Collective excitations of massive flavor branes
HHIP-2016-04/TH
Collective excitations of massive flavor branes
Georgios Itsios , , ∗ , Niko Jokela , † , and Alfonso V. Ramallo , ‡ Department of Physics, University of OviedoAvda. Calvo Sotelo 18, 33007 Oviedo, Spain Department of Physics and Helsinki Institute of PhysicsP.O.Box 64FIN-00014 University of Helsinki, Finland Departamento de F´ısica de Part´ıculasUniversidade de Santiago de Compostelaand Instituto Galego de F´ısica de Altas Enerx´ıas (IGFAE)E-15782 Santiago de Compostela, Spain
Abstract
We study the intersections of two sets of D-branes of different dimensionalities. Thisconfiguration is dual to a supersymmetric gauge theory with flavor hypermultipletsin the fundamental representation of the gauge group which live on the defect of theunflavored theory determined by the directions common to the two types of branes.One set of branes is dual to the color degrees of freedom, while the other set addsflavor to the system. We work in the quenched approximation, i.e. , where the flavorbranes are considered as probes, and focus specifically on the case in which the quarksare massive. We study the thermodynamics and the speeds of first and zero soundat zero temperature and non-vanishing chemical potential. We show that the systemundergoes a quantum phase transition when the chemical potential approaches itsminimal value and we obtain the corresponding non-relativistic critical exponents thatcharacterize its critical behavior. In the case of (2 + 1)-dimensional intersections, wefurther study alternative quantization and the zero sound of the resulting anyonicfluid. We finally extend these results to non-zero temperature and magnetic field andcompute the diffusion constant in the hydrodynamic regime. The numerical results wefind match the predictions by the Einstein relation. ∗ [email protected] † niko.jokela@helsinki.fi ‡ [email protected] a r X i v : . [ h e p - t h ] F e b ontents p -D q systems with charge 5 p = 4 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.5 The λ = 2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 A.1 Fluctuations at zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38A.2 Fluctuations at non-zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 41
B Transverse correlators and the conductivity 45C Conductivity by the Karch-O’Bannon method 49 Introduction
There is hope that the gauge/gravity holographic duality could serve to characterize newtypes of compressible states of matter, i.e. , states with non-zero charge density which varycontinuously with the chemical potential. Indeed, holography provides gravitational de-scriptions of strongly interacting systems without long-lived quasiparticles, situations whichcannot be accommodated within the standard Landau’s Fermi liquid theory. Although thefield theories with known holographic dual are very different from those found so far inNature, there are good reasons to believe that these studies could reveal generic universalfeatures of strongly interacting quantum systems [1].In this paper we approach this problem in a top-down model of intersecting branes ofdifferent dimensionalities. We will consider a stack of N c color D p -branes which intersect N f flavor D q -branes ( q ≥ p ) along n common directions. This configuration, which wewill denote by ( n | p ⊥ q ), is dual to a ( p + 1)-dimensional SU ( N c ) gauge theory with N f fundamental hypermultiplets (quarks) living on a ( n + 1)-dimensional defect [2]. In thecontext of holography, we will work in the large N c ’t Hooft limit with N f (cid:28) N c . In thislimit the quarks are quenched and the D q -branes can be treated as probes, whose actionis the Dirac-Born-Infeld (DBI) action, in the gravitational background created by the D p -branes. The embedding of the flavor branes is parameterized by a function which measuresthe distance between the two types of branes. The field theory dual of this distance is themass of the hypermultiplet. Moreover, in order to engineer a system with non-zero baryoniccharge density, we must switch on a suitable gauge field on the worldvolume of the flavorbrane [3]. We will also study the influence of a magnetic field directed along two of thespatial directions of the worldvolume.In [4] we studied the collective excitations of generic brane intersections corresponding tomassless quarks and we uncovered a certain universal structure. The purpose of this article isto extend the results of [4] to the case in which the quarks have a non-zero mass. We will studyfirst the system at zero temperature and non-zero chemical potential. This is the so-calledcollisionless quantum regime, in which the dynamics is dominated by the zero sound mode.This mode is a collective excitation, first found in the holographic context in [5, 6]. Theseresults were generalized to non-zero temperature in [7,8] and to non-vanishing magnetic fieldin [9, 10] (see [11–28] for studies on different aspects of the holographic zero sound). In [4]we developed a general formalism which included all possible intersections ( n | p ⊥ q ) and,in particular, we found an index λ (depending on n , p , and q ) which determines the speedof zero sound for massless quarks. This is intimately related with the fact that λ determinesthe scaling dimension of the charge density or to put it slightly differently, λ acts as thepolytropic index in the equation of state for the holographic matter.In the case of massive quarks the embedding of the D q -brane is non-trivial and must bedetermined in order to extract the different physical properties. When the charge density isnon-vanishing, the brane reaches the horizon of the geometry, i.e. , we have a black hole em-bedding. This embedding depends on a function which parameterizes the shape of the flavorbrane in the background geometry and, in general, must be found by numerical integration of3he equations of motion of the probe. However, in the case of intersections ( n | p ⊥ q ) whichpreserve some amount of supersymmetry at zero temperature T and chemical potential µ some remarkable simplification occurs. Indeed, as shown in [29], in these intersections onecan choose a system of coordinates such that the embedding function is a cyclic variable ofthe DBI Lagrangian when T = 0 and µ (cid:54) = 0. As a consequence, the embedding functionand the physical properties of the configuration, can be found analytically. In particular,one can study the zero temperature thermodynamics of these systems and find the speed offirst sound. This was done in refs. [12, 13] for the D3-D q intersections (3 | ⊥ | ⊥ | ⊥ n | p ⊥ q ) intersection with i.e. , when n = ( p + q − /
2. These cases correspond to those brane intersections whichare supersymmetric in flat space at low energies as the gravitational and Ramond-Ramondforces cancel out. Here the index λ can only take three different values λ = 2 , ,
6, cor-responding to codimension 2 (D p -D p ), codimension 1 (D p -D( p + 2)), and codimension 0(D p -D( p + 4)) intersections, respectively. As in the conformal D3-background, the speedsof first and zero sounds coincide. Moreover, we find the same kind of universality as in themassless case: the speed is the same for those intersections which have the same λ index,or codimension. However, in the massive case the speed of sound depends continuously onthe chemical potential, i.e. , on the charge density, and vanishes when the chemical potentialreaches its minimal value, which corresponds to a vanishing charge density d . Actually, asargued in [30] for the D3-D7 and D3-D5 intersections, there is a quantum phase transitionas d → d (cid:54) = 0 degenerate into a Minkowskiembedding with zero charge density. Here we will find the critical exponents for the general n of common dimensions of the color and flavor branes is equal totwo, the matter hypermultiplets live on a (2+1)-dimensional theory. In this case one canperform an alternative quantization of the quasinormal modes, which consists in imposingmixed Dirichlet-Neumann boundary conditions at the UV. As shown in [31], this alternativequantization amounts to transforming the charged excitations into particles of fractionalstatistics, i.e. , anyons (see also [28, 32–34] for the analysis of different aspects of the holo-graphic anyonic systems). In [4] we studied the zero sound mode as a function of the constantthat measures the degree of mixing the UV boundary conditions. We found that the anyoniczero sound is generically gapped and that this gap can be fine-tuned to zero if a suitablemagnetic field is switched on. This choice corresponds to the case, where the anyons expe-rience no effective magnetic field. In this paper we generalize these results to the case inwhich the quarks are massive.In this article we also study the hydrodynamic regime that is reached when the tem-perature is high enough. The dominant collective mode in this regime is a diffusion mode,which has a purely imaginary dispersion relation characterized by a diffusion constant D .4hen the temperature is non-zero the embedding function is no more a cyclic coordinate ofthe DBI action and cannot therefore be found analytically. Thus, we study this T (cid:54) = 0 caseby using numerical methods, after performing a convenient change of variables. Moreover,this numerical analysis allow us to check the analytic results found at zero temperature, bytaking