the current in lower dimensions atzero temperature. Thus, for any given charge carrier with a fixed mass M , one can alwaysproduce a detectable anomalous current by increasing the temperature. We look forward tothe experimental detection of this novel anomalous current. For simplicity, we focus on freecomplex scalars in this paper. It is interesting to generalize the results of this paper to Diracfields. It is also interesting to study the holographic anomalous current at finite temperaturefollowing the approach of [7, 29]. We hope these problems could be addressed in future. Acknowledgements
R. X. Miao acknowledges the supports from Guangdong Basic and Applied Basic ResearchFoundation (No.2020A1515010900) and NSFC grant (No. 11905297). One can check that the integrand functions of (38) and (39) approach zero as M eff → ∞ . eferences [1] M. J. Duff, Class. Quant. Grav. , 1387 (1994)[2] L. S. Brown, Phys. Rev. D , 1469 (1977).[3] L. Casarin, H. Godazgar and H. Nicolai, Phys. Lett. B , 94-99 (2018)[arXiv:1809.06681 [hep-th]].[4] M. N. Chernodub, Phys. Rev. Lett. , no. 14, 141601 (2016) [arXiv:1603.07993 [hep-th]].[5] M. N. Chernodub, A. Cortijo and M. A. H. Vozmediano, Phys. Rev. Lett. , no. 20,206601 (2018) [arXiv:1712.05386 [cond-mat.str-el]].[6] C. S. Chu and R. X. Miao, Phys. Rev. Lett. , no. 25, 251602 (2018) [arXiv:1803.03068[hep-th]].[7] C. S. Chu and R. X. Miao, JHEP , 005 (2018) [arXiv:1804.01648 [hep-th]].[8] For a review, see for example, D. E. Kharzeev, “The Chiral Magnetic Effect andAnomaly-Induced Transport,” Prog. Part. Nucl. Phys. (2014) 133 [arXiv:1312.3348[hep-ph]];K. Landsteiner, “Notes on Anomaly Induced Transport,” Acta Phys. Polon. B (2016)2617 [arXiv:1610.04413 [hep-th]].[9] A. Vilenkin, “Parity nonconservation and neutrino transport in magnetic fields,” Astro-phys. J. (1995) 700.[10] A. Vilenkin, “Equilibrium Parity Violating Current In A Magnetic Field,” Phys. Rev.D (1980) 3080.[11] M. Giovannini and M. E. Shaposhnikov, “Primordial hypermagnetic fields and triangleanomaly,” Phys. Rev. D (1998) 2186 [hep-ph/9710234].[12] A.Y. Alekseev, V. V. Cheianov, and J. Froehlich, Phys. Rev. Lett. (1998) 3503 [cond-mat/9803346].[13] K. Fukushima, “Views of the Chiral Magnetic Effect,” Lect. Notes Phys. (2013) 241[arXiv:1209.5064 [hep-ph]].[14] D. Kharzeev and A. Zhitnitsky, “Charge separation induced by P-odd bubbles in QCDmatter,” Nucl. Phys. A , 67 (2007) [arXiv:0706.1026 [hep-ph]].[15] J. Erdmenger, M. Haack, M. Kaminski and A. Yarom, “Fluid dynamics of R-chargedblack holes,” JHEP (2009) 055 [arXiv:0809.2488 [hep-th]].[16] N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Dutta, R. Loganayagam andP. Surowka, “Hydrodynamics from charged black branes,” JHEP (2011) 094[arXiv:0809.2596 [hep-th]]. 1317] D. T. Son and P. Surowka, “Hydrodynamics with Triangle Anomalies,” Phys. Rev. Lett. (2009) 191601 [arXiv:0906.5044 [hep-th]].[18] K. Landsteiner, E. Megias and F. Pena-Benitez, “Gravitational Anomaly and Trans-port,” Phys. Rev. Lett. (2011) 021601 [arXiv:1103.5006 [hep-ph]].[19] S. Golkar and D. T. Son, “(Non)-renormalization of the chiral vortical effect coefficient,”JHEP , 169 (2015) [arXiv:1207.5806 [hep-th]].[20] K. Jensen, R. Loganayagam and A. Yarom, “Thermodynamics, gravitational anomaliesand cones,” JHEP , 088 (2013) [arXiv:1207.5824 [hep-th]].[21] C. Chu and R. Miao, JHEP , 151 (2019) [arXiv:1812.10273 [hep-th]].[22] C. Chu, Fortsch. Phys. , no.8-9, 1910005 (2019) [arXiv:1903.02817 [hep-th]].[23] R. Miao and C. Chu, JHEP , 046 (2018) [arXiv:1706.09652 [hep-th]].[24] R. Miao, JHEP , 098 (2019) [arXiv:1808.05783 [hep-th]].[25] M. N. Chernodub, V. A. Goy and A. V. Molochkov, Phys. Lett. B , 556 (2019)[arXiv:1811.05411 [hep-th]].[26] M. Chernodub and M. A. Vozmediano, Phys. Rev. Research. , 032002 (2019)[arXiv:1902.02694 [cond-mat.str-el]].[27] V. E. Ambrus and M. Chernodub, [arXiv:1912.11034 [hep-th]].[28] J. Zheng, D. Li, Y. Zeng and R. Miao, Phys. Lett. B , 134844 (2019)[arXiv:1904.07017 [hep-th]].[29] R. X. Miao, JHEP , 025 (2019) doi:10.1007/JHEP02(2019)025 [arXiv:1806.10777 [hep-th]].[30] C. S. Chu and R. X. Miao, Phys. Rev. D , no.4, 046011 (2020) [arXiv:2004.05780[hep-th]].[31] C. S. Chu and R. X. Miao, JHEP , 134 (2020) [arXiv:2005.12975 [hep-th]].[32] P. J. Hu, Q. L. Hu and R. X. Miao, Phys. Rev. D , no.12, 125010 (2020)[arXiv:2004.06924 [hep-th]].[33] M. Kawaguchi, S. Matsuzaki and X. G. Huang, JHEP , 017 (2020) [arXiv:2007.00915[hep-ph]].[34] M. Kurkov and D. Vassilevich, Phys. Rev. Lett. , no.17, 176802 (2020)[arXiv:2002.06721 [hep-th]].[35] M. Kurkov and D. Vassilevich, JHEP , 072 (2018) [arXiv:1801.02049 [hep-th]].1436] I. Fialkovsky, M. Kurkov and D. Vassilevich, Phys. Rev. D , no.4, 045026 (2019)[arXiv:1906.06704 [hep-th]].[37] D. V. Vassilevich, Phys. Rept. , 279 (2003)[38] D. M. McAvity and H. Osborn, Class. Quant. Grav.8