Angular Momentum Generation by Parity Violation
CCALT 68-2857, IPMU13-0193, MIT-CTP/4501
Angular momentum generation by parity violation
Hong Liu, Hirosi Ooguri,
2, 3 and Bogdan Stoica ∗ Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA California Institute of Technology, 452-48, Pasadena, California 91125, USA Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa 277-8583, Japan (Dated: July 23, 2018)We generalize our holographic derivation of spontaneous angular momentum generation in 2 + 1dimensions in several directions. We consider cases when a parity-violating perturbation responsiblefor the angular momentum generation can be nonmarginal (while in our previous paper we restrictedto a marginal perturbation), including all possible two-derivative interactions, with parity violationstriggered both by gauge and gravitational Chern-Simons terms in the bulk. We make only a minimalassumption about the bulk geometry that it is asymptotically AdS, respects the Poincar´e symmetryin 2 + 1 dimensions, and has a horizon. In this generic setup, we find a remarkably concise anduniversal formula for the expectation value of the angular momentum density, to all orders in theparity violating perturbation.
PACS numbers: 11.25.Tq
I. INTRODUCTION
The spontaneous generation of angular momentum andof an edge current are typical phenomena in parity-violating physics (see, for example, [1–5]). For a giveninteracting system, whether spontaneous generation ofangular momentum does occur, and if yes, the precisevalue, are important dynamical questions for which auniversal answer (applicable to generic parity-violatingsystems) does not appear to exist. A famous exampleis helium 3-A, in which case there has been a long con-troversy about the value of its angular momentum (seee.g. [1, 6]). The controversy highlights the importance offinding exactly solvable models, especially strongly inter-acting systems, through which one could extract genericlessons. Holographic systems are ideal laboratories forthis purpose.In a previous paper [7], we initiated exploration ofthese phenomena in holographic systems. There, fortechnical simplicity, we restricted to parity violation ef-fected by turning on a marginal pseudoscalar opera-tor, and considered only the Schwarzschild and Reissner-Nordstr¨om geometries. In this paper, we generalize theresults to parity violation through a relevant scalar op-erator, and to general bulk black hole geometries.More explicitly, we consider a (2 + 1)-dimensionalboundary field theory with a U (1) global symmetry,which is described by classical gravity (together with var-ious matter fields) in a four-dimensional, asymptoticallyanti–de Sitter spacetime (AdS ). We consider two rep-resentative bulk mechanisms for parity violation, with a ∗ Electronic address: [email protected] See [8–18] for other discussions of parity-violating effects in holo-graphic systems and in (2 + 1)-dimensional field theories. gravitational Chern-Simons interaction [19] α CS (cid:90) ϑ R ∧ R, (1.1)or an axionic coupling [20, 21] β CS (cid:90) ϑ F ∧ F, (1.2)where R is the Riemann curvature two-form, F is fieldstrength for the bulk gauge field A a dual to the U (1)global current, and ϑ is a pseudoscalar dual to a bound-ary relevant pseudoscalar operator O . α CS and β CS aresome constants.The parity symmetry is broken explicitly if a sourceis turned on for O corresponding to turning on a non-normalizable mode for the pseudoscalar field ϑ . Alter-natively, the parity can be spontaneously broken when O develops an expectation value in which case the bulkfield ϑ is normalizable. In both situations if we put thesystem in a finite box (i.e., parity-violation terms arenonzero only inside the box), the spontaneous generationof angular momentum is always accompanied by an edgecurrent. We emphasize that the source or expectationvalue for O is taken to be homogeneous along boundarydirections. An angular momentum density is generated,despite the boundary quantum state and the correspond-ing bulk geometry being homogeneous and isotropic.It may appear puzzling how a homogeneous andisotropic bulk geometry can give rise to a nonzero angularmomentum, as directly applying the standard AdS/CFTdictionary to such a geometry will clearly yield a zerovalue. The key idea, following [7], is to consider a smalland slightly inhomogeneous perturbation δϑ around thebackground value of ϑ , which results in a nonzero mo-mentum current density δT i . To leading order in the We use latin letters in the middle of the alphabet ( i, j, k, . . . ) a r X i v : . [ h e p - t h ] J un derivative expansion along the boundary directions, T i depends linearly on (cid:15) ij ∂ j δϑ . Now let us consider a con-figuration of δϑ which is homogeneous along boundaryspatial directions inside a big box but vanishes outside.Then δT i is only