Anomalous transport, massive gravity theories and holographic momentum relaxation
AAnomalous transport, massive gravity theories andholographic momentum relaxation
Eugenio Megías ∗ a , b a Departamento de Física Atómica, Molecular y Nuclear andInstituto Carlos I de Física Teórica y Computacional, Universidad de Granada,Avenida de Fuente Nueva s/n, 18071 Granada, Spain b Departamento de Física Teórica, Universidad del País Vasco UPV/EHU,Apartado 644, 48080 Bilbao, SpainE-mail: [email protected]
Quantum anomalies give rise to new non-dissipative transport phenomena in relativistic fluidsinduced by external electromagnetic fields and vortices. These phenomena can be studied inholographic models with Chern-Simons couplings dual to anomalies in field theory. We performa computation in AdS/CFT of the anomalous transport coefficients in a holographic massivegravity model, and find that the anomalous conductivities turn out to be independent of theholographic disorder couplings of the model. To arrive at this result we suggest a new definitionof the energy-momentum tensor in presence of the gauge-gravitational Chern-Simons coupling.We also compute the electric DC conductivity and find that it can vanish for certain values of thedisorder couplings.
XIII Quark Confinement and the Hadron Spectrum - Confinement201831 July - 6 August 2018Maynooth University, Ireland ∗ Speaker. © Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - t h ] N ov nomalous transport, massive gravity theories and holographic momentum relaxation Eugenio Megías
1. Introduction
The modern understanding of hydrodynamics is as an effective field theory [1]. The equationsof motion of the hydrodynamical systems correspond to the conservation laws of the energy-momentum tensor and charged currents. In presence of quantum anomalies, however, the currentsare no longer conserved, and this has important consequences for the transport properties of thesystems. This is the case of gauge anomalies, which are responsible for new dissipationless transportphenomena, such as the chiral magnetic and chiral vortical effects (see e.g. [2, 3] for some reviews).In addition to anomalies, there are other sources of non-conservation of the currents, such as disorder effects in condensed matter systems, see e.g. [4]. From a field theory perspective, disorderis related to explicit breaking of translational invariance, and this leads to momentum dissipation anda modification of the conservation law of the energy-momentum tensor. A possible way of studyingthis in holography is by giving a mass to the graviton. A particular realization of this idea is basedon the introduction of massless scalar fields in AdS space with spatially linear profiles [5]. Somedissipative transport coefficients, like the DC electric conductivity, have been studied with theseholographic massive gravity models leading to a decreasing value when disorder increases [6, 7].In this work we will address the generalization of these models to study non-dissipative transportproperties. A reanalysis of the form of the holographic energy-momentum tensor will be needed toget a consistent result.
2. Hydrodynamics of relativistic fluids
The basic ingredients to study hydrodynamics are the (anomalous) conservations laws ofthe energy-momentum tensor and charged currents. These are supplemented by the constitutiverelations, i.e. expressions of the energy-momentum tensor and the currents in terms of fluidquantities, which are organized in a derivative expansion, also called hydrodynamic expansion [1], (cid:104) T µν (cid:105) = ( ε + P ) u µ u ν + P g µν + ( σ B ε ) a ( B µ a u ν + B ν a u µ ) + σ V ε ( Ω µ u ν + Ω ν u µ ) + · · · , (2.1) (cid:104) J µ a (cid:105) = n a u µ + σ ab (cid:16) E µ b − T P µν D ν (cid:16) µ b T (cid:17) (cid:17) + σ B ab B µ b + σ V a Ω µ + · · · . (2.2)Here ε is the energy density, P is the pressure, n a are the charge densities, u µ is the local fluidvelocity and P µν = g µν + u µ u ν is the transverse projector to the fluid velocity. External electric andmagnetic fields are covariantized as E µ a = F µν a u ν and B µ a = (cid:15) µνρσ u ν F a ,ρσ , where the field strengthsof the gauge fields in an Abelian theory are defined as F a ,µν = ∂ µ A a ,ν − ∂ ν A a ,µ . In addition to theequilibrium contributions, there are extra terms in the constitutive relations which lead to dissipativeand anomalous transport effects. While the electric conductivities σ ab in Eq. (2.2) are responsiblefor dissipative transport, we find in this equation two examples of anomalous transport: i) the chiralmagnetic effect , which is responsible for the generation of an electric current parallel to a magneticfield [8], and ii) the chiral vortical effect in which the electric current is induced by a vortex in thefluid Ω µ = (cid:15) µνρσ u ν ∂ ρ u σ [9, 2]. Apart from the charge flow in a relativistic fluid, there exists alsoenergy flow and consequently analogous anomaly related transport effects in the energy current J i ε ≡ (cid:104) T i (cid:105) , cf. Eq. (2.1). At first order in derivatives the notion of fluid velocity is ambiguous, and needs to be fixed by prescribing a choice nomalous transport, massive gravity theories and holographic momentum relaxation Eugenio Megías