the T → B . We compare the results for the diffusion constant obtained from the fluctuationanalysis at T (cid:54) = 0 with the ones predicted by the Einstein relation, which gives D in termsof the DC conductivity σ and the charge susceptibility χ . Both σ and χ can be obtainedfrom the embedding function. We find a very good agreement between the numerical resultsfor D and the value given by the Einstein relation.The rest of this paper is organized as follows. In section 2 we formulate our top-downholographic model, solve the equations of motion of the probe at T = 0 and µ (cid:54) = 0, andstudy the thermodynamics at zero temperature. In particular, in this section we find thespeed of first sound and compute the charge susceptibility at T = 0. In section 3 we writethe equations of motion for the fluctuations of the probe at zero temperature. In section4 we analyze the zero sound and find analytically the dispersion relation of this collectivemode. Section 5 is devoted to the study of the scaling behavior near the quantum criticalpoint. In section 6 we study the zero sound mode in an anyonic fluid. Section 7 contains ourresults at non-zero temperature and magnetic field. We summarize our results and discusssome possible future research directions in section 8.We complement and give further details of our analysis in several appendices. AppendixA.1 contains a detailed derivation of the Lagrangian of the fluctuations at zero temperaturewhich is used in section 3. In appendix A.2 we work out the equations of motion of thefluctuations at T (cid:54) = 0. In appendix B we find the correlator of two transverse currents andextract the DC conductivity in the absence of magnetic field. Finally, in appendix C weprovide an alternative derivation of the conductivity, valid also when B (cid:54) = 0. p -D q systems with charge Let us begin our analysis by introducing our setup and studying its properties at zero tem-perature and magnetic field. We will consider a generic D p -brane metric at zero temperatureof the type: ds = g tt ( r ) dt + g xx ( r ) (cid:2) ( dx ) + · · · + ( dx p ) (cid:3) + g rr ( r ) d(cid:126)y · d(cid:126)y , (2.1)where (cid:126)y = ( y , . . . , y − p ) are the coordinates transverse to the D p -brane and the functions g tt , g xx , and g rr depend on the transverse radial direction r = √ (cid:126)y · (cid:126)y . We now embed N f D q -brane probes, with N f (cid:28) N c , extended along the directions( t, x , . . . , x n , y , . . . , y q − n ) . (2.2)5e will refer to this configuration as a ( n | p ⊥ q ) intersection ( n is the number of commonspatial directions of the D p and D q ). This intersection is represented by the array: x · · · x n x n +1 · · · x p y · · · y q − n y q − n +1 · · · y − p Dp : × · · · × × · · · × − · · · − − · · · − Dq : × · · · × − · · · − × · · · × − · · · − We shall denote by (cid:126)z the coordinates (cid:126)y transverse to the D q -brane: (cid:126)z = ( z , . . . , z n − p − q ) , (2.3)with z m = y q − n + m for m = 1 , . . . , n − p − q . Moreover, we define ρ as the radial coordinatefor the subspace spanned by ( y , . . . , y q − n ): ρ = ( y ) + · · · + ( y q − n ) . (2.4)Let us make a short comment on the global symmetries. The original D p -background hasa rotational symmetry in the y i directions, this corresponds to the SO (9 − p ) R-symmetry.When we add N f coincident probe D q -branes we introduce U ( N f ) flavor symmetry. TheD p -D q -intersection ( n | p ⊥ q ) breaks the original R-symmetry, which can be easily read offfrom the isometries. We end up with the global symmetry SO ( n, × U ( N f ) × SO ( p − n ) p × SO ( q − n ) q × SO (9 + n − p − q ). The last group will be further broken when we considermassive D q -brane embeddings.Since, d(cid:126)y = dρ + ρ d Ω q − n − + d(cid:126)z , (2.5)the background metric in these coordinates can be written as: ds = g tt ( r ) dt + g xx ( r ) (cid:104) ( dx ) + · · · + ( dx n ) + ( dx n +1 ) + · · · + ( dx p ) (cid:105) + g rr ( r ) (cid:104) dρ + ρ d Ω q − n − + d(cid:126)z (cid:105) . (2.6)Let us consider a stack of D q -branes with a non-trivial profile in the transverse space. Wewill choose our transverse coordinates in such a way that this profile can be parameterized as (cid:126)z = ( z ( ρ ) , , . . . , z ( ρ ) instead of z ( ρ ) and we will denoteby r = r ( ρ ) the function: r ( ρ ) = (cid:112) ρ + z ( ρ ) . (2.7)The induced metric on the D q -brane worldvolume at zero temperature is: ds q +1 = g tt ( ρ ) dt + g xx ( ρ ) (cid:2) ( dx ) + · · · + ( dx n ) (cid:3) + g rr ( ρ ) (cid:2) (1 + z (cid:48) ) dρ + ρ d Ω q − n − (cid:3) , (2.8)with z (cid:48) = dz/dρ . Let us compute the DBI action of the D q -brane in the case in which thereis a worldvolume gauge field F with components ρt . Thus, we will take F to be given by: F = A (cid:48) t dρ ∧ dt . (2.9)6here A (cid:48) t = ∂ ρ A t and we have chosen a gauge for A such that A ρ = 0. This means thatwe aim to study holographic matter at non-zero baryon charge density by introducing achemical potential for the diagonal U (1) ⊂ U ( N f ). The DBI action becomes: S Dq = − N f T Dq (cid:90) d q +1 ξ e − φ (cid:112) − det( g + 2 πα (cid:48) F ) = (cid:90) dt d n x dρ L DBI , (2.10)with the Lagrangian density L DBI given by: L DBI = −N e − φ ρ q − n − g n xx g q − n − rr (cid:113) g rr | g tt | (1 + z (cid:48) ) − (2 πα (cid:48) ) A (cid:48) t , (2.11)where N is the normalization factor N = N f T Dq Vol( S q − n − ) , (2.12)and where the tension of the D q -brane and the volume of the unit sphere are T Dq = 1(2 π ) q √ α (cid:48) q +1 g s , Vol( S q − n − ) = 2 π q − n Γ (cid:0) q − n (cid:1) . (2.13)For a D p -brane background at zero temperature, the metric and the dilaton are given by: − g tt = g xx = (cid:16) rR (cid:17) − p , g rr = (cid:16) Rr (cid:17) − p , e − φ = (cid:16) Rr (cid:17) (7 − p )( p − . (2.14)This background satisfies g rr | g tt | = 1 and the Lagrangian density L DBI can be written as: L DBI = −N ρ q − n − (cid:16) rR (cid:17) (2 n − p − q +4)(7 − p )4 (cid:113) z (cid:48) − (2 πα (cid:48) ) A (cid:48) t . (2.15)In the following we will scale out the constant R , i.e. , we will take directly R = 1. Toavoid clutter, we also redefine the gauge field by absorbing the factors of the string length2 πα (cid:48) A µ → A µ . Moreover, we will restrict ourselves to the case in which the embeddingfunction z ( ρ ) is a cyclic variable, i.e. , when L DBI depends on z (cid:48) and not on z . The onlydependence on z in (2.15) is the one induced by the power of r multiplying the DBI squareroot. Therefore z ( ρ ) is cyclic only when the following condition between n , p , and q issatisfied: n = p + q − . (2.16)One can check that this happens only in the supersymmetric intersections with p | p ⊥ p + 4), ( p − | p ⊥ p + 2), and ( p − | p ⊥ p ). In the following we will restrictourselves to these cases. Let us define λ as: λ = 2( q − n −
1) = q − p + 2 . (2.17)Notice that λ = 6 , , p -D( p +4), D p -D( p +2), and D p -D p , respectively.We can then write the Lagrangian density as: L DBI = −N ρ λ (cid:112) z (cid:48) − A (cid:48) t . (2.18)7he cyclic nature of z and A t implies the following conservation laws:1 N ∂ L DBI ∂z (cid:48) = − ρ λ z (cid:48) (cid:112) z (cid:48) − A (cid:48) t ≡ − c N ∂ L DBI ∂A (cid:48) t = ρ λ A (cid:48) t (cid:112) z (cid:48) − A (cid:48) t ≡ d , (2.19)with c and d being constants of integration. These relations can be inverted as: z (cid:48) = c (cid:112) ρ λ + d − c , A (cid:48) t = d (cid:112) ρ λ + d − c . (2.20)When c = d = 0, both z ( ρ ) and A t ( ρ ) are constant and we have a Minkowski embedding.Let us suppose that c does not vanish. Then, it follows from (2.20) that A (cid:48) t and z (cid:48) are relatedas: A (cid:48) t = dc z (cid:48) . (2.21)When c = d (cid:54) = 0 both z ( ρ ) and A t ( ρ ) diverge at ρ = 0. Therefore, we discard thisconfiguration and we will assume in the following that d > c . In this case, from theexpression of z (cid:48) and A (cid:48) t written in (2.20) it is easy to conclude that the point ρ = 0 isreached. In what follows we will assume that this condition holds. We will integrate theequation for A t ( ρ ) by imposing that A t (0) = 0. We have: A t ( ρ ) = d (cid:90) ρ d ¯ ρ (cid:112) ¯ ρ λ + d − c . (2.22)This integral can be computed analytically and expressed in terms of the hypergeometricfunction as: A t ( ρ ) = d (cid:0) d − c (cid:1) − λ ρ (cid:2) ρ λ + d − c (cid:3) λ F (cid:16) λ ,
12 + 1 λ ; 1 + 1 λ ; ρ λ ρ λ + d − c (cid:17) . (2.23)Similarly, the embedding function z ( ρ ) can be written as: z ( ρ ) = c (cid:0) d − c (cid:1) − λ ρ (cid:2) ρ λ + d − c (cid:3) λ F (cid:16) λ ,
12 + 1 λ ; 1 + 1 λ ; ρ λ ρ λ + d − c (cid:17) . (2.24)Notice that when d > c the brane reaches the Poincar´e horizon of the metric at ρ = z = 0and we have a black hole embedding. The two constants d and c are related to the chargedensity and condensate of the dual theory, respectively. Let us first consider the intersections with λ >
2. We also restrict to T = 0, as at non-zerotemperature not much can be said analytically. In this case the functions A t ( ρ ) and z ( ρ )8n (2.23) and (2.24) approach a constant value in the UV region ρ → ∞ . According to thestandard AdS/CFT dictionary, the flavor chemical potential µ is the UV value of A t : µ = A t ( ρ → ∞ ) = d (cid:0) d − c (cid:1) − λ F (cid:16) λ ,