nonvanishing at the edge of the box,but remarkably such an edge current generates an angu-lar momentum proportional to the volume of the box δJ = (cid:15) ij (cid:90) d x x i δT j ∝ V box δϑ (1.3)resulting in a nonzero angular momentum density δ L which survives even when we take the size of the boxto infinity. Thus in the homogeneous limit, the angularmomentum density δ L arises from the global effect of anedge current, which explains why it is not visible fromthe standard local analysis of the stress tensor.When ϑ is dual to a marginal operator, δϑ is inde-pendent of radial direction of AdS and δ L is given by δϑ times a constant which can be easily integrated tofind the value of L for a finite ϑ . But for ϑ dual to aboundary relevant operator, δϑ has a nontrivial radialevolution (which simply reflects that a relevant operatorflows), and the relation between δ L and δϑ involves asomewhat complicated radial integral over various bulkfields. Remarkably, this relation can be written as a to-tal variation in the space of gravity solutions, which canthen be easily integrated to yield a closed expression for L at a finite ϑ .More explicitly, we consider a most general bulk metricconsistent with translational and rotational symmetriesalong boundary directions, which can be written in aform ds = (cid:96) z (cid:16) − f ( z ) dt + h ( z ) dz + (cid:0) dx i (cid:1) (cid:17) (1.4)with z = 0 as the boundary. Matter fields include ϑ ( z ), A t ( z ), and possibly others. We denote z as the hori-zon of the metric. Note that in the coordinate choiceof (1.4) z is inversely proportional to the square rootof the entropy density s , i.e. z ∝ s − , and serves asan IR cutoff scale of the boundary system. For the ax-ionic coupling (1.2) we find that the angular momentumdensity can be written as L = − β CS (cid:96) κ µ ϑ ( z ) + 2 β CS (cid:96) κ z (cid:90) dz ( A t ( z ) − µ ) ϑ (cid:48) ( z )(1.5)where µ is the chemical potential, (cid:96) is the AdS radius,and κ = 8 πG . For gravitational CS coupling (1.1), we to denote two-dimensional spatial indices on the boundary. Physically, it can be interpreted as characterizing the correlationlength of the boundary system. find that L = − π α CS (cid:96) κ T ϑ ( z ) + α CS (cid:96) κ z (cid:90) dz (cid:18) f (cid:48) f h (cid:19) ϑ (cid:48) (1.6)where T is the temperature.Equations (1.5)–(1.6) are universal in the bulk sensethat they have the same form in terms of bulk gaugefields or metric components, independent of the specificform of bulk actions, geometries and possible other mat-ter fields. But they are not universal in the boundarysense as it appears that they cannot be further reducedto expressions in terms of boundary quantities only.When ϑ is dual to a marginal operator at the boundary, ϑ ( z ) is constant in the bulk and its value can be identi-fied as the coupling of O . Then for both (1.5) and (1.6), L is given by the first term, reproducing our earlier re-sults in [7]. These expressions are now universal also inthe boundary sense, valid for any boundary theory witha gravity dual. In Sec. IV, we will present a preliminaryexplanation of this universal behavior from the perspec-tive of the boundary conformal field theory (CFT). Wehope to explore this point in future.For ϑ dual to a relevant operator, ϑ ( z ) can be in-terpreted as the running coupling for the correspond-ing boundary operator O , with z as the renormalizationgroup (RG) length scale. In this case, the first termof (1.5) and (1.6) is proportional to the running cou-pling evaluated at the IR cutoff scale z . The secondterm of (1.5) and (1.6) has the form of the beta function(given by ϑ (cid:48) ) for O integrated over the RG trajectory allthe way to the IR cutoff. This indicates that in the caseof a relevant operator, despite being an IR quantity, theangular momentum receives contribution from all scales.The simplicity of the integration kernel in these equationsmay suggest a possible simple boundary interpretationwhich should be explored further.Another interesting phenomenon associated to parityviolation in 2 + 1 dimensions is the Hall viscosity [22]. Itturns out that, in quantum Hall states, there is a closerelation between the Hall viscosity and the angular mo-mentum density [5, 23–25]. It would be interesting to un-derstand how universal such a relation is. In a forthcom-ing paper, we will discuss this issue from the holographicperspective. We will apply the prescription of [12] toidentify models where the Hall viscosity is nonzero andcompare its value with the angular momentum density.For the remainder of this paper, we will use the follow-ing. Latin letters stand for (3+1)-dimensional spacetimeindices, greek letters stand for (2+1)-dimensional indiceson the boundary, latin letters in the middle of the alpha-bet ( i, j, k, . . . ) stand for 2-dimensional spatial indices