A convenient way to compute the anomalous conductivities are the Kubo formulae, which arebased on retarded correlators of the charged currents and the energy-momentum tensor, and they areobtained within linear response theory [2]. Using this formalism for a theory of free chiral fermions,it has been found that the 1-loop calculation of the chiral magnetic and vortical conductivities receivecontributions of the axial anomaly [8], and also of the mixed gauge-gravitational anomaly [10].More explicitly, the conductivities read σ B ab = π d abc µ c , σ V a = π d abc µ b µ c + T b a , (2.3)where d abc = tr ( T a { T b , T c }) L − tr ( T a { T b , T c }) R and b a = tr ( T a ) L − tr ( T a ) R are the group theoreticfactors related to the axial and gauge-gravitational anomalies, respectively. The Kubo formulae alsopredict that the chiral vortical conductivity coincides with the chiral magnetic conductivity for theenergy current σ V a = ( σ B ε ) a . Apart from the Kubo formalism, the anomalous transport coefficientshave been studied in a wide variety of methods, either in field theory or in holography, leading tosimilar results: these include diagrammatic methods [11], fluid/gravity correspondence [12, 13],and the partition function formalism [14, 15, 16]. In this work we will study anomalous transportphenomena in a holographic massive gravity model in the context of linear response theory. Theelectric DC conductivity will be computed as well in this model, and compared with previousstudies. We will deal with a single U ( ) symmetry.
3. Massive gravity and holographic momentum relaxation
In massive gravity theories, the momentum relaxation is described through the Stückelbergmechanism with Goldstone modes corresponding to scalar fields, X I , which are related to spatialtranslations. A recent holographic implementation of this idea in 4-dim has been presented inRefs. [6, 7]. In this work we will consider explicitly the model of [6], but the same conclusions areobtained when considering the model of [7]. In order to study anomalous transport in holographic massive gravity theories, one shouldconsider the theory in odd dimensions, as only in this case one can introduce in the action thecorresponding Chern-Simons (CS) terms that account for the effects of quantum anomalies. Forthe moment we will focus on non-anomalous properties, and leave the study of the CS terms forSec. 4. The action of the model in 5-dim is S = ∫ d x √− g (cid:20) R + − ∂ M X I ∂ M X I − F − J ∂ M X I ∂ N X I F N L F LM (cid:21) + S GH , (3.1)with scalars X I = k δ Ii x i that break translational invariance. The parameter k controls the degreeof breaking of translational invariance, and in particular when k = of frame. Here we choose a frame in which we demand that the definition of the fluid velocity is not influenced whenswitching on an external magnetic field or having a vortex. In the following we will adopt the following notation: capital letters ’ M ’ denote 5-dim indexes, greek letters ’ µ ’denote 4-dim indexes in the holographic boundary of AdS, and lower-case latin letters ’ i ’ denote spatial directions in thatboundary. nomalous transport, massive gravity theories and holographic momentum relaxation Eugenio Megías gravity theory. The coupling J represents the effects of disorder on the charged sector of thetheory. S GH is the usual Gibbons-Hawking boundary term. In the following we will consider asbackground a charged black hole solution with AdS asymptotics of the form ds = u (cid:16) − f ( u ) dt + dx + d y + dz (cid:17) + du u f ( u ) , A t = φ ( u ) . (3.2)The solutions of the equations of motion are then f ( u ) = (cid:18) − uu h (cid:19) (cid:18) + uu h − k u − µ u u h (cid:19) , φ ( u ) = µ (cid:18) − uu h (cid:19) , (3.3)where we identify µ with the chemical potential. The temperature T = π √ u h (cid:16) − k u h − µ u h (cid:17) doesnot depend on the charge disorder coupling J . The electric conductivity measures the electric current J µ induced by an electric field E µ , cf.Eq. (2.2). There are several methods to compute the DC conductivity in holography, but one of themost straightforward is the one proposed in Ref. [17] within linear response theory. Let us considerthe small perturbations in the metric and gauge field A z = (cid:15) (− Et + a z ( u )) , g tz = (cid:15) u h zt ( u ) , g uz = (cid:15) u h zu ( u ) , (3.4)where E is the external electric field in the z -direction, and assume that the perturbations do notintroduce additional sources, i.e. a z ( ) = h zt ( ) = h zu ( ) =
0. The equations of motion for theperturbations can be solved by demanding regularity of the metric. Noting that the electric currentin the z -direction is J z = u → ( f ( u ) a (cid:48) z ( u )) , and without going into the details of the solution, weobtain the DC conductivity [18, 19] σ DC = J z E = √ u h (cid:18) − k J u h (cid:19) (cid:20) + (cid:18) − k J u h (cid:19) µ k ( + J µ u h ) (cid:21) . (3.5)Note that in the case J =
0, the DC conductivity is σ DC = ( + µ / k )/√ u h >
1, so that it isbounded from below. On the other hand, the conductivity vanishes for k J u h =
2, so that inthis regime the system behaves as an insulator. Moreover, σ DC can become even negative in somerange of the parameters indicating an instability. We show in Fig. 1 (left) the value of the DCconductivity as a function of the parameter k for different values of the parameter J . We alsodisplay in Fig. 1 (right) the regime of parameters where the DC conductivity vanishes.