12 + 1 λ ; 1 + 1 λ ; 1 (cid:17) = d (cid:16) d − c (cid:17) − λ γ , (2.25)where γ is the constant γ = 1 √ π Γ (cid:16) − λ (cid:17) Γ (cid:16) λ (cid:17) (2.26)and we used the identity F ( A, B ; C ; 1) = Γ( C ) Γ( C − A − B )Γ( C − A ) Γ( C − B ) .The mass parameter m of the embedding is defined as m = z ( ρ → ∞ ). It follows from(2.24) that: m = c (cid:16) d − c (cid:17) − λ γ . (2.27)Let us invert (2.25) and (2.27) and compute c and d in terms of µ and m . First, we noticethat: µ − m = (cid:16) d − c (cid:17) λ γ . (2.28)Since d ≥ c , eq. (2.28) implies that µ ≥ m for the embeddings we are considering.Moreover, from (2.28) we get d − c as a function of µ and m and, using this result in (2.25)and (2.27), we obtain c = m γ − λ (cid:16) µ − m (cid:17) λ − , d = µ γ − λ (cid:16) µ − m (cid:17) λ − . (2.29)When λ > µ = m in (2.29) corresponds to c = d = 0, i.e. , to the Minkowski embeddingswith vanishing density discussed above. Actually, as illustrated in Fig. 1, the topology ofthe embeddings changes when m → µ , where a quantum phase transition takes place. Theorder parameter of this transition is the charge density (see [29] for further details).Let us now evaluate the on-shell action of the probe. Using (cid:112) z (cid:48) − A (cid:48) t (cid:12)(cid:12)(cid:12) on − shell = ρ λ (cid:112) ρ λ + d − c , (2.30)we find S on − shell = −N (cid:90) ∞ ρ λ (cid:112) ρ λ + d − c dρ , (2.31)which is divergent and must be regulated. We will do it by subtracting the same integralwith the integrand evaluated at the UV ( ρ → ∞ ). We arrive at S regon − shell = −N (cid:90) ∞ ρ λ (cid:34) ρ λ (cid:112) ρ λ + d − c − (cid:35) dρ = 2 N λ + 2 (cid:16) d − c (cid:17) λ + γ . (2.32)9 ρ m zm Figure 1: In this figure we plot the different embeddings for λ = 4 and m/µ = 0 . , . , . m/µ = 1 would correspondto the constant horizontal line z/m = 1.The zero-temperature grand canonical potential Ω is given by minus the regulated on-shellaction: Ω = − S regon − shell = − N λ + 2 (cid:16) d − c (cid:17) λ + γ . (2.33)In terms of m and µ the grand canonical potential can be written as:Ω = − N λ + 2 γ − λ (cid:16) µ − m (cid:17) λ +24 , (2.34)where we have used (2.28). Moreover, the charge density is: ρ ch = − ∂ Ω ∂µ = µ N γ − λ (cid:16) µ − m (cid:17) λ − = N d , (2.35)which confirms our identification of the constant d . It is worth noting that the formulasthat we will write down do not have the factor of the (infinite) volume of the gauge theorydirections V R n , rather all thermodynamic quantities are densities per unit volume. Next, wecompute the energy density as: (cid:15) = Ω + µ ρ ch = N ( λ + 2) γ − λ (cid:16) µ − m (cid:17) λ − ( λµ + 2 m ) . (2.36)To calculate the speed of first sound u s we make use of the equation u s = ∂P∂(cid:15) = ∂P∂µ (cid:16) ∂(cid:15)∂µ (cid:17) − , (2.37)10here P is the pressure. Let us first compute the derivative appearing in the numerator.Since P = − Ω, we get from (2.34): ∂P∂µ = µ N γ − λ (cid:16) µ − m (cid:17) λ − . (2.38)Moreover, from (2.36) we have: ∂(cid:15)∂µ = µ N γ − λ (cid:16) µ − m (cid:17) λ − (cid:16) λ µ − m (cid:17) . (2.39)These yield u s = 2 µ − m λ µ − m , (2.40)which is the result we were looking for. As a check notice that (2.40) gives u s = 2 /λ for m = 0, which is the universal result found in [4]. Moreover, the speed of sound (2.40)depends on the integers ( n, p, q ) through the combination λ , i.e. , u s is the same for conformaland non-conformal brane backgrounds with the same index λ . In particular, for the D3-D7and D3-D5 supersymmetric intersections we have: u s = µ − m µ − m , for D3 − D7 ,u s = µ − m µ − m , for D3 − D5 . (2.41)These results agree with the calculation in [12, 13]. Notice that the speed of sound vanishesin the zero density limit with µ = m , which is a clear sign of a quantum phase transition.Let us now consider the case λ = 2, which corresponds to the ( p − | p ⊥ p ) intersections.In these systems A t ( ρ ) and z ( ρ ) grow logarithmically when ρ → ∞ and the AdS/CFTdictionary must be adapted accordingly. Indeed, in this case the chemical potential andthe mass are obtained from the subleading terms of A t and z in the UV . Moreover, theon-shell action has additional logarithmic divergences, which must be eliminated with newcounterterms [35,36]. As the result of this analysis one gets that the grand canonical potentialfor black hole embeddings takes the form Ω = − a ( µ − m ), where a is a positive constant[37]. Repeating the calculation of u s performed above, it is straightforward to verify that u s = 1 in this λ = 2 case. Notice that this value is exactly the one obtained by taking λ = 2in (2.40). For the massless SO ( n, × SU ( N f ) × U (1) × SO (3 − λ/ p × SO (1 + λ/ q × SO (5 − n ). The SO (1 + λ/ q part ro-tates a sphere of λ/ m = 0. Fluctuations
We now allow fluctuations of both the gauge field along the Minkowski directions of theintersection and of the scalar function in the form: A ν = A (0) ν + a ν ( ρ, x µ ) , z = z ( ρ ) + ξ ( ρ, x µ ) , (3.1)where A (0) = A (0) ν dx ν = A t dt is the one-form for the unperturbed gauge field (2.23) and z is the embedding function written in (2.24). The total gauge field strength is: F = F (0) + f , (3.2)with F (0) = dA (0) and f = da . The dynamics of the fluctuations is determined by theLagrangian density L that results after expanding the DBI action to second order in theperturbations a µ and ξ . The detailed calculation of L is performed in appendix A.1. TheLagrangian can be neatly written in terms of open string metric G ab , which is symmetric andhas the following non-vanishing components: G tt = − ρ λ + d ( ρ + z ) − p ρ λ G ρρ = ( ρ + z ) − p ρ λ + d − c ρ λ G x i x j = δ ij ( ρ + z ) − p . (3.3)The Lagrangian density for the fluctuations can be written as: L = −N ρ λ (cid:112) ρ λ + d − c (cid:34) G ac G bd f cd f ab + 12 r − p (cid:16) − c ρ λ (cid:17) G ab ∂ a ξ ∂ b ξ − c d r − p ρ λ ( ∂ t ξ ) − cdr − p ρ λ G ab ∂ a ξ f tb (cid:35) , (3.4)where we have defined r = r ( ρ ) as: r ( ρ ) = (cid:112) ρ + z ( ρ ) . (3.5)In (3.4) the tensor indices a, b, c, d run over the directions ( ρ, x µ ).Let us now explicitly write down the equations of motion derived from the Lagrangian(3.4). We will choose the gauge in which a ρ = 0. Moreover, we will consider fluctuationfields a ν which depend on ρ , t , and x . Then, it is possible to restrict to the case in which a ν (cid:54) = 0 only when ν = t, x ≡ x , and x ≡ y . The equation of motion for a ρ when a ρ = 0leads to the following transversality condition: (cid:16) d ρ λ (cid:17) ∂ t a (cid:48) t − c dρ λ ∂ t ξ (cid:48) − ∂ x a (cid:48) x = 0 . (3.6)12et us Fourier transform the gauge and scalar fields to momentum space as: a ν ( ρ, t, x ) = (cid:90) dω dk (2 π ) a ν ( ρ, ω, k ) e − iω t + ikx ξ ( ρ, t, x ) = (cid:90) dω dk (2 π ) ξ ( ρ, ω, k ) e − iω t + ikx . (3.7)In momentum space the transversality condition (3.6) takes the form: (cid:16) d ρ λ (cid:17) ω a (cid:48) t − c dρ λ ω ξ (cid:48) + k a (cid:48) x = 0 . (3.8)Let us now define the electric field E as the gauge-invariant combination: E = k a t + ω a x . (3.9)Using (3.8) we can obtain a (cid:48) t and a (cid:48) x in terms of E (cid:48) and ξ (cid:48) as follows: a (cid:48) t = k ρ λ E (cid:48) − c d ω ξ (cid:48) ρ λ k − ( ρ λ + d ) ω , a (cid:48) x = ωρ λ k − ( ρ λ + d ) ω (cid:2) c d kξ (cid:48) − ( ρ λ + d ) E (cid:48) (cid:3) . (3.10)The equation of motion for a t derived from (3.4) is: ∂ ρ (cid:34) (cid:112) ρ λ + d − c ρ λ (cid:104) ( ρ λ + d ) a (cid:48) t − c d ξ (cid:48) (cid:105)(cid:35) − r − p (cid:112) ρ λ + d − c (cid:104) ( ρ λ + d ) ∂ x ( ∂ t a x − ∂ x a t ) + c d ∂ x ξ (cid:105) = 0 . (3.11)The equation for a x is: ∂ ρ (cid:16)(cid:112) ρ λ + d − c a (cid:48) x (cid:17) − r − p (cid:112) ρ λ + d − c (cid:2) ( ρ λ + d ) ∂ t ( ∂ t a x − ∂ x a t ) + cd∂ t ∂ x ξ (cid:3) = 0 . (3.12)By using (3.10), eqs. (3.11) and (3.12) reduce (in momentum space) to the following equationin terms of the electric field E : ∂ ρ (cid:104) (cid:112) ρ λ + d − c ( ω − k ) ρ λ + ω d (cid:2) ( ρ λ + d ) E (cid:48) − cd k ξ (cid:48) (cid:3)(cid:105) + ( ρ λ + d ) E − cd k ξr − p (cid:112) ρ λ + d − c = 0 . (3.13)The equation for a y in momentum space is: ∂ ρ (cid:16)(cid:112) ρ λ + d − c a (cid:48) y (cid:17) + ( ρ λ + d ) ω − ρ λ k r − p (cid:112) ρ λ + d − c a y = 0 . (3.14)Finally, the equation for the scalar ξ in momentum space can be written as: ∂ ρ (cid:34) (cid:112) ρ λ + d − c ρ λ (cid:104) ( ρ λ − c ) ξ (cid:48) + cd a (cid:48) t (cid:105)(cid:35) + [( c − ρ λ ) k + ( ρ λ + d − c ) ω ] ξ − cd k Er − p (cid:112) ρ λ + d − c = 0 . (3.15)13y using (3.10) we can rewrite this equation in terms of the electric field E : ∂ ρ (cid:34) (cid:112) ρ λ + d − c ( ω − k ) ρ λ + ω d (cid:104) [( c − ρ λ ) k + ( ρ λ + d − c ) ω ] ξ (cid:48) − cd k E (cid:48) (cid:105)(cid:35) + [( c − ρ λ ) k + ( ρ λ + d − c ) ω ] ξ − cd k Er − p (cid:112) ρ λ + d − c = 0 . (3.16)In the next section we study these equations of motion in the regime in which the frequency ω and the momentum k are small and of the same order. We will find a sound mode, thezero sound, and we will be able to determine analytically its dispersion relation followingthe matching technique introduced in [5]. We now study the zero sound of the massive embeddings by matching the near-horizon andlow frequency behavior of the fluctuations. The technique we employ consists in performingthese two limits in different order [5].