onthe boundary and ∂ ≡ ∂ x + ∂ y . The metric is denotedvia g ab with signature ( − , + , + , +) in the bulk, and via h αβ on the boundary; the Einstein summation conven-tion and geometric units with (cid:126) = c = 1 are assumed,unless otherwise specified; we denote κ = 8 πG .After posting this paper on the arXiv e-print server,it was pointed out by K. Landsteiner and by a referee ofthis paper that the spontaneous generation of the edgecurrent and of the angular momentum in 2 + 1 dimen-sions discussed in this paper may be related to the chiralmagnetic effect [27–29] and axial magnetic effect [30–32]in 3 + 1 dimensions. Prompted by their suggestions, wefound that the effects in 3+1 dimensions and 2+1 dimen-sions are indeed related by dimensional reduction whenthe parity-violating perturbation is marginal , which wasthe focus of our previous paper [7]. For completeness,we added Sec. IV to discuss the relation. The purpose ofthis paper is to generalize our results to the case whenthe parity-violating perturbation is relevant , and the dis-cussion in Sec. IV is not immediately applicable. Theremay exist a generalization of the chiral magnetic effectand axial magnetic effect in 3 + 1 dimensions which cor-respond to dimensional oxidation of the effects studied inthis paper. II. AXIONIC COUPLING
In this section we consider a scalar field ϑ coupled to aMaxwell field via an axionic coupling, ϑ ∗ F ab F ab . We firstexplicitly work out the angular momentum for a simplesetup, and then generalize the results to general gravitytheories. A. Angular momentum
1. Small perturbations
Consider the action S = 12 κ (cid:90) d x √− g (cid:20) R −
12 ( ∂ϑ ) − V ( ϑ ) − (cid:96) F ab F ab − (cid:96) β CS ϑ ∗ F ab F ab (cid:21) , (2.1)with β CS a coupling constant, and ϑ dual to a relevant(or marginal) pseudoscalar boundary operator. We as-sume that the background geometry is asymptoticallyAdS with (cid:96) the AdS radius. The equations of motion are R ab − ∂ a ϑ∂ b ϑ − g ab V ( ϑ ) − (cid:96) (cid:18) F ca F cb − g ab F (cid:19) = 0 , (2.2)1 √− g ∂ a (cid:0) g ab √− g∂ b ϑ (cid:1) − V (cid:48) ( ϑ ) − β CS (cid:96) ∗ F F = 0(2.3) ∂ a (cid:2) √− g (cid:0) F ab + β CS ϑ ∗ F ab (cid:1)(cid:3) = 0 . (2.4)A most general solution describing the boundary in astatic, homogeneous, isotropic state can be written as g (0) ab dx a dx b = (cid:96) z (cid:16) − f ( z ) dt + h ( z ) dz + (cid:0) dx i (cid:1) (cid:17) , ϑ = ϑ ( z ) , A a = A t ( z ) δ ta . (2.5)The AdS boundary lies at z = 0 with f ( z ) → , h ( z ) → , z → A t ( z = 0) = µ (2.7)where µ is the chemical potential. We assume that thereis a horizon as z = z , where f ( z ) has a simple zero and h ( z ) has a simple pole. The temperature is given by T = 14 π (cid:113) f (cid:48) ( z ) h − (cid:48) ( z ) . (2.8)Here are some background equations of motion whichwill be important below. The t component of the back-ground Maxwell equation can be integrated to give, A (cid:48) t ( z ) = Q (cid:112) f ( z ) h ( z ) (2.9)with Q the charge density. The tt and ii components ofthe background Einstein equations can be used to obtain4 (cid:112) f hQ = (cid:18) f (cid:48) z √ f h (cid:19) (cid:48) . (2.10)As discussed in the Introduction, to compute the an-gular momentum, we consider a small and slightly inho-mogeneous perturbation δϑ ( z, x i ) around the backgroundvalue ϑ ( z ). Such a perturbation will clearly also induceperturbations of the metric and gauge field, g ab = g (0) ab + (cid:96) z δg ab , (2.11) A a = A t ( z ) δ ta + δA a ( z, x i ) . (2.12)The metric and gauge field perturbations will be assumedto be normalizable, while δϑ can be either normalizableor non-normalizable. We will also make the followinggauge choice, δA z = 0 , δg zt = 0 . (2.13)To find the angular momentum, we first compute T ti ,which in turn requires us to find δg ti . Since δϑ is smallwe can work at the linear order in all perturbations, andsince we will eventually take δϑ to be homogeneous, itwill be enough to keep only terms with at most oneboundary spatial derivative (for details on the derivativeexpansion in holographic fluid dynamics see for instance[33, 34]).We now proceed with the computation in detail. The ti component of the Einstein equations reads (cid:18) f (cid:48) f + h (cid:48) h + 4 z (cid:19) ∂ z δg ti − ∂ z δg ti = 8 z A (cid:48) t ∂ z δA i (2.14)while the i component of the Maxwell equations reads ∂ z (cid:18) √ f ∂ z δA i √ h + Qδg ti (cid:19) + β CS (cid:15) ij (cid:16)(cid:112) f hQ∂ j δϑ − ϑ (cid:48) ∂ j δA t (cid:17) = 0 . (2.15)Due to the presence of δϑ and δA t in (2.15), Eqs. (2.14)–(2.15) do not close between themselves, which impliesthat solving δg ti explicitly will be a very complicatedtask, if possible at all. Fortunately as we will see itturns out to be unnecessary to do so.Integrating (2.15) from the horizon to z we find that ∂ z δA i ( z, x k ) = (cid:112) h ( z ) (cid:112) f ( z ) (cid:34) − Qδg ti ( z, x k )+ β CS (cid:15) ij z (cid:90) z dw ( ϑ (cid:48) ∂ j δA t − A (cid:48) t ∂ j δϑ ) (cid:35) (2.16)where we have assumed that ∂ z δA i ( z, x k ) is nonsingularat the horizon.Plugging Eqs. (2.16) into (2.14), and using (2.10) wefind that ∂ z (cid:34) f ( z ) z (cid:112) h ( z ) ∂ z (cid:18) δg ti ( z, x k ) f ( z ) (cid:19)(cid:35) (2.17)= 4 β CS (cid:15) ij A (cid:48) t ( z ) z (cid:90) z dw [ A (cid:48) t ∂ j δϑ − ϑ (cid:48) ∂ j δA t ] . The above equation implies that despite the mixingbetween δA i and δg ti , the combination f δg ti remains“massless.” Writing g ab = g (0) ab + g (1) ab with g (1) ab = (cid:96) z δg ab ,we note that f δg ti in fact corresponds to ( g (1) ) ti .Integrating Eq. (2.18) from the boundary z = 0 to thehorizon z , we find that f ( z ) z (cid:112) h ( z ) ∂ z (cid:18) δg ti ( z, x k ) f ( z ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z =0 = 4 β CS (cid:15) ij ×× (cid:90) z dzA (cid:48) t ( z ) z (cid:90) z dw [ A (cid:48) t ∂ j δϑ − ϑ (cid:48) ∂ j δA t ] (2.18)where we have used that at the horizon δg ti ( z , x i ) = 0 (2.19)and ∂ z δg ti is regular there. Equation (2.19) is analogousto the well-known statement that A t vanishes at blackhole horizons, and is similarly most transparent in Eu-clidean signature, where a nonzero δg ti at the shrinkingtime cycle indicates a delta-function contribution to theEinstein tensor. It can be also shown directly from con-sistency of various components of Einstein equations (seeAppendix A). Note that equations for δϑ and δA t are rather complicated. Thisis especially the case for more general action (2.41). Now consider the left-hand side of (2.18). With δg ti normalizable, i.e. δg ti ( z, x l ) = G (3) i ( x l ) z + O ( z ) . (2.20)we find 3 G (3) i = 4 β CS (cid:15) ij (cid:90) z dzA (cid:48) t ( z ) z (cid:90) z dw × (2.21) × [ A (cid:48) t ( w ) ∂ j δϑ ( w ) − ϑ (cid:48) ( w ) ∂ j δA t ( w )] . Using the standard formulas as in [35–37] (see also Ap-pendix B and the Appendix of [7]), the boundary stress-energy tensor is δT ti = 3 (cid:96) κ G (3) i = − (cid:15) ij ∂ j δ Φ (2.22)where δ Φ = 2 β CS (cid:96) κ z (cid:90) dzA (cid:48) t ( z ) z (cid:90) z dw [ A (cid:48) t δϑ − ϑ (cid:48) δA t ]= 2 β CS (cid:96) κ (cid:90) z dw [ A (cid:48) t δϑ − ϑ (cid:48) δA t ] ( A t − µ ) . (2.23)In the second equality above we have exchanged the or-der of integration to perform one integral and used that A t (0) = µ .Now consider a configuration of δϑ which is homoge-neous along boundary spatial directions inside a big boxbut vanishes outside. The above δT i is nonvanishingonly at the edge of the box, but generates an angularmomentum proportional to the volume of the box, re-sulting in an angular momentum density δ L = 2 δ Φ (2.24)We now take the box size to infinity, with δϑ and δA t homogeneous everywhere with no dependence on x i .
2. Angular momentum density
Equations (2.22)–(2.24) apply to infinitesimal varia-tions δϑ and δA t around (2.5). To compute L for (2.5),we need to integrate (2.23) along some trajectory in thespace of field configurations from a configuration with ϑ = 0 (and thus L = 0) to (2.5), i.e. schematicallyΦ = (cid:90) ϑϑ =0 δ Φ (2.25)from which we then find T ti = − (cid:15) ij ∂ j Φ , L = 2Φ . (2.26)At first sight this appears to be an impossible task assolving δA t in terms of δϑ is complicated and so is inte-gration over field space as A t in general also has nontriv-ial ϑ dependence.Remarkably, Eq. (2.23) can be written as a totalderivative δ in the field configuration space. Choosinga trajectory in configuration space with a fixed µ (i.e. δµ = 0) we can rewrite (2.23) as δ Φ = β CS (cid:96) κ (cid:90) z dw [ B (cid:48) δϑ − ϑ (cid:48) δB ]= β CS (cid:96) κ (cid:90) z dw [( Bδϑ ) (cid:48) − δ ( Bϑ (cid:48) )] (2.27)where B = A t − µA t and in the second line we haveused that for arbitrary functions F and G ( F δG ) (cid:48) − δ ( F G (cid:48) ) = F (cid:48) δG − δF G (cid:48) . (2.28)Recall that A t is zero at the horizon and equal to µ atthe boundary. Evaluating the total derivative and tak-ing δ operation outside the integral for the second term,Eq. (2.27) becomes δ Φ = β CS (cid:96) κ δ (cid:20) − µ ϑ (0) + (cid:90) z dw ( A t − µA t ) ϑ (cid:48) (cid:21) . (2.29)Note that in exchanging the order of δ with the integra-tion, there is a term proportional to δz , which, however,vanishes as A t ( z ) = 0. Now (2.29) is a total variationand we conclude thatΦ = β CS (cid:96) κ (cid:20) − µ ϑ (0) + (cid:90) z dw ( A t − µA t ) ϑ (cid:48) (cid:21) . (2.30)The above equation can also be slightly rewritten asΦ = β CS (cid:96) κ (cid:20) − µ ϑ ( z ) + (cid:90) z dw ( A t − µ ) ϑ (cid:48) (cid:21) . (2.31)Note that Eqs. (2.30)–(2.31) also apply to inhomogeneousconfigurations as far as the spatial variations are