4. Anomalous transport in massive gravity theories
To study the non-dissipative transport properties of the 5-dim massive gravity model, we shouldintroduce anomalous effects in the theory. As mentioned above, anomalies are mimicked in the This bound is similar to the one proven for holographic matter in ( + ) -dim in Ref. [20]. nomalous transport, massive gravity theories and holographic momentum relaxation Eugenio Megías k Μ Σ D C T (cid:61) (cid:74) (cid:61) (cid:74) (cid:61) (cid:74) (cid:61) Figure 1:
Left panel: DC conductivity at zero temperature as a function of k (normalized to the chemicalpotential µ ). Right panel: Region in the plane ( J , T / µ, k / µ ) where the DC conductivity of Eq. (3.5) vanishes. gravity side through CS terms in the action. Then, we can extend the model of Eq. (3.1) by addingthe following gauge and mixed gauge-gravitational CS terms [13, 21] S CS = ∫ d x √− g (cid:15) µνρστ A µ (cid:16) κ F νρ F στ + λ R α βνρ R β αστ (cid:17) . (4.1)and S CSK = − λ ∫ ∂ d x √ γ n µ (cid:15) µνρστ A ν K ρβ D σ K βτ , (4.2)so that the total action is S tot = S + S CS + S CSK . Eq. (4.2) is a convenient counterterm that allows toreproduce the gravitational anomaly at general hypersurface, γ µν is the induced metric and K µν isthe extrinsic curvature on the holographic boundary of an asymptotically AdS space defined by anoutward pointing unit normal vector n µ . Let us consider for the moment the standard Fefferman-Graham coordinates, ds = dr + γ µν dx µ dx ν . The variation of the on-shell action with a timelike hypersurface at a fixed r is δ S tot = ∫ ∂ √− γ (cid:0) t µν δγ µν + u µν δ K µν (cid:1) + δ S matter , (4.3)where we keep γ µν and K µν as independent variables, i.e. the extrinsic curvature acts like anexternal source conjugate to the operator u µν . We will define the holographic energy-momentumtensor as [19, 22] T µν = t µ ν + u µρ K ρν , with t µν = t µν + t µνλ , (4.4)where t µν = − √− γ ( K µν − K γ µν ) is the standard Brown-York contribution, and t µνλ = − λ √− γ(cid:15) ρστ ( µ (cid:16) D σ K ν ) τ F r ρ + γ ν ) β (cid:219) K βσ F τρ − F τρ K ν ) β K βσ (cid:17) , (4.5) u µν = λ √− γ(cid:15) ρστ ( µ F ρσ K ν ) τ , (4.6)4 nomalous transport, massive gravity theories and holographic momentum relaxation Eugenio Megías where A ( µν ) : = ( A µν + A νµ ) , and dot denotes differentiation with respect to r . This result nat-urally follows from the Ward identity of the energy-momentum tensor in presence of the gauge-gravitational CS term [19]. While ( t ) µν is divergent and needs to be regularized by the standardcounterterms [24], the contributions ( t λ ) µν and u µρ K ρν are already finite before the holographicrenormalization is performed. As we will see in the following, the extra contributions ( t λ ) µν and u µρ K ρν are essential to get the physically correct results for the anomalous transport coefficients. To compute this conductivity in linear response theory, we consider the perturbation A y = (cid:15) Bx , A z = (cid:15) a z ( u ) , g tz = (cid:15) u h zt ( u ) . (4.7)If we apply the usual holographic dictionary to the solution of the equations of motion of theperturbations, and compute the energy-momentum tensor from these solutions, one finds T z = g (cid:48) tz ( u = ) = (cid:0) κ µ + λ π T − λ k (cid:1) B [24], which corresponds only to the contribution ( t ) µν .Using the background, one can easily calculate the corrections to the energy-momentum tensor dueto ( t λ ) µν and u µρ K ρν , leading to ( t λ ) µν = u µρ K νρ = λ k B δ µ ( δ z ) ν . Finally, from the newdefinition of the energy-momentum tensor given by Eq. (4.4), we find (cid:174) J = κ µ (cid:174) B , (cid:174) J ε = (cid:16) κ µ + λ π T (cid:17) (cid:174) B , (4.8)which are the usual expressions for the chiral magnetic effect in the charge