Let us first consider the equations of motion (3.13) and (3.16) near the Poincar´e horizon ρ ≈
0. To perform this analysis we define the functions χ and χ as: χ = ( ρ λ + d ) E − cd k ξχ = [( c − ρ λ ) k + ( ρ λ + d − c ) ω ] ξ − cd k E . (4.1)For small ρ we just neglect the terms containing ρ λ in the χ i ’s. Then, these functions takethe form: χ ≈ d E − cd k ξ , χ ≈ [ c k + ( d − c ) ω ] ξ − cd k E , (4.2)and, therefore, are related to E and ξ by linear combinations with constant coefficients. Inorder to write the near-horizon equations for χ and χ , let us study the behavior of theembedding function z ( ρ ) for small ρ . From (2.24) we easily obtain: z ≈ c √ d − c ρ . (4.3)It follows that r ( ρ ) behaves near ρ ≈ r ≈ d √ d − c ρ . (4.4)14sing these results it is straightforward to demonstrate that, for small ρ , the χ i ’s satisfy theequation: χ (cid:48)(cid:48) i + Λ ρ − p χ i = 0 , (4.5)where Λ is the following rescaled frequency:Λ = (cid:16) d − c d (cid:17) − p ω . (4.6)Eq. (4.5) is the same equation as in the massless case (with ω → Λ). When p <
5, thesolution of this equation with incoming boundary condition at the horizon is given by thefollowing Hankel function: χ i ( ρ ) = ρ H (1) − p (cid:16) − p ρ p − (cid:17) , ( p < . (4.7)For d (cid:54) = c the equations in (4.2) can be inverted and one can obtain E and ξ as linearcombinations (with constant coefficients) of χ and χ . Thus, E and ξ behave as in (4.7).Moreover, when p < ω is small, we have: E ( ρ ) = A ρ + A c p Λ − p + · · · , ξ ( ρ ) = B ρ + B c p Λ − p + · · · , ( p < , (4.8)where A and B are constants and the coefficient c p is: c p = π (5 − p ) p − − p (cid:104) Γ (cid:16) − p (cid:17)(cid:105) (cid:104) i − cot (cid:16) π − p (cid:17)(cid:105) , ( p < . (4.9) Let us now start by taking the low frequency limit of the fluctuation equations (3.13) and(3.16). One can show that in this limit one can neglect the terms without derivatives. Then,the fluctuation equations reduce to: ∂ ρ (cid:104) (cid:112) ρ λ + d − c ( ω − k ) ρ λ + ω d (cid:2) ( ρ λ + d ) E (cid:48) − cd k ξ (cid:48) (cid:3)(cid:105) = 0 ∂ ρ (cid:34) (cid:112) ρ λ + d − c ( ω − k ) ρ λ + ω d (cid:104) [( c − ρ λ ) k + ( ρ λ + d − c ) ω ] ξ (cid:48) − cd k E (cid:48) (cid:105)(cid:35) = 0 . (4.10)These equations can be immediately integrated once to give:( ρ λ + d ) E (cid:48) − cd k ξ (cid:48) = C ( ω − k ) ρ λ + ω d (cid:112) ρ λ + d − c c d k E (cid:48) − [( c − ρ λ ) k + ( ρ λ + d − c ) ω ] ξ (cid:48) = C ( ω − k ) ρ λ + ω d (cid:112) ρ λ + d − c , (4.11)15here C and C are integration constants. Solving for E (cid:48) and ξ (cid:48) , we get: E (cid:48) = [( ω − k ) ρ λ + ( k − ω ) c + ω d ] C − cd k C ( ρ λ + d − c ) ξ (cid:48) = cd k C − ( ρ λ + d ) C ( ρ λ + d − c ) . (4.12)In order to perform a further integration, let us define the following functions: J ( ρ ) ≡ (cid:90) ∞ ρ ¯ ρ λ ( ¯ ρ λ + d − c ) d ¯ ρ , J ( ρ ) ≡ (cid:90) ∞ ρ d ¯ ρ ( ¯ ρ λ + d − c ) . (4.13)For λ > J ( ρ ) = 2 λ − ρ − λ F (cid:16) , − λ ; 32 − λ ; − d − c ρ λ (cid:17) J ( ρ ) = 23 λ − ρ − λ F (cid:16) , − λ ; 52 − λ ; − d − c ρ λ (cid:17) . (4.14)Moreover, by construction J ( ρ → ∞ ) = J ( ρ → ∞ ) = 0. It follows that: E ( ρ ) = E (0) − ( ω − k ) C J ( ρ ) − (cid:104) [( k − ω ) c + ω d ] C − cd k C (cid:105) J ( ρ ) ξ ( ρ ) = ξ (0) + C J ( ρ ) + d (cid:2) d C − c k C (cid:3) J ( ρ ) , (4.15)where E (0) and ξ (0) are the values of E and ξ at the boundary ρ → ∞ . Let us now expand E ( ρ ) and ξ ( ρ ) near ρ ≈
0. With this purpose it is better to deal directly with the integralsdefining J and J . One can easily prove that: J ( ρ ) = 2 λ γ ( d − c ) λ − + O ( ρ ) J ( ρ ) = λ − λ γ ( d − c ) λ − − ρ ( d − c ) + O ( ρ ) , (4.16)where γ is the constant defined in (2.26). Using these expansions we can represent E ( ρ ) nearthe horizon as: E ( ρ ) = E (0) + b C + b C + ( a C + a C ) ρ + · · · , (4.17)where the coefficients b i and a i are given by: b = − γλ ( d − c ) λ − (cid:104) ( λc − d ) k + λ ( d − c ) ω (cid:105) b = λ − λ γ ( d − c ) λ − c d ka = ( d − c ) ω + c k ( d − c ) a = − c d k ( d − c ) . (4.18)16imilarly, ξ ( ρ ) can be expanded as: ξ ( ρ ) = ξ (0) + ˜ b C + ˜ b C + (˜ a C + ˜ a C ) ρ + · · · , (4.19)where the different coefficients are:˜ b = − λ − λ γ ( d − c ) λ − c d k = − b ˜ b = γλ ( d − c ) λ − ( λ d − c )˜ a = c d k ( d − c ) = − a ˜ a = − d ( d − c ) . (4.20) We now match (4.17) and (4.19) with (4.8). By identifying the terms linear in ρ , we canwrite the constants A and B of (4.8) in terms of the coefficients of (4.18) and (4.20): A = a C + a C = (cid:2) ( d − c ) ω + c k (cid:3) C − c d k C ( d − c ) B = ˜ a C + ˜ a C = c d k C − d C ( d − c ) . (4.21)By eliminating A and B and comparing the constant terms in (4.8), (4.17), and (4.19) weget the boundary values of E and ξ as functions of C and C : E (0) ξ (0) = Λ − p c p a − b Λ − p c p a − b Λ − p c p ˜ a − ˜ b Λ − p c p ˜ a − ˜ b C C . (4.22)We now require the vanishing of the sources E (0) and ξ (0) , which only happens non-triviallyif the determinant of the matrix written in (4.22) is zero. This leads to the following relation:( a ˜ a + a ) Λ − p c p − ( a ˜ b + ˜ a b + 2 a b ) Λ − p c p + b ˜ b + b = 0 . (4.23)From (4.23) we can find the dispersion relation of the zero sound modes. Indeed, let usassume that ω ∼ k ∼ O ( (cid:15) ). Then, Λ ∼ O ( (cid:15) ) and the orders of the different coefficients in(4.23) are: b ∼ a ∼ O ( (cid:15) ) , b ∼ a ∼ O ( (cid:15) ) , ˜ b ∼ ˜ a ∼ O ( (cid:15) ) , ˜ b ∼ ˜ a ∼ O ( (cid:15) ) . (4.24)At leading order the only contribution comes from the last two terms in (4.23), which there-fore reduces to: b ˜ b + b = 0 . (4.25)17y using the values of the constants b , b , and ˜ b from (4.18) and (4.20) it is straightforwardto verify that (4.25) leads to the following dispersion relation: ω = ω = 2( d − c ) λ d − c k . (4.26)Let us write this result in terms of the reduced mass parameter m , defined as: m = mµ . (4.27)One easily checks that: cd = m . (4.28)and the leading dispersion relation can be written as: ω = c s k , (4.29)where c s is the speed of zero sound, given by c s = 2 1 − m λ − m . (4.30)Notice that, non-trivially, c s is equal to the speed of first sound written in (2.40).Let us now compute the next order in the dispersion relation. We write: ω = ω + δω . (4.31)At first-order in δω , we get: δω = − p − − p ) c p λ dγ ( d − c ) − p − p − λ ( λ d − c ) − p − p ) +1 k − p − p . (4.32)In terms of m this expression becomes: δω = − p − − p ) c p λµ (1 − m ) − p − p − ( λ − m ) − p − p ) +1 k − p − p , (4.33)where we used the following relation of µ , d , and m : µ = γ d λ (1 − m ) λ − . (4.34)Let us use the expression of c p in (4.9) and separate the imaginary and real parts:Im δω = − π λµ (5 − p ) p − − p (cid:104) Γ (cid:16) − p (cid:17)(cid:105) p − − p ) (1 − m ) − p − p − ( λ − m ) − p − p ) +1 k − p − p Re δω = π λµ (5 − p ) p − − p (cid:104) Γ (cid:16) − p (cid:17)(cid:105) cot (cid:16) π − p (cid:17) p − − p ) (1 − m ) − p − p − ( λ − m ) − p − p ) +1 k − p − p . (4.35)18n particular, for p = 3 the real part of Re δω vanishes at the order we are working in (4.35)and the complete dispersion relation is given by: ω p =3 = ± √ (cid:34) − m λ − m (cid:35) k − i λµ − m ( λ − m ) k . (4.36)In order to compare with the results in [8, 11], let us substitute µ by its expression in termsof the density d (eq. (4.34)). We find ω p =3 = ± √ (cid:34) − m λ − m (cid:35) k − i λ d λ Γ (cid:0) (cid:1) Γ (cid:16) − λ (cid:17) Γ (cid:16) λ (cid:17) (1 − m ) − λ ( λ − m ) k . (4.37)In particular, for the D3-D5 system we take λ = 4 and arrive at the following dispersionrelation: ω D − D = ± (cid:34) − m − m (cid:35) k − i d Γ (cid:0) (cid:1)(cid:104) Γ (cid:16) (cid:17)(cid:105) (1 − m ) (2 − m ) k . (4.38)In Fig. 2 we check (4.38) by comparing it with the results obtained by numerical integration m μ c s m μ - - - - Im ω ^ k Figure 2: We depict the speed of zero sound (left) and the attenuation divided by momentumsquared (right) for the D3-D5 intersection with ˆ d = 10 ( ˆ d , ˆ ω , and ˆ k are defined in (7.17) and(7.35)). The dots have been obtained by integrating numerically the fluctuation equations(A.48) and (A.49). The continuous line corresponds to the analytic expression (4.38).at non-zero (but small) temperature. As it can be appreciated in this figure, the agreementis very good, both for the speed of zero sound c s and for the attenuation ( i.e. , the imaginarypart of ω ). p = 4 case As pointed out around (4.8), the p = 4 case is special and we have to modify our analysis.Indeed, the expansion of the Hankel function H (1)1 ( x ) near x = 0 contains logarithmic terms,19hich implies that E ( ρ ) and ξ ( ρ ) behave near the horizon at low frequency as: E ( ρ ) = A ρ + A c Λ + A Λ log (cid:0) ρ Λ (cid:1) + · · · ξ ( ρ ) = B ρ + B c Λ + B Λ log (cid:0) ρ Λ (cid:1) + · · · , (4.39)where c is the constant: c = i π + 1 − γ E . (4.40)In (4.40) γ E = 0 . · · · is the Euler-Mascheroni constant. Let us now try to obtain theexpansion (4.39) by performing the limits in the opposite order. As in [12], we have tocompute the next correction to (4.17) and (4.19) near the horizon. First we notice that theequations satisfied by E ( ρ ) and ξ ( ρ ) near ρ = 0 are just obtained by taking p = 4 in (4.5): E (cid:48)(cid:48) = − Λ ρ E , ξ (cid:48)(cid:48) = − Λ ρ ξ . (4.41)Neglecting the right-hand side in (4.41) and integrating twice, we arrive at a linear solutionas in (4.17) and (4.19). To go beyond this approximation we plug the values of E and ξ into the right-hand side of (4.41) and perform the integration. In the low-frequency limit ω (cid:28) ρ , we have: E ( ρ ) = E (0) + b C + b C + ( a C + a C ) ρ + ( a C + a C )Λ log ρ + · · · ξ ( ρ ) = ξ (0) + ˜ b C + ˜ b C + (˜ a C + ˜ a C ) ρ + (˜ a C + ˜ a C )Λ log ρ + · · · . (4.42)Let us now match (4.39) and (4.42). By comparing the linear and logarithmic terms of theseequations we arrive at the same values of A and B as those written in (4.21). Moreover,using these values of A and B and identifying the constant terms, we find the followingmatrix relation between ( E (0) , ξ (0) ) and ( C , C ): E (0) ξ (0) = Λ ( c − log Λ ) a − b Λ ( c − log Λ ) a − b Λ ( c − log Λ ) ˜ a − ˜ b Λ ( c − log Λ ) ˜ a − ˜ b C C . (4.43)As in the p < a ˜ a + a ) Λ ( c − log Λ ) − ( a ˜ b + ˜ a b + 2 a b ) Λ ( c − log Λ )+ b ˜ b + b = 0 . (4.44)Notice that (4.44) is obtained from (4.23) by taking p = 4 and changing c p → c − log Λ on the latter. Using this observation it is straightforward to find the dispersion relationencoded in (4.44). At leading order in ω ∼ k (4.44) reduces to (4.25), which means that theleading dispersion relation is just given by (4.29) and (4.30). Moreover, the next-to-leadingcontribution δω is: δω = − √ λµ ( c − log Λ ) (1 − m ) ( λ − m ) k . (4.45)20he imaginary part of δω is easily deduced from (4.45):Im δω = − π √ λµ (1 − m ) ( λ − m ) k . (4.46)Notice that (4.46) is the same as in the first equation in (4.35) for p = 4. Similarly, the realpart of δω can be written as:Re δω = √ λµ (cid:34) γ E − (cid:32) − m ) λ − m k (cid:33)(cid:35) (1 − m ) ( λ − m ) k . (4.47) λ = 2 case For λ = 2 the integral J ( ρ ), defined in (4.13), is not convergent and, therefore, the expres-sions written in (4.15) for E ( ρ ) and ξ ( ρ ) at low frequency are not correct. In order to obtainthe solution of (4.12) for λ = 2, let us define the integral ¯ J ( ρ ) as:¯ J ( ρ ) ≡ (cid:90) ∞ ρ d ¯ ρ (cid:104) ¯ ρ ( ¯ ρ + d − c ) − ρ (cid:105) = ρ (cid:112) ρ + d − c − ρ (cid:112) ρ + d − c + ρ . (4.48)Then, (4.12) for λ = 2 can be integrated as: E ( ρ ) = E (0) − ( ω − k ) C [ ¯ J ( ρ ) − log ρ ] − (cid:104) [( k − ω ) c + ω d ] C − cd k C (cid:105) J ( ρ ) ξ ( ρ ) = ξ (0) + C [ ¯ J ( ρ ) − log ρ ] + d (cid:2) d C − c k C (cid:3) J ( ρ ) , (4.49)where E (0) and ξ (0) are constants. When ρ is very large the integrals ¯ J ( ρ ) and J ( ρ ) vanishby construction and thus E ( ρ ) and ξ ( ρ ) behave at the UV as: E ( ρ ) = E (0) + ( ω − k ) C log ρ + · · · ξ ( ρ ) = ξ (0) − C log ρ + · · · , ( ρ → ∞ ) . (4.50)As argued in [35], when the logarithmic behavior displayed in (4.50) is present, the sourcesare identified with the coefficients of the logarithms, which should vanish. It is clear fromthe behavior of ξ ( ρ ) in (4.50) that we must require that C = 0. Moreover, the logarithmicterm in E ( ρ ) is absent either when C = 0 or when: ω = ± k . (4.51)If C = C = 0 it follows from (4.49) that the functions E ( ρ ) and ξ ( ρ ) are constant andalso the matching with the near-horizon results in (4.8) imply that both E and ξ mustvanish. Therefore, the only non-trivial solution is given by the dispersion relation (4.51),which corresponds to a zero sound mode without dissipation and speed c s = 1. Notice thatthis result coincides with the value of the speed of first sound in (2.40) for λ = 2. Moreover,when C = 0 and ω = k , eq. (4.49) reduces to: E ( ρ ) = E (0) − ω d C J ( ρ ) , ξ ( ρ ) = ξ (0) − c d k C J ( ρ ) . (4.52)21aking ρ → E (0) and ξ (0) are related to the constant C as: E (0) = C d d − c ω + c p C dd − c ω − p )5 − p ξ (0) = C c dd − c k + c p C cd − c k ω − p . (4.53)Notice that (4.53) coincides with (4.22) when λ = 2, C = 0 and ω = k . In particular,these relations imply that the ratio of E (0) and ξ (0) is given by: E (0) ξ (0) = dc k . (4.54)The analysis performed so far in this section is valid for p <
4. When p = 4 we have to gobeyond the leading term in ω , as in section 4.4, in order to match the logarithmic terms inthe near-horizon expansion. It is easy to check that the λ = 2 solution written above can becorrected to match the ρ → As already mentioned, the probe D-brane systems analyzed above undergo a quantum phasetransition as µ → m and the density d vanishes. It was shown in [30] that the critical pointsof the D3-D7 and D3-D5 intersections are described by a non-relativistic scale invariant fieldtheory exhibiting hyperscaling violation. In this section we extend these results to the caseof non-conformal backgrounds ( i.e. , for p (cid:54) = 3) and we compute the corresponding criticalexponents.Let us thus follow the approach of [30] and study the behavior of the system near thequantum critical point at µ = m . Accordingly, we consider a chemical potential of the form: µ = m + ¯ µ , (5.1)where ¯ µ is considered to be small. At leading order in ¯ µ we can expand the differentthermodynamic functions of (2.34), (2.36), and (2.29) as:Ω = − P ≈ − λ +64 λ + 2 γ − λ N (cid:0) m ¯ µ ) λ +24 (cid:15) = f ≈ λ − γ − λ N m λ +64 ¯ µ λ − d ≈ λ − γ − λ m λ +24 ¯ µ λ − . (5.2)where f is the free energy density. The non-relativistic energy density e is defined as in [30]: e = (cid:15) − ρ ch m = (cid:15) − N d m , (5.3)22here ρ ch = N d is the physical charge density. By using (2.36) and (2.29) we get: e = N γ − λ (cid:0) µ − m (cid:1) λ − (cid:104) λµ + 2 m λ + 2 − µ m (cid:105) . (5.4)Expanding at leading order in ¯ µ , we arrive at: e ≈ λ − λ − λ + 2 N γ − λ (cid:0) m ¯ µ (cid:1) λ +24 . (5.5)Comparing this result with the one for the pressure in (5.2), we obtain the following relationbetween e and P : e = λ − P . (5.6)According to the analysis in [30], the relation between e and P at zero temperature near thequantum critical point is: e = n − θz P , (5.7)where θ is the hyperscaling violation exponent and z is the dynamical critical exponent. Eq.