suffi-ciently small.When ϑ is dual to a marginal operator, ϑ is constantin the bulk with ϑ (0) = ϑ ( z ) = ϑ , and the second termin (2.30) or (2.31) drops out. We then recover the resultof [7], L = − β CS (cid:96) κ µ ϑ. (2.32)For a general relevant operator, the second term in (2.30)or (2.31) is nonzero and the angular momentum densitywill receive contribution from integration over the bulkfull spacetime. In terms of boundary language, the an-gular momentum receives contributions from degrees offreedom at all scales. Also note that for a relevant op-erator ϑ (0) = 0, so in (2.30) the sole contribution comesfrom the second term. T (cid:61) (cid:61) (cid:61) MT (cid:61) M (cid:144) (cid:61) M (cid:144) Μ (cid:144) M L a x (cid:144) M FIG. 1: (Color online) Angular momentum density as a func-tion of µ /M for axionic coupling and non-normalizablescalar field in a quadratic potential with m = −
3. An explicit example
We now consider an explicit example. For simplicitywe take V ( ϑ ) = m ϑ with m = −
2. Thus ϑ is dualto a relevant boundary operator O in d = 3 with ∆ =2. We will consider a solution (2.5) in which ϑ is non-normalizable, i.e. ϑ has the asymptotic behavior near theboundary ϑ ( z ) = M z + O ( z ) , z → M is a parameter of dimension mass. The solu-tion (2.5) then describes a boundary theory flow uponturning on a relevant perturbation (cid:82) d x M O , with M interpreted as the bare coupling. Since we are consid-ering the system at a finite density/finite temperature,the flow is cut off at some infrared scale characteristicof finite density/finite temperature physics. In the coor-dinate system we are using in (2.5), such a scale shouldcorrespond to location of the horizon z ∝ s − with s the entropy density.We present plots of the axionic angular momentum asa function of µ /M in Figs. 1 and 2 and as a function of µ /M T in Figs. 3 and 4. We exhibit the two terms en-tering Eq. (1.5), as well as the total angular momentum,in Figs. 2 and 4. We note that in the large T regimethe angular momentum density grows as L ax ∝ µ M/T .This is expected from the general structure of Eq. (1.5)since roughly speaking the angular momentum is propor-tional to A t and ϑ , while the gauge field is proportionalto µ plus corrections and the scalar field is proportionalto M/T plus corrections. When T →
0, the angular mo-mentum tends to a finite constant. We also remark thatout of the three contributions represented in Figs. 2 and4, the second term in Eq. (1.5) varies almost linearlywith µ /M T over the interval we have considered. (cid:45) (cid:45) Μ (cid:144) M L a x (cid:144) M FIG. 2: (Color online) Angular momentum density as a func-tion of µ /M for axionic coupling and non-normalizablescalar field in a quadratic potential with m = − T = 5 M (orange), T = M (blue) and T = M/ T (cid:61)
5M T (cid:61) (cid:61) MT (cid:61) M (cid:144) (cid:61) M (cid:144) Μ (cid:144) MT L a x (cid:144) M FIG. 3: (Color online) Angular momentum density as a func-tion of µ /MT for axionic coupling and non-normalizablescalar field in a quadratic potential with m = − B. Electric edge current
Another interesting phenomenon associated to the ax-ionic coupling is a spontaneous generation of the electriccurrent dual to the bulk gauge field. As we will see inSec. IV, this is closely related to the angular momentumgeneration when the scalar field ϑ is dual to a marginaloperator.The expectation value of the current is defined in termsof the normalizable mode of the bulk gauge field as δj i = − (cid:96) κ lim z → δA i z = − (cid:96) κ ( δA i ) (cid:48) (cid:12)(cid:12)(cid:12) z =0 . (2.34)Evaluating (2.16) at the boundary we obtain (with δg ti (cid:45) Μ (cid:144) MT L a x (cid:144) M FIG. 4: (Color online) Angular momentum density as a func-tion of µ /MT for axionic coupling and non-normalizablescalar field in a quadratic potential with m = − T = 5 M (orange), T = M (blue) and T = M/ normalizable) ∂ z δA i ( z = 0 , x k ) = − β CS (cid:15) ij ∂ j z (cid:90) dw (cid:104) ϑ (cid:48) δA t − A (cid:48) t δϑ (cid:105) (2.35)leading to δj i = − (cid:15) ij ∂ j δχ (2.36)with δχ = 2 (cid:96) β CS κ (cid:90) z dw (cid:104) ϑ (cid:48) δA t − A (cid:48) t δϑ (cid:105) . (2.37)Again using (2.28), the above equation can be written asa total variation in the space of configurations δχ = − (cid:96) β CS κ δ µϑ (0) + z (cid:90) dw ϑ (cid:48) A t = 2 (cid:96) β CS κ δ z (cid:90) dw ϑA (cid:48) t . (2.38)We thus find an electric current j i = − (cid:15) ij ∂ j χ (2.39)with χ = 2 (cid:96) β CS κ z (cid:90) dw ϑA (cid:48) t . (2.40) C. Bulk universality
The results obtained in the previous subsections ex-tend without modification to most general two-derivativetheories of the form S = 12 κ (cid:90) d x √− g (cid:20) R − G IJ ( ϑ K ) ∂ a ϑ I ∂ a ϑ J − V ( ϑ K ) − (cid:96) Z P Q ( ϑ K ) F Pab F Qab − (cid:96) β CS C P Q ( ϑ K ) ∗ F P ab F Qab (cid:21) . (2.41)In the above I, J, K label different scalar fields, while
P, Q label different vector fields, and G IJ , Z P Q and C P Q are functions of scalar fields ϑ K . They are symmetricand assumed to be invertible. We consider a metric ofthe form (2.5) with ϑ I = ϑ I ( z ) , A Pa = A Pt ( z ) δ ta , A Pt (0) = µ P (2.42)where µ P is the chemical potential for boundary con-served current J P dual to A Pa .The discussion exactly parallels that of Sec. II A so be-low we will simply list the counterparts of the key equa-tions there.Background equations of motion (2.9)–(2.10) now be-come A Pt (cid:48) ( z ) = ( Z − ) P R Q R (cid:112) f ( z ) h ( z ) (2.43)with Q R the charge density for J R and4 (cid:112) f h ( Z − ) P R Q P Q R = (cid:18) f (cid:48) z √ f h (cid:19) (cid:48) . (2.44)As before we consider general small perturbations gen-erated by a small and slow-varying δϑ I ( z, x i ) and makethe gauge choice δA Pz = 0 , δg zt = 0 . (2.45)Equations (2.14) and (2.15) then generalize respectivelyto (cid:18) f (cid:48) f + h (cid:48) h + 4 z (cid:19) ∂ z δg ti − ∂ z δg ti = 8 z Z P Q A Pt (cid:48) ∂ z A Qi (2.46)and ∂ z (cid:32) (cid:112) f ( z ) Z P Q ∂ z δA Qi (cid:112) h ( z ) + Q P δg ti (cid:33) (2.47)+ β CS (cid:15) ij ∂ j (cid:16) δC P Q A Qt (cid:48) − C P Q (cid:48) δA Qt (cid:17) = 0 . Then identical manipulations as before lead to (2.26)with Φ = β CS (cid:96) κ (cid:20) − µ P µ Q C P Q (0) (2.48)+ z (cid:90) dw (cid:16) A Pt A Qt ( w ) − µ P A Qt ( w ) (cid:17) C P Q (cid:48) ( w ) (cid:21) or equivalentlyΦ = β CS (cid:96) κ (cid:20) − µ P µ Q C P Q ( z ) (2.49)+ z (cid:90) dw (cid:0) A Pt − µ P (cid:1) (cid:16) A Qt − µ Q (cid:17) C P Q (cid:48) (cid:21) . III. GRAVITATIONAL CHERN-SIMONS TERM
In this section, we consider the induced stress tensorand angular momentum density for bulk theories whereparity violation is generated by the gravitational Chern-Simons coupling, ϑ ∗ RR . We will first consider a simpleexample with a relevant scalar operator and then gener-alize the discussion to generic theories. The discussion issimilar to that of the last section, so we will be briefer. A. Relevant scalar field
Consider the action S = 12 κ (cid:90) d x √− g (cid:20) R −
12 ( ∂ϑ ) − V ( ϑ ) − α CS (cid:96) ϑ ∗ RR (cid:21) (3.1)where α CS is a constant and ϑ is dual to a relevant (ormarginal) pseudoscalar boundary operator. In (3.1) ∗ RR = ∗ R abcd R bacd , ∗ R abcd = 12 (cid:15) cdef R abef (3.2)and (cid:15) abcd is the totally antisymmetric tensor with (cid:15) z =1 / √− g . The equations of motion are R ab − ∂ a ϑ∂ b ϑ − g ab V ( ϑ ) = α CS (cid:96) C ab , √− g ∂ a (cid:0) g ab √− g∂ b ϑ (cid:1) − α CS (cid:96) ∗ RR = 0 (3.3)where C ab ≡ ∇ c ( ∇ d ϑ ∗ R c ( ab ) d ).We again consider a solution of the form (2.5) (with-out the gauge field). The strategy is the same as be-fore. We consider a small and slowly varying pertur-bation δϑ ( z, x i ) and work out the momentum response δT ti to order O ( (cid:15) ) where the power (cid:15) counts the numberof spatial derivatives of δϑ . We then write the resultingexpression as a total variation in the space of field config-urations which enables us to find the angular momentumassociated with (2.5).We will choose a gauge where δg tz = δg xx = δg yy = 0.With the Einstein equations schematically readingLHS ab = α CS (cid:96) C ab (3.4)we note that in this gauge,LHS ti = − z (cid:112) f ( z ) h ( z ) ∂ z (cid:34) f ( z ) z (cid:112) h ( z ) ∂ z (cid:18) δg ti ( z, x k ) f ( z ) (cid:19)(cid:35) (3.5)and δg zi , δg xy are all at least of order O ( (cid:15) ). We then findto O ( (cid:15) ), C ti can be written as C ti = z (cid:96) √ f h (cid:15) ij ∂ j δ Ψ (3.6)with δ Ψ = K (cid:48) + (cid:18) f (cid:48) f h (cid:19) (cid:48) δϑ − f (cid:48) ϑ (cid:48) δg tt f h + f (cid:48) ϑ (cid:48) δg (cid:48) tt f h + f (cid:48) ϑ (cid:48) δg zz f h (3.7)and K = f f (cid:48) h (cid:48) + h (cid:0) f (cid:48) − f f (cid:48)(cid:48) (cid:1) f h δϑ − f (cid:48) ϑ (cid:48) δg zz h + f (cid:48) ϑ (cid:48) δg tt f h − ϑ (cid:48) δg (cid:48) tt h . (3.8)Then following similar manipulations as in (2.18)–(2.22)we find that δT ti = − (cid:15) ij ∂ j δ Φ (3.9)with δ Φ = α CS (cid:96) κ (cid:90) z δ Ψ dz. (3.10)Note that δg tt = − δf and δg zz = δh and δ Ψ can befurther written as δ Ψ = K (cid:48) + (cid:18) f (cid:48) δϑ f h (cid:19) (cid:48) − δ (cid:18) f (cid:48) f h ϑ (cid:48) (cid:19) . (3.11)It can then be immediately checked that the boundaryterms coming from K are all zero with the assumptionof the asymptotic behavior f ( z ) = 1+ z α + · · · , h ( z ) = 1+ z β + · · · (3.12)where α > β >
0. We then note further that z (cid:90) dz (cid:34)(cid:18) f (cid:48) δϑ f h (cid:19) (cid:48) − δ (cid:18) f (cid:48) f h ϑ (cid:48) (cid:19)(cid:35) (3.13)= δ z (cid:90) dz (cid:18) f (cid:48) f h (cid:19) (cid:48) ϑ − δ (cid:32) f (cid:48) f h (cid:12)(cid:12)(cid:12)(cid:12) z (cid:33) ϑ ( z )where the second term is proportional to δT , and thusvanishes if we choose a path in configuration space suchthat δT = 0. Collecting the above we thus find δ Φ is atotal variation withΦ = − α CS (cid:96) κ z (cid:90) dz (cid:18) f (cid:48) f h (cid:19) (cid:48) ϑ (3.14)= − π α CS (cid:96) κ T ϑ ( z ) + α CS (cid:96) κ z (cid:90) dz (cid:18) f (cid:48) f h (cid:19) ϑ (cid:48) (3.15) The angular momentum is thus given by L = 2Φ . (3.16)For a marginal ϑ , ϑ is independent of z and only the firstterm in (3.22) is present. We then find a universal resultwhich is independent of specific forms of f and h L = − π α CS (cid:96) κ T ϑ. (3.17) B. Generalizations
The above discussion can be immediately generalizedto theories of the form S = 12 κ (cid:90) d x √− g (cid:20) R − G IJ ( ϑ K ) ∂ a ϑ I ∂ a ϑ J − V ( ϑ K ) − (cid:96) Z P Q ( ϑ K ) F Pab F Qab − α CS (cid:96) C ( ϑ K ) ∗ RR (cid:21) . (3.18)Fixing the gauge A Pz = 0, one finds that ∂ z δA Pi ( z, x k ) = − Q P (cid:115) h ( z ) f ( z ) δg ti ( z, x k ) . (3.19)From (3.19) one then finds that the ti component of theEinstein equations can again be written asLHS ti = α CS (cid:96) C ti (3.20)with LHS ti given by (3.5) and C ti by (3.6)–(3.8) exceptthat everywhere in C ti the pseudoscalar ϑ is replaced by C ( ϑ I ). In this case we thus find thatΦ = − α CS κ z (cid:90) dz (cid:18) f (cid:48) f h (cid:19) (cid:48) C ( ϑ I ) (3.21)= − π α CS (cid:96) κ T C ( ϑ I ( z ))+ α CS (cid:96) κ z (cid:90) dz (cid:18) f (cid:48) f h (cid:19) C ( ϑ I ) (cid:48) (3.22)We also note in passing that in this case there is noelectric edge current as δj Pi = − (cid:96) κ lim z → δA Pi z = 0 (3.23)where we have used (3.19) and that δg ti ∼ O ( z ). C. An explicit example
We now examine an explicit example. For simplic-ity we once again consider the setup of Sec. II A 3 with T (cid:144) M L g r (cid:144) M FIG. 5: Angular momentum density as a function of
T /M forgravitational Chern-Simons coupling and non-normalizablescalar field in a quadratic potential with m = −
2, at µ = 0.The total angular momentum is represented by solid lines,the first term in Eq. (1.6) by dot-dashed lines and the secondterm in Eq. (1.6) by dashed lines. V ( ϑ ) = m ϑ , m = − ϑ non-normalizable with M the scalar source. We exhibit plots of the gravitationalangular momentum as a function of T /M in Fig. 5, withthe two terms entering Eq. (1.6) presented separately.We remark that the plots are almost linear, which canbe understood from the general structure of Eq. (1.5) asfollows: the geometric factor under the integral is roughlyproportional to T to leading order, while the scalar fieldis proportional to M/T at leading order, making the over-all leading order dependence L gr ∝ M T . IV. RELATION TO THE CHIRAL MAGNETICEFFECT AND THE AXIAL MAGNETIC EFFECT
When the scalar field is marginal it is possible to relateour results to the chiral magnetic effect and to the axialmagnetic effect in 3 + 1 dimensions [27–32] via dimen-sional reduction, as we now explain.In 3 + 1 dimensions, the gauge anomaly, ∂ α j α = b CS (cid:15) αβγδ F αβ F γδ , (4.1)is known to cause spontaneous generation of the corre-sponding current, j i = b CS µ(cid:15) ijk F jk , (4.2)and of the momentum density, T i = b CS µ (cid:15) ijk F jk , (4.3)where i, j, k = 1 , , j i and the axial magnetic effect for T i . (The for-mulas derived in [28] in the Landau frame contain terms in higher powers of µ . The formulas in the above are inthe laboratory frame [29].)In comparison, the Chern-Simons term in our bulk ac-tion in 3 + 1 dimensions, S CS = − β CS (cid:96) κ (cid:90) d x √− gϑ ∗ F ab F ab , (4.4)gives rise to an anomalous divergence of the current j α on the boundary in 2 + 1 dimensions as ∂ α j α = 2 β CS (cid:96) κ (cid:15) αβγ ∂ α ϑF βγ , (4.5)where F βγ is the background gauge field for the boundaryCFT. Since it is the dimensional reduction of the chiralanomaly (4.1) in 3+1 dimensions, where the scalar field ϑ in the bulk is identified with the extra component ϑ = A and F i = ∂ i ϑ , we expect effects corresponding to thechiral magnetic effect (4.2) and to the axial magneticeffect (4.8) to be j i = 2 b CS µ(cid:15) ij ∂ j ϑ,T i = b CS µ (cid:15) ij ∂ j ϑ, (4.6)where we should identify b CS = β CS (cid:96) /κ .We can also include effects due to the axial-gravitational anomalies. In 3 + 1 dimensions, the axial-gravitational anomaly, ∂ α j α = a CS π (cid:15) γδηθ R αβηθ R βαγδ , (4.7)is known to generate the momentum current T i = a CS T (cid:15) ijk F jk , (4.8)but not the current j α itself. The corresponding effect in2 + 1 dimensions should be T i = a CS T (cid:15) ij ∂ j ϑ, (4.9)with the identification, a CS = 2 π α CS (cid:96) /κ .The dimensional reduction of the chiral magnetic effectand axial magnetic effect, (4.6) and (4.9), are in agree-ment with Eqs. (2.39) and (2.40) and consistent withresults in our previous paper [7], where the scalar field ϑ is dual to a marginal operator on the boundary CFT.The main results in this paper, however, are for ϑ dualto a relevant operator, which cannot be obtained by di-mensional reduction of a massless gauge field in 4 + 1dimensions. There may be a generalization of the chiralmagnetic effect and of the axial magnetic effect in 3 + 1dimensions which would correspond to dimensional oxi-dation of the effects studied in this paper, and we leavethis possibility for future investigation.0 Acknowledgments
We thank S. S. Gubser, O. Saremi and D. T. Son foruseful discussion, and we are grateful to N. Yunes for col-laboration on the early stages of this project. We wouldlike to thank K. Landsteiner and the referee for their use-ful comments on the paper. H. O. and B. S. are supportedin part by U.S. DOE Grant No. DE-FG03-92-ER40701.The work of H. O. is also supported in part by a Si-mons Investigator award from the Simons Foundation,the WPI Initiative of MEXT of Japan, and JSPS Grant-in-Aid for Scientific Research No. C-23540285. He alsothanks the hospitality of the Aspen Center for Physicsand the National Science Foundation, which supports theCenter under Grant No. PHY-1066293, and of the Si-mons Center for Geometry and Physics. The work ofB. S. is supported in part by a Dominic Orr GraduateFellowship. B. S. would like to thank the hospitality ofthe Kavli Institute for the Physics and Mathematics ofthe Universe and of the Yukawa Institute for TheoreticalPhysics. H. L. is supported in part by funds providedby the U.S. DOE under cooperative research agreementDE-FG0205ER41360 and thanks the hospitality of IsaacNewton Institute for Mathematical Sciences.
Note added. —When this paper was almost complete,we received the paper [26], in which holographic mod-els with nonzero angular momentum and Hall viscosityare discussed. Their models are different from those dis-cussed in this paper.
Appendix A: Boundary condition at the horizon
The zt component of the Einstein equations reads f (cid:48) ( z ) ∂ i δg ti ( z, x i ) − f ( z ) ∂ i δg (cid:48) ti ( z, x i ) = 0 (A1)which can be integrated to give ∂ i δg ti ( z, x i ) = f ( z ) W ( x i ) . (A2)Since f (0) = 1 and we choose δg ti to be a normalizableperturbation, we must have W ( x i ) = 0 so we conclude ∂ i δg ti ( z, x i ) = 0 . (A3)Using the ii component of the background Einstein equa-tions the ti component of the Einstein equations reads+ 2 zf h (cid:15) ij ∂ j ( ∂ x δg ty − ∂ y δg tx ) − zf hδg (cid:48)(cid:48) ti + ( zf h (cid:48) + zhf (cid:48) + 4 f h ) δg (cid:48) ti − z f hA (cid:48) t δA (cid:48) i ( z, x i ) = 0 . (A4)Using (A3) (cid:15) ij ∂ j ( ∂ x δg ty − ∂ y δg tx ) = − ∂ δg ti with ∂ = ∂ i ∂ i this is − zf h ∂ δg ti + ( zf h (cid:48) + zhf (cid:48) + 4 f h ) δg (cid:48) ti − zf hδg (cid:48)(cid:48) ti − z f hA (cid:48) t δA (cid:48) i = 0 . (A5) We now count the divergences in (A5), using that nearthe horizon h ( z ) = K f ( z ) + K + K ( z − z ) + . . . , (A6) h (cid:48) ( z ) = K f (cid:48) ( z ) f ( z ) + K + . . . , (A7)with K an arbitrary constant. Since the gauge and scalarfields do not diverge at the horizon we obtain the lhs ofthe Einstein equations to be − ∂ δg ti ( z , x i ) = 0 (A8)and imposing the boundary condition δg ti ( z , x i ) → δg ti ( z , x i ) = 0 . (A9) Appendix B: Regularization and renormalization
Consider the action S = 12 κ (cid:90) d x √− g (cid:20) R − G IJ ( ϑ K ) ∂ a ϑ I ∂ a ϑ J − V ( ϑ K ) − (cid:96) Z P Q ( ϑ K ) F Pab F Qab + S cs (cid:21) , (B1)where either S cs = − (cid:96) β CS C P Q ( ϑ K ) ∗ F P ab F Qab (B2)or S cs = − α CS (cid:96) ϑ I =0 ∗ RR. (B3)Note the gravitational Chern-Simons term can always bewritten in this form via field redefinition.
A priori , there are four possible contributions thatneed to be accounted for: the usual Gibbons-Hawking-York boundary term, a term arising from the variationof the (axionic or gravitational) Chern-Simons term S cs ,potential additional terms that must be added for theDirichlet boundary-value problem to be well-defined andlocal counterterms (see the Appendix of [7] for details).Thus, we can write T bdy αβ = 12 κ (cid:16) K αβ − h αβ K + T cs αβ + T reg αβ − T ct αβ (cid:17) . (B4)The CFT stress-energy tensor is obtained by comput-ing the boundary stress-energy tensor T bdy αβ on a planeat finite z parallel to the boundary, multiplying by anappropriate power of z ( z − in our case for the stress-energy tensor with both indices down) and taking the z → √− hH ( ϑ I ) (B5)to the action, with H some function and h ab the inducedmetric, h ab = g ab − n a n b , n a = 1 g zz δ za . (B6)The scalar field counterterms contribute T ct ,ϑti ∼ H ( ϑ I ) h ti (B7) to the ti component of T bdy αβ . 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