and energy currents [2]. Vorticity Ω i = (cid:15) ijk ∂ j u k can be introduced through a gravitomagnetic field B ig = (cid:15) ijk ∂ j ( A g ) k .In the rest frame u µ = ( , , , ) , the gravitomagnetic field is in the mixed component of the metric,i.e. ds = − dt + ( A g ) i dtdx i + d (cid:174) x , and it follows (cid:174) B g = (cid:174) Ω [25]. Let us consider the ansatz A y = (cid:15) B g u µ x , A z = (cid:15) a z ( u ) , g ty = (cid:15) f ( u ) u B g x , g tz = (cid:15) u h zt ( u ) . (4.9)After solving the equations of motion for the perturbations, one finds from the u → a z ( u ) and h zt ( u ) the following response due to a gravitomagnetic field (cid:174) J = (cid:16) κ µ + λ π T (cid:17) (cid:174) B g , (cid:174) J ε = (cid:18) κ µ + λ π T µ (cid:19) (cid:174) B g . (4.10)Remembering that (cid:174) B g = (cid:174) Ω , these are the usual responses of a chiral fluid due to vorticity. The results of this section can be summarized in the following values for the conductivities [19] σ B = κ µ, σ V = κ µ + λ π T , (4.11) In the case of holographic pure gravitational anomalies dual to 2-dim field theories a similar correction has beenfound in Ref. [23]. To compare to a free theory of N f chiral fermions we can identify κ = N f /( π ) and λ = N f /( π ) . nomalous transport, massive gravity theories and holographic momentum relaxation Eugenio Megías σ B ε = κ µ + λ π T , σ V ε = κ µ + λ π T µ . (4.12)Note that these values are the same as in massless gravity, i.e. they don’t have any dependenceon the holographic disorder couplings ( k , J ) . This means that the anomalous conductivities arenot affected by translational breaking effects, and this constitutes one of the most important resultsof this work. From this property, together with the result of Sec. 3.2, we conclude that there is aregime in the theory in which the DC conductivity vanishes, but the anomalous conductivities donot vanish. Finally, one important aspect to remark is that the equality between σ B ε and σ V followsnon-trivially from the definition of the energy-momentum tensor in Eq. (4.4), as the term u µρ K ρν induces a contribution which exactly cancels a dependence σ B ε ∝ λ k [19].
5. Conclusions
Massive gravity theories have been introduced in the literature as holographic duals of disorderin condensed matter systems. In this work we have studied non-dissipative transport propertiesinduced by external electromagnetic fields and vortices in these theories, in particular the chiralmagnetic and chiral vortical effects. We found, as expected, that the corresponding conductivitiesare unchanged by the holographic disorder effects. This property, for the case of the response in theenergy current, follows non-trivially from a careful study of the energy-momentum tensor, whichturns out to be modified by the presence of the mixed gauge-gravitational Chern-Simons term in theaction [19]. This solves the puzzle found in Ref. [18]. In addition, we have studied the electric DCconductivity, and found an interesting regime in which it vanishes, but the anomalous conductivitiesdo not vanish. This leads to the possibility of studying the anomalous transport effects of thesesystems in this regime in a clean way.
Acknowledgments
This work is based on Refs. [18, 19]. I would like to thank J. Fernández-Pendás and especiallyK. Landsteiner for enlightening discussions. Research supported by the Spanish MINEICO andEuropean FEDER funds grants FPA2015-64041-C2-1-P and FIS2017-85053-C2-1-P, by the Juntade Andalucía grant FQM-225, and by the Basque Government grant IT979-16. The research of E.M.is also supported by the Ramón y Cajal Program of the Spanish MINEICO, and by the Universidaddel País Vasco UPV/EHU, Bilbao, Spain, as a Visiting Professor.
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