(5.7) is a consequence of the scaling dimensions of e , P , ¯ µ , and d , namely: [ e ] = [ P ] = z + n − θ ,[¯ µ ] = z , and [ d ] = n − θ . Thus, in our case we have the following relation between θ and z : θ = n − λ − z . (5.8)Notice that the relation (5.8) between θ and z coincides with the ones found in [30] for theD3-D7 system (taking n = 3 and λ = 6) and for the D3-D5 intersection (taking n = 2and λ = 4). In order to determine z we look at the speed of sound (2.40) for µ ≈ m . Atfirst-order in ¯ µ it is given by: u s ≈ λ − µm , (5.9)and the corresponding dispersion relation is: ω ≈ (cid:114) λ − µm k . (5.10)Matching the scaling dimensions of both sides of (5.10) as in [30], using that [ ω ] = z and[ k ] = 1, we conclude that: z = 2 . (5.11)Therefore θ takes the value: θ = n − λ . (5.12)Taking into account that for the SUSY D p -D q intersections we are considering n = p + q − , λ = q − p + 2 , (5.13)we can rewrite the expression of θ simply as: θ = p − . (5.14)23otice that for a D3-D q intersection the previous formula gives θ = 1, in agreement with [30].Eq. (5.14) is the generalization of this result for any p .Let us now consider the system at finite temperature T . According to the analysis of [38],when T is small the free energy density can be approximated as: f ( µ, m, T ) = f ( µ, m, T = 0) + π ρ ch T + O ( T ) . (5.15)Then, the non-relativistic free energy density is given by: f non − rel ( µ, m, T ) = f ( µ, m, T ) − ρ ch m = e + π ρ ch T + O ( T ) . (5.16)At leading order in ¯ µ we have: f non − rel ( µ, m, T ) = 2 λ − λ − λ + 2 N γ − λ (cid:0) m ¯ µ (cid:1) λ +24 (cid:104) π λ + 2 λ − T ¯ µ + O (cid:16)(cid:16) T ¯ µ (cid:17) (cid:17)(cid:105) . (5.17)In the quantum critical region the non-relativistic free energy density should scale as: f non − rel ∼ (cid:0) ¯ µ (cid:1) − α g (cid:16) T ¯ µ νz (cid:17) , (5.18)where α is the exponent which characterizes the scaling of the specific heat capacity C and ν is the exponent corresponding to the correlation length ξ ( i.e. , C ∼ ( T − T c ) − α and ξ ∼ ( T − T c ) − ν near a phase transition at T = T c ). Comparing (5.18) and (5.17) it followsthat, in our case, we have: 2 − α = λ + 24 , ν z = 1 . (5.19)Since z = 2 for our system, the exponents α and ν are: α = 6 − λ , ν = 12 . (5.20)Using the expression of λ in terms of p and q written in (5.13), we can recast α simply as: α = 1 − q − p . (5.21)These results again coincide with the ones in [30] for the D3-D7 and D3-D5 intersections.Remarkably, the exponents obtained above satisfy the hyperscaling-violation relation:( n + z − θ ) ν = 2 − α . (5.22) In this section we will restrict ourselves to the study of intersections which are (2 + 1)-dimensional. In this case one can impose mixed Dirichlet-Neumann boundary conditions to24he fluctuation modes, i.e. , one can adopt an alternative quantization scheme [39, 40]. Theequations of motion are the same for different quantizations, only the boundary conditionsin the UV are different. On the dual field theory side this corresponds to having an anyonicfluid [31–34]. Let us impose the following boundary condition at the UV:lim ρ →∞ (cid:104) n ρ λ f ρ µ − (cid:15) µαβ f αβ (cid:3) = 0 , (6.1)where n is a constant that characterizes the boundary condition (the normal quantizationcondition considered so far corresponds to n = 0). As in [4], it is straightforward to provethat (6.1) is equivalent to require:lim ρ →∞ E = − i n lim ρ →∞ (cid:2) ρ λ a (cid:48) y (cid:3) , lim ρ →∞ a y = i n ω − k lim ρ →∞ (cid:2) ρ λ E (cid:48) (cid:3) . (6.2)Notice that, even if the equations of motion (3.13) and (3.14) for E and a y are decoupled,the mixed boundary conditions (6.2) introduce a coupling between them. Therefore, toimplement (6.2) we have to study the equation of motion of a y , written in (3.14). Near thehorizon ρ ≈ a (cid:48)(cid:48) y + Λ ρ − p a y = 0 , (6.3)which is just the same as (4.5). For p < p < a y ( ρ ) = C ρ + C c p Λ − p + · · · , ( p < , (6.4)with C being a constant. We now perform the two limits in the opposite order. For lowfrequencies (3.14) reduces to: ∂ ρ (cid:104)(cid:112) ρ λ + d − c a (cid:48) y (cid:105) = 0 , (6.5)whose integration is straightforward: a y ( ρ ) = a (0) y − C J ( ρ ) , (6.6)where a (0) y = a y ( ρ → ∞ ), C is a constant of integration, and J ( ρ ) is the following integral(for λ > J ( ρ ) = (cid:90) ∞ ρ d ¯ ρ ( ¯ ρ λ + d − c ) = 2 λ − ρ − λ F (cid:16) , − λ ; 32 − λ ; − d − c ρ λ (cid:17) . (6.7)Let us now expand a y ( ρ ) in powers of ρ . First, one can check that, for small ρ , the integral J ( ρ ) can be approximated as: J ( ρ ) ≈ µd − ρ √ d − c , (6.8)25here µ is the chemical potential (2.25). Therefore, for small ρ , a y can be approximated as: a y ( ρ ) ≈ a (0) y − C µd + C ρ √ d − c . (6.9)Let us now match (6.4) and (6.9). From the linear terms, we get the following relationbetween the constants C and C : C = C √ d − c . (6.10)Using this relation, and identifying the constant terms in (6.4) and (6.9), we get the followingrelation between a (0) y and C : a (0) y = (cid:104) µd + c p √ d − c Λ − p (cid:105) C . (6.11)Let us now rewrite the boundary conditions (6.2) at low frequency and momentum. Fromthe expressions of E and a y in this regime (eqs. (4.12) and (6.6)), we conclude that theybehave in the UV as: E (cid:48) (cid:12)(cid:12) ρ →∞ ≈ ( ω − k ) ρ − λ C , a (cid:48) y (cid:12)(cid:12) ρ →∞ ≈ ρ − λ C . (6.12)Taking this into account, we can recast the boundary conditions for the alternative quanti-zation as a relation between the constants E (0) , a (0) y , C , and C . Indeed, let us define E (0) n and a (0) y, n as: E (0) n ≡ E (0) + i n C , a (0) y, n = a (0) y − i n C . (6.13)Then, (6.2) is equivalent to the conditions: E (0) n = a (0) y, n = 0 . (6.14)The UV values E (0) n , ξ (0) , and a (0) y, n can be related to the constants C , C , and C . In matrixform this relation becomes: E (0) n ξ (0) a (0) y, n = Λ − p c p a − b Λ − p c p a − b i n Λ − p c p ˜ a − ˜ b Λ − p c p ˜ a − ˜ b − i n µd + c p √ d − c Λ − p C C C , (6.15)where a , a , b , b and ˜ a , ˜ a , ˜ b , ˜ b are given in (4.18) and (4.20), respectively. To have a non-trivial solution of the condition E (0) n = ξ (0) = a (0) y, n = 0 we must require that the determinantof the matrix in (6.15) be zero. This leads to: (cid:16)(cid:104) Λ − p c p a − b (cid:105)(cid:104) Λ − p c p ˜ a − ˜ b (cid:105) − (cid:104) Λ − p c p a − b (cid:105)(cid:104) Λ − p c p ˜ a − ˜ b (cid:105)(cid:17) × (cid:104) µd + c p √ d − c Λ − p (cid:105) + n (cid:104) ˜ b − Λ − p c p ˜ a (cid:105) = 0 . (6.16)26t leading order in frequency and momentum this equation simplifies as: b ˜ b + b + d n µ ˜ b = 0 . (6.17)Since: b ˜ b + b = γ λ (cid:0) d − c (cid:1) λ − (cid:104) d − c ) k − ( λd − c ) ω (cid:105) , (6.18)then (6.17) implies the following gapped dispersion relation: ω = ω = 2( d − c ) λ d − c k + (cid:16) d n µ (cid:17) . (6.19)In terms of the reduced mass parameter m , defined in (4.27), we have ω = 2 1 − m λ − m k + (cid:16) d n µ (cid:17) . (6.20)One can also calculate the next order term in the dispersion relation. Indeed, one can checkthat ω = ω + δω , where δω is given by: δω = − p − − p ) c p λµ (1 − m ) − p − p − ( λ − m ) − p − p ) +1 k − p − p − n λ − c p kµ (cid:16) γµ (cid:17) λ ω p − − p (1 − m ) + λ . (6.21) k Re ω k Re ω Figure 3: We plot the dispersions in the D3-D5 model ( p = 3 , λ = 4) at ˆ d = 10 andˆ B = 3 · . In both plots the red points stand for numerical results whereas the blue curvesare the analytic from (6.23); we emphasize that the analytic result (6.23) is an educatedguess, but reproduces the numerics precisely. (left) We vary the quantization parameter n = 0 , n crit , n crit (top-down) at fixed mµ = 0 .
5. (right) The quantization parameter is chosento be critical n = n crit . Different lines correspond to varying mµ = 0 . , . , . x x plane. Actually, itwas found in [4] that the effect of a magnetic field B effectively changes the parameter n as n → n − Bd . (6.22)27n the present massive case we cannot verify analytically the substitution rule (6.22) sincethe embedding function z ( ρ ) is not a cyclic variable in the presence of a B field. Therefore,we conjecture that the dispersion relation of the zero sound with general anyonic boundaryconditions and magnetic field is given (at leading order) by: ω = 2 1 − m λ − m k + 1 µ (cid:0) d n − B (cid:1) . (6.23)Thus, the spectrum is generically gapped for non-vanishing B and n . However, it can bemade gapless by adjusting the alternative quantization parameter n to the critical value: n crit ≡ Bd . (6.24)This particular case corresponds to one, where the anyonic fluid experiences zero net effectivemagnetic field, thus the resulting spectrum is also gapless. In Fig. 3 we compare the resultsobtained from the numerical integration of the fluctuation equations to our analytic formula(6.23). We see that the agreement is very good and, in particular, the numerics confirm thatthe spectrum becomes gapless at n = n crit . Let us now consider the D p -D q intersections ( n | p ⊥ q ) at non-zero temperature and magneticfield. First, we introduce a more convenient system of coordinates. Let us represent thedifferent components of the Cartesian coordinates (cid:126)y transverse to the D p -brane as: y m = r cos θ η m , m = 1 , · · · , q − n ,y l = r sin θ ξ l , l = q − n + 1 · · · , − p , (7.1)where η m and ξ l satisfy: q − n (cid:88) m =1 (cid:0) η m (cid:1) = − p (cid:88) l = q − n +1 (cid:0) ξ l (cid:1) = 1 . (7.2)Clearly, the η m ( ξ l ) are the coordinates of a ( q − n − n − p − q )-sphere). As: − p (cid:88) l = q − n +1 (cid:0) y l (cid:1) = r sin θ , q − n (cid:88) m =1 (cid:0) y m (cid:1) = r cos θ , (7.3)we identify the coordinates z and ρ used so far with: z = r sin θ , ρ = r cos θ . (7.4)It is straightforward to check that d(cid:126)y · d(cid:126)y = dr + r (cid:2) dθ + cos θ d Ω (cid:107) + sin θ d Ω ⊥ (cid:3) , (7.5)28here d Ω (cid:107) = d Ω q − n − is the line element of the ( q − n − q -brane worldvolumeand d Ω ⊥ = d Ω n − p − q is the metric of the (8 + n − p − q )-sphere transverse to the D q -brane.The ten-dimensional metric of a black D p -brane in these coordinates is: ds = (cid:16) rR (cid:17) − p (cid:2) − f p ( r ) dt + d(cid:126)x (cid:3) + (cid:16) Rr (cid:17) − p (cid:104) dr f p ( r ) + r (cid:0) dθ + cos θ d Ω (cid:107) + sin θ d Ω ⊥ (cid:1)(cid:105) , (7.6)where R is a constant radius and the blackening factor f p is: f p ( r ) = 1 − (cid:16) r h r (cid:17) − p , (7.7)and r h is the horizon radius, that is related to the temperature as follows: T = 7 − p π r − p h . (7.8)Let us consider a D q -brane probe extended along t, x . . . , x n , r and the ( q − n − θ = θ ( r ), the inducedmetric is (for R = 1): ds q +1 = r − p (cid:2) − f p dt +( dx ) + · · · +( dx n ) (cid:3) + r p − (cid:2) (1+ r f p ˙ θ ) dr f p + r cos θ d Ω (cid:107) (cid:3) , (7.9)with ˙ θ = dθ/dr . In what follows we will take n , p and q to be related as in (2.16). Moreover,we will add a magnetic field in the x x directions. The Ansatz for the worldvolume gaugefield strength in this case becomes: F = ˙ A t dr ∧ dt + B dx ∧ dx . (7.10)The DBI Lagrangian density for this Ansatz is: L = −N √ H (cos θ ) λ (cid:113) − ˙ A t + r f p ˙ θ , (7.11)where λ is given by (2.17) and we have introduced a new function H , defined as: H ≡ r λ + r λ + p − B . (7.12)In this Lagrangian A t is a cyclic variable. Its equation of motion can be integrated once togive: (cos θ ) λ √ H ˙ A t (cid:113) − ˙ A t + r f p ˙ θ = d , (7.13)with d being an integration constant. From this last equation we obtain ˙ A t as:˙ A t = d (cid:113) r f p ˙ θ (cid:113) d + H (cid:0) cos θ (cid:1) λ . (7.14)29fter eliminating A t , we find the following equation for the embedding function θ ( r ): ∂ r (cid:34) r f p (cid:115) d + H (cos θ ) λ r f p ˙ θ ˙ θ (cid:35) + λ H (cos θ ) λ − sin θ (cid:113) r f p ˙ θ (cid:113) d + H (cid:0) cos θ (cid:1) λ = 0 . (7.15)The equation of motion (7.15) has explicit dependence on the blackening factor f p , whichhas factors of the horizon radius r h . This feeds in temperature dependence via (7.8). Thehorizon radius r h can be scaled out by an appropriate change of variables, followed by aredefinition of the density d and the magnetic field B . Indeed, let us define the reducedradial variable ˆ r as follows: ˆ r = rr h . (7.16)It is then straightforward to verify that, in terms of ˆ r , the embedding equation is just (7.15)with r h = 1 and d and B substituted by the scaled quantities ˆ d and ˆ B , defined as:ˆ d = dr λ h , ˆ B = Br − p h . (7.17)We will integrate (7.15) by imposing that the D q -brane intersects the horizon r = r h at somevalue θ h ≡ θ ( r = r h ), i.e. , we will require that our embedding is a black hole embedding. At the UV r → ∞ the function θ ( r ) behaves generically as: θ ( r ) ∼ mr + C r λ + · · · = ˆ m ˆ r + ˆ C ˆ r λ + · · · ( r → ∞ ) , (7.18)where m and C are related to the mass and condensate, respectively. Notice that we haveintroduced in (7.18) the scaled quantities ˆ m and ˆ C , related to m and C as:ˆ m = mr h , ˆ C = C r λ h . (7.19)It is also interesting to write the chemical potential µ in terms of the scaled quantities. Wehave: ˆ µ = µr h , (7.20)where ˆ µ is given by the following integral:ˆ µ = ˆ d (cid:90) ∞ d ˆ r (cid:113) r p − (ˆ r − p − (cid:0) dθd ˆ r (cid:1) (cid:113) ˆ d + (ˆ r λ + ˆ r λ + p − ˆ B )(cos θ ) λ . (7.21)Notice that ˆ m/ ˆ µ = m/µ , i.e. , the horizon radius r h drops out when one computes themass/chemical potential ratio as both of the quantities have the same dimension. It is interesting to write the zero temperature results of section 2 in terms of the ( r, θ ) variables used inthis section. Let θ ∗ be the angle at the horizon when T = 0, i.e. , θ ∗ = θ ( r = 0). Then, tan θ ∗ = c/ √ d − c .Other useful relations at zero temperature are m = γ d λ tan θ ∗ (cos θ ∗ ) λ and µ = γ d λ / (cos θ ∗ ) − λ , whichimply sin θ ∗ = m/µ . .1 Charge susceptibility Let us consider now the case B = 0 and compute the charge susceptibility χ , which is definedas: χ = ∂ρ ch ∂µ . (7.22)Taking into account that the charge density ρ ch is related to d as ρ ch = N d , we can rewritethe last expression as: χ − = 1 N ∂µ∂d = 1 N (cid:90) ∞ r h ∂ ˙ A t ∂d dr . (7.23)By a direct calculation using (7.14) for B = 0 , we get: ∂ ˙ A t ∂d = √ ∆ r λ (cid:0) cos θ (cid:1) λ d + r λ (cid:0) cos θ (cid:1) λ (cid:34) d (cid:32) λ θ ∂θ∂d + r f p ˙ θ ∆ ∂ ˙ θ∂d (cid:33)(cid:35) , (7.24)where ∆ is defined as:∆ ≡ − ˙ A t + r f p ˙ θ = r λ (cos θ ) λ r f p ˙ θ d + r λ (cos θ ) λ . (7.25)Therefore, the charge susceptibility can be written as: χ − = 1 N (cid:90) ∞ r h dr √ ∆ r λ (cid:0) cos θ (cid:1) λ d + r λ (cid:0) cos θ (cid:1) λ (cid:34) d (cid:32) λ θ ∂θ∂d + r f p ˙ θ ∆ ∂ ˙ θ∂d (cid:33)(cid:35) . (7.26)Let us consider some particular cases of (7.26). First of all, we consider the massless case,in which θ = 0 and the integral in (7.26) can be performed explicitly. We get: N χ − = 2 λ − r − λ h F (cid:16) , − λ ; 32 − λ ; − d r λh (cid:17) , m = 0 . (7.27)Another interesting limiting case is when T = 0 . In this case we can obtain χ withoutusing (7.26). Indeed, we can compute the derivative of µ from the second equation in (2.29).Computing ∂d/∂µ for constant m , we get: ∂d∂µ = γ − λ (cid:0) µ − m (cid:1) λ − (cid:104) λ µ − m (cid:105) , T = 0 , (7.28)where γ is the constant defined in (2.26). Then, it follows that: χ = N γ − λ λ µ − m (cid:0) µ − m (cid:1) − λ , T = 0 . (7.29)Notice that (for λ < m = µ .31 .0 0.5 1.0 1.5 m μ D Figure 4: We plot the rescaled diffusion constant ˆ D = r − p h D for the D2-D6 model atˆ d = 1. The solid blue curve is obtained purely from the background for ˆ B = 0, i.e. , byevaluating the integral (7.31). The red points, on the other hand, are numerical results fromthe fluctuation analysis by solving the coupled equations of motion in (A.48)-(A.50) for zeromagnetic field. We see that the two different methods agree perfectly. The lower black curveis the result obtained from (7.42) for ˆ B = 1 /
2, while the red points are the result of thenumerical integration of the fluctuation equations for this value of ˆ B . Again, both methodsagree perfectly. The diffusion constant D can be related to the charge susceptibility χ by means of theso-called Einstein relation, which reads: D = σ χ − , (7.30)where σ is the DC conductivity. The value of σ can be extracted from the analysis of thetwo-point correlators of the transverse currents. This analysis is carried out in appendix B.The final result for σ is written in (B.34). Plugging this value of σ and the susceptibilitywritten in (7.26) into (7.30), we arrive at the following expression for D : D = r p − h (cid:113) r λh (cos θ h ) λ + d (cid:90) ∞ r h dr √ ∆ r λ (cid:0) cos θ (cid:1) λ d + r λ (cid:0) cos θ (cid:1) λ × (cid:34) d (cid:32) λ θ ∂θ∂d + r f p ˙ θ ∆ ∂ ˙ θ∂d (cid:33)(cid:35) . (7.31)32et us now extract the low temperature behavior of D by using the T = 0 susceptibilitywritten in (7.29). As σ ∼ N d r p − h for low T , we get: D ≈ γ λ (cid:0) µ − m (cid:1) − λ λ µ − m d r p − h , ( T ∼ . (7.32)The expression (7.31) for D can be compared with the values obtained by analyzing thespectrum of diffusive modes of the probe in the hydrodynamical regime (see section 7.3below). This comparison is shown in Fig. 4 for the D2-D6 intersection. We have obtained avery good agreement between the two methods in all intersections studied. Let us now consider a fluctuation of the embedding angle and of the gauge field of the form: θ ( x µ , r ) = θ ( r ) + ζ ( x µ , r ) , A ( x µ , r ) = A (0) ( r ) + a ( x µ , r ) , (7.33)where a ( x µ , r ) = a ν ( x µ , r ) dx ν and A (0) = A (0) ν dx ν = A t dt + B x dx is the one-form for theunperturbed gauge field. We will choose the gauge in which a r = 0 and we will considerfluctuation fields a ν and ζ depending only on r , t , and x . In this case it is possible torestrict to the case in which a ν (cid:54) = 0 only when ν = t , x ≡ x , and x ≡ y . In appendix A.2we obtain the Lagrangian density for the fluctuations and we perform a detailed analysisof the corresponding equations of motion. This analysis is performed in momentum space.Accordingly, let us Fourier transform a ν and ζ as: a ν ( r, t, x ) = (cid:90) dω dk (2 π ) a ν ( r, ω, k ) e − iω t + ikx ζ ( r, t, x ) = (cid:90) dω dk (2 π ) ζ ( r, ω, k ) e − iω t + ikx . (7.34)At very low temperature the numerical analysis of the coupled fluctuation equations (A.48),(A.49), and (A.50) allows to find sound modes, i.e. , the zero sound. The correspondingdispersion relation is given in terms of the rescaled frequency and momentum ˆ ω and ˆ k ,related to ω and k as: ˆ ω = ωr − p h , ˆ k = kr − p h . (7.35)For vanishing magnetic field the numerical results are in very good agreement with theanalytic equations of section 4, as it was illustrated already in Fig. 2 for the conformal D3-D5 intersection. This agreement is confirmed in Fig. 5 for the non-conformal cases D2-D4and D2-D6.At higher temperatures the system is in a hydrodynamic regime, in which the dominantmode is a diffusion mode with purely imaginary frequency. The spectrum of these diffusion33 .2 0.4 0.6 0.8 1.0 m μ c s Figure 5: We demonstrate how well the analytics and numerics match also for the non-conformal brane intersections, by focusing on p = 2 and at low temperature and vanishingmagnetic field. We depict the speed of zero sound for two intersections D2-D4 ( λ = 4)and D2-D6 ( λ = 6) at ˆ d = 10 λ . The numerics are represented by red points whereas theanalytics, following (4.30), are continuous curves. Lower dataset is for λ = 6 and higher for λ = 4. In extracting the slope we kept ˆ k ˆ d − pλ = 0 . ω = − i ˆ D ˆ k , (7.36)where ˆ D is the rescaled diffusion constant, related to D as:ˆ D = r − p h D . (7.37)The value of ˆ D predicted by Einstein relation can be straightforwardly obtained from (7.31).Indeed, one must simply take r h = 1 and change d by ˆ d in (7.31). The low temperature limitof ˆ D can also be obtained easily from (7.32). We get:ˆ D ≈ γ λ (cid:0) ˆ µ − ˆ m (cid:1) − λ λ ˆ µ − m ˆ d , ( T ∼ . (7.38)In Fig. 6 we show the temperature dependence of ˆ D for the D2-D6 model. The temperatureis decreased by increasing ˆ d . The results displayed in Fig. 6 indeed show that ˆ D approachesthe value written in (7.38) as ˆ d → ∞ .Let us now consider the dependence on the magnetic field. The results of section 6 (andthose of refs. [4, 9, 10]) strongly suggest that the spectrum of the zero sound is gapped and34 .2 0.4 0.6 0.8 1.0 m μ D Figure 6: We depict the diffusion constant at various temperatures and vanishing magneticfield for the D2-D6 model ( p = 2 , λ = 6) as obtained by solving the fluctuation equations.The continuous curves correspond to ˆ d = 10 , , (bottom-up). As a reference we havealso included the T → d = 10 , depicted as a dashed black curve,showing how well the ˆ d = 10 numerical curve is converging to it. Higher values of ˆ d wouldbe overlapping even more.that the gap is just B/µ . Therefore we are led to conjecture the following expression of theleading order dispersion relation of the zero sound: ω = 2 1 − m λ − m k + B µ , (7.39)where we have just added the gap to the gapless value of ω . Notice that (7.39) impliesthat the gap is independent of the quark mass m for fixed chemical potential µ . We haveexplicitly verified this feature numerically in Fig. 7 for the D2-D6 system.Let us next analyze the dependence of the diffusion constant on the magnetic field B .In order to write the expression of D which follows from the Einstein relation, we need toknow the value of the DC conductivity σ when B (cid:54) = 0. In principle, this conductivity couldbe obtained from the analysis of the transverse correlators, as was done in appendix B for B = 0. However, the fluctuation equations couple the transverse and longitudinal modeswhen B (cid:54) = 0 and and it is not clear to us how to deal with this coupling. For this reason wewe have computed σ by applying the method of ref. [41]. The details of this calculation areexplained in appendix C. The final result for σ is: σ = N (cid:113) r λh (1 + r p − h B ) (cos θ h ) λ + d r − ph + B r − p h . (7.40)35 .1 0.2 0.3 0.4 0.5 k Re ω k Re ω Figure 7: We present numerical evidence that our conjecture for the dispersions at lowtemperature (7.39) is supported at finite magnetic field strength. We demonstrate this inthe case of the D2-D6 model ( p = 2, λ = 6) at ˆ B = 10 for two different cases: in the canonicaland in the grand canonical ensemble. From the latter case we clearly find that the massgap of the zero sound is indeed independent of the mass ( ˆ m ) of the fundamentals. Differentcurves in both panels correspond to mµ = 0 . , . , . d = 10 . (Right) We keep the chemical potential fixed ˆ µ = 200.It is now straightforward to write down the expression of D which follows from (7.30).Indeed, let us define ∆ B as: ∆ B = H (cos θ ) λ (1 + r f p ˙ θ ) d + H (cos θ ) λ , (7.41)where H is the quantity defined in (7.12). Then, the Einstein relation gives the followingvalue of the diffusion constant: D = (cid:113) r λh (1 + r p − h B ) (cos θ h ) λ + d r − p h + r p − h B (cid:90) ∞ r h dr (cid:112) ∆ B H (cid:0) cos θ (cid:1) λ d + H (cid:0) cos θ (cid:1) λ × (cid:34) d (cid:32) λ θ ∂θ∂d + r f p ˙ θ ∆ B ∂ ˙ θ∂d (cid:33)(cid:35) . (7.42)In Fig. 4 we compare the predictions of (7.42) for the D2-D6 model and the numerical resultsobtained by direct integration of the coupled fluctuation equations (A.48)-(A.50). As can beappreciated in this figure, the agreement between the two methods is very good. In this paper we studied the collective excitations of flavor D q -branes in the supergravitybackground generated by color D p -branes. The two set of branes are separated in theirtransverse directions, which corresponds to adding massive flavors in the dual field theory.36e first studied this D p -D q model at T = 0 and µ (cid:54) = 0 in the quenched approximation. Thenon-zero chemical potential is generated by a suitable worldvolume gauge field on the probe.We then generalized these results for T (cid:54) = 0 and non-vanishing magnetic field.At zero temperature and non-vanishing chemical potential the supersymmetric D p -D q intersections with p = 3. These results allow to characterize the quantum phase transition thatoccurs when µ = m and d = 0. In this point several thermodynamic quantities vanish andthe system displays a non-relativistic scaling behavior with hyperscaling violation. We havebeen able to compute the corresponding critical exponents.We also analyzed the massive flavor brane systems at non-zero temperature and magneticfield. We verified numerically that, when the magnetic field is non-vanishing, the zero soundspectrum becomes gapped, with the gap given by B/µ . Moreover, when T is large enoughthe system enters into a hydrodynamic regime, which is dominated by a diffusion mode. Wedetermined numerically the corresponding diffusion constant and verified the validity of theEinstein relation.When the intersection is (2 + 1)-dimensional we performed an alternative quantizationof the fluctuations, which corresponds to adding degrees of freedom with fractional statistics(anyons). In those systems the zero sound is generically gapped, although it becomes gaplessif the magnetic field is chosen appropriately. In fact, this choice corresponds to a fluid ofanyons experiencing zero effective magnetic field, thus the occurrence of gapless mode wasexpected. Our understanding of the anyonic fluid is still lacking, though. In order to describeits properties better one would need to make a definite choice for the SL (2 , Z ) transformationas this is needed to make an identification of the resulting charge density of the anyons.Moreover, as there is a residual gauge freedom in adding boundary terms to the action,the calculation of the free energy depends crucially on the chosen SL (2 , Z ) transformation.The variational principle is still well-defined, which allowed us in the current analysis toinvestigate the transport properties and collective phenomena of the anyon fluid in terms ofthe statistics, proportional to the quantization parameter n .There are several other open topics which deserve further investigation. The D p -branemetrics with p (cid:54) = 3 violate hyperscaling [42] with θ = − ( p − / (5 − p ). It would beworth to explore the relation between this scaling of the background and the one foundabove for the probe. Another interesting problem for the future would be the analysis ofmore general D p -D q intersections. Contrary to the supersymmetric cases studied here, themassive embeddings of a general D p -D q model are generically unstable and one must turn onfluxes on the worldvolume of the probe to stabilize them (see, for example [43–45]). Theseadditional worldvolume gauge fields give an important contribution to the Wess-Zuminoterm of the probