Holographic Anomalous Current at a Finite Temperature
HHolographic Anomalous Current at a Finite Temperature
Jian-Guo Liu, Rong-Xin Miao ∗ School of Physics and Astronomy, Sun Yat-Sen University, 2 Daxue Road, Zhuhai 519082,China
Abstract
Weyl anomaly leads to novel anomalous currents in a spacetime with boundaries.Recently it is found that the anomalous current can be significantly enhanced by the hightemperature for free theories, which could make the experimental measurement easier. Inthis paper, we investigate holographic anomalous currents at a finite temperature. It isfound that the holographic current is still enhanced by the high temperature in dimensionshigher than three. However, the temperature dependence is quite different from that offree theories. This may be due to the fact that the holographic CFT is strongly coupledand there is non-zero resistance in the holographic model. Remarkably, the temperaturedependence of holographic anomalous currents is universal in the high temperature limit,which is independent of the choices of background magnetic fields. ∗ Email: [email protected] a r X i v : . [ h e p - t h ] F e b ontents Due to Weyl anomaly [1], an external electromagnetic field can induce novel anomalouscurrents in a conformally flat space [2, 3] and a spacetime with boundaries [4, 5]. Similarto Casimir effect [6, 7, 8], the anomalous currents arise from the effect of the backgroundgravitational field and the boundary on the quantum fluctuations of the vacuum. Otheranomaly-induced transports [9] include chiral magnetic effect (CME) [10, 11, 12, 13, 14] andchiral vortical effect (CVE) [15, 16, 17, 18, 19, 20, 21]. See also [22, 23, 24, 25, 26, 27, 28, 29,30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41] for related works.In this paper, we focus on the anomalous current in the spacetime with a boundary [4, 5].In four dimensions, it takes a universal form < J µ > = − βF µν n ν x + ..., x ∼ , (1)near the boundary. Here β is the beta function, F µν are the field strength, x is the properdistance to the boundary, n µ are the normal vectors and ... denotes higher order terms in O ( x ). Note that (1) applies to not only the conformal field theory (CFT) but also the generalquantum field theory (QFT). That is because it is derived from Weyl anomaly [4], which iswell-defined for the general QFT [1, 42, 43].Unfortunately, the anomalous current is heavily suppressed by the mass and the distanceto the boundary [26, 33], which makes it difficult to be measured in laboratory. Remarkably,recently it is found that the anomalous current can be greatly enhanced by the high temper-ature for free theories [44], which could make the experimental measurement easier. For freetheories, the anomalous current is proportional to the temperature in the high temperature1imit lim T →∞ < J > ∼ T. (2)This means, for any given charge carrier with fixed mass, one can always produce a detectableanomalous current by increasing the temperature.In this paper, we investigate the holographic anomalous current at a finite temperature.We find that the holographic current is still enhanced by the high temperature in dimensionshigher than three. However, the temperature dependence is quite different from that of freetheories. See (3,16,22) for example. The holographic current decreases with the temperaturein three dimensions, while it increases with the temperature in dimensions higher than four.In four dimensions, the absolute value of current first decreases and then increases withthe temperature. In the high temperature limit, the holographic anomalous current takes auniversal form, lim T →∞ < J > ∼ (cid:40) T d − , d (cid:54) = 4 , log( T ) , d = 4 , (3)which is independent of the choices of external magnetic fields.The paper is organized as follows. In section 2, by choosing a suitable external magneticfield, we derive an exact expression of the holographic anomalous current. We find that theholographic anomalous current is enhanced by the high temperature in dimensions higherthan three. In section 3, we discuss the general background magnetic fields and derive theholographic anomalous current in the high temperature limit. In section 4, we present somenumerical results. Finally, we conclude with some open questions in section 5. The main purpose of this paper is to explore the temperature dependence of holographicanomalous currents [5, 30]. To warm up, let us first study a special case that the backgroundmagnetic field is given by B = 2 b a x , where b a are a constant and x is the distance to theboundary. As it is shown below, we can derive exact expressions of anomalous currents forthis case.For simplicity, we focus on the probe limit, where the background spacetime is given byAdS-Schwarzschild black hole, ds = dz − zdzdh − (1 − z d z dh ) dt + dx + dy a z . (4)2igure 1: Geometry of the holographic model. The bulk is an AdS-Schwarzschild black hole,the bulk boundary Q , the AdS boundary M and the horizon H are located at x = 0, z = 0and z = z h , respectively.Here the bulk boundary Q , the AdS boundary M and the horizon H are located at x = 0, z = 0 and z = z h , respectively. See Fig. 1 for the geometry of our holographic model. Seealso [45] for more explanations of the geometry in AdS/BCFT.The temperature of the black hole (4) is T = d πz h . (5)For simplicity, we choose the following gauges for the bulk Maxwell’s fields [5] A z = A t = A x = 0 , A a = A a ( z, x ) . (6)Now the bulk Maxwell’s equations become (cid:16) Z d + d − (cid:17) A a (1 , ( Z, x ) + Z (cid:16) Z d − (cid:17) A a (2 , ( Z, x ) − Zz h A a (0 , ( Z, x ) = 0 , (7)where we have made the coordinate transformation Z = z/z h . Following [45], we imposeNeumann boundary condition (NBC) on the bulk boundary Q∂ x A a ( Z, x ) | x =0 = 0 , (8)3hich is equivalent to the absolute BC, i.e., F xa | x =0 = 0.We take the following ansatz of A a , A a ( Z, x ) = b a x + d a + ¯ A a ( Z ) , (9)which automatically obeys the NBC (8). Here b a , d a are constants and the backgroundmagnetic field on the AdS boundary is B = F xa | Z → = 2 b a x. (10)Substituting (9) into (7), we get Z (cid:16) Z d − (cid:17) ¯ A (cid:48)(cid:48) a ( Z ) + (cid:16) Z d + d − (cid:17) ¯ A (cid:48) a ( Z ) − b a Zz h = 0 , (11)which can be solved as¯ A a = c a + c a Z d − F ( , d − d ;2 − d ; Z d ) d − + b a Z zh F ( , d ; d +2 d ; Z d ) d − , d (cid:54) = 4 c a + c a log (cid:16) Z +11 − Z (cid:17) − b a z h (cid:16) Li (cid:0) Z (cid:1) − (cid:0) Z (cid:1) + 4 log( Z ) log (cid:16) Z +11 − Z (cid:17)(cid:17) , d = 4 (12)where F is the hypergeometric function, Li is the polylogarithm function and c a , c a areintegral constants. Note that c a can be absorbed into the definition of d a (9). Thus we canset c a = 0 without loss of generality. We impose the natural boundary condition on thehorizon A a ( Z, x ) | Z =1 is finite , (13)which fixes the left integral constant as c a = (cid:40) b a z h − d , d (cid:54) = 4 , , d = 4 . (14)The holographic current can be read off from the asymptotic solutions of bulk vectorsnear the AdS boundary z = 0 A a = A a ( z = 0) + ... + z d − (cid:18) J a d − f a ln z (cid:19) + ..., (15)where J a is the holographic current and f a is an irrelevant function which appears only ineven dimensions. Expanding the solutions (12) near z = 0 and comparing with (15), wefinally obtain the holographic anomalous current J a = (cid:40) d − π d − (4 − d ) d d − b a T d − , d (cid:54) = 4 ,b a (cid:0) − πT ) (cid:1) , d = 4 . (16)In dimensions higher than four, the holographic current increases with the temperature, whilein dimensions lower than four, the anomalous current decreases with the temperature. Infour dimensions, the absolute value of current first decreases and then increases with thetemperature. 4 Holographic current II: perturbative result
In the above section, we focus on a special kind of external magnetic field (10) and derivean exact expression of the holographic anomalous current. In this section, we discuss thecurrents induced by general background magnetic fields. In the high temperature limit, wefind that the temperature dependence of holographic currents is universal and is still givenby (16).The general bulk Maxwell’s fields obeying NBC (8) take the following form A a ( Z, x ) = (cid:90) dkF ( k ) cos( kx ) ˆ A a ( Z ) , (17)where F ( k ) is an arbitrary function as long as it defines a convergent integral. From (17),we read off the background magnetic fields on the AdS boundary B = − (cid:90) dkkF ( k ) sin( kx ) ˆ A a (0) . (18)Substituting (17) into (7), we obtain Z (cid:16) Z d − (cid:17) ˆ A (cid:48)(cid:48) a ( Z ) + (cid:16) Z d + d − (cid:17) ˆ A (cid:48) a ( Z ) + k Zz h ˆ A a ( Z ) = 0 . (19)Note that the above equation is quite similar to (11), only the last term is different. Unfor-tunately, unlike (11), the above equation cannot be solved analytically. For simplicity, weinvestigate the perturbation solutions in this section, and leave the discussions of numericalsolutions to next section.In the high temperature limit, z h = d/ (4 πT ) is a small parameter. Thus we can expandthe bulk vectors ˆ A a ( Z ) in powers of z h ,ˆ A a ( Z ) = ˆ A (0) a ( Z ) + z h ˆ A (1) a ( Z ) + O ( z h ) . (20)Let us solve (19) order by order in O ( z h ). One can check that ˆ A (0) a ( Z ) must be a constantˆ A (0) a ( Z ) = c a (21)in order to satisfy the EOM (19) and the natural boundary condition (13) at the same time.Substituting (20,21) into (19), we get the EOM of ˆ A (1) a ( Z ), which is exactly the same as (11)provided that we identify − b a z h with c a k . Now following the approach of sect. 2, we canderive the holographic current as J a = − c a + O (1 /T )2 (cid:90) dkF ( k ) k cos( kx ) (cid:40) d − π d − (4 − d ) d d − T d − , d (cid:54) = 4 , (cid:0) − πT ) (cid:1) , d = 4 . (22)It is remarkable that the temperature dependence of the holographic current is universal,which is independent of the choices of background vector fields in the high temperature limit.5 Holographic current III: numerical result
In this section, we discuss holographic anomalous currents at general temperatures. Themain task is to solve (19) numerically. To do so, we need to specify the boundary conditionnear the horizon Z = 1. Assume that ˆ A a ( Z ) takes the following form near the horizonˆ A a = a a + a a (1 − Z ) + a a (1 − Z ) + a a (1 − Z ) + O (1 − Z ) , (23)where a ia are constants to be determined. Imposing the natural boundary condition (13), wesolve a a = k z h d a a ,a a = (cid:0) dk z h + k z h (cid:1) d a a ,a a = k z h (cid:0) − d − d + d ( d + 5) k z h + k z h (cid:1) d a a , (24)where a a is a free parameter, which can be set to be a a = 1. Now the boundary conditionsof the differential equation (19) becomesˆ A a (1 − (cid:15) ) = a a + a a (cid:15) + a a (cid:15) + a a (cid:15) + O ( (cid:15) ) , ˆ A (cid:48) a (1 − (cid:15) ) = − a a − a a (cid:15) − a a (cid:15) + O ( (cid:15) ) , (25)where (cid:15) is a small constant chosen for the numerical calculations, and a ia are given by (24)with a a = 1.Now it is straightforward to numerically solve the differential equation (19) together withthe BCs (25). Once we obtain the solutions, we can derive the holographic anomalous currentfrom the asymptotic solutions (15) near the AdS boundary. Let us draw some figures to showthe temperature dependence of the holographic currents. Without loss of generality, we set k = a a = 1 for all the figures. See Fig. 2, Fig. 3, Fig. 4 for examples. It is found that theholographic anomalous currents in these figures match exactly the results (22) of sect.3 inthe high temperature limit. This can be regarded as a double check of our results. Besides,the current decreases with the temperature in three dimensions, while it increases with thetemperature in dimensions higher than four. In four dimensions, the absolute value of currentfirst decreases and then increases with the temperature. In this paper, we investigate the holographic anomalous current at a finite temperature. Forthe external magnetic field B ∼ x , we derive an exact expression of the holographic current.As for general background magnetic fields, we obtain perturbative and numerical results.6igure 2: Holographic anomalous current at finite temperature in three dimensions (Left);Compare the numerical result with that of high temperature limit − k πT (Right). The holo-graphic current decreases with temperature in three dimensions.Figure 3: Holographic anomalous current at finite temperature in four dimensions (Left);Compare the numerical result with that of high temperature limit k (cid:0) log( πT ) − (cid:1) (Right).The absolute value of current first decreases and then increases with the temperature in fourdimensions.It is remarkable that the temperature dependence of the holographic anomalous current isuniversal in the high temperature limit, which is independent of the choices of backgroundmagnetic fields. Similar to the case of free theories, the holographic anomalous current is stillenhanced by the high temperature in dimensions higher than three. However, the temperaturedependence is quite different from that of free theories. The reasons may lie in the fact thatthe holographic CFT is strongly coupled and there is non-zero resistance in the holographicmodel [46]. In this paper, we focus on the probe limit, where the background spacetimeis given by the AdS-Schwarzschild black hole. It is interesting to study the back-reactionsand the charged black holes to see if the universal temperature dependence of holographiccurrents still holds or not. It is also interesting to investigate the low temperature regionscarefully. Finally, the temperature dependence of the anomalous Fermi condensation [31, 32]is also a problem worth exploring. We leave these problems to future work.7igure 4: Holographic anomalous current at finite temperature in five dimensions (Left);Compare the numerical result with that of high temperature limit πk T (Right). Theholographic current increases with temperature in five dimensions. Acknowledgements
We thank Miaoxin Liu for helpful discussions and contributions. R. X. Miao acknowledgesthe supports from NSFC grant (No. 11905297) and Guangdong Basic and Applied BasicResearch Foundation (No.2020A1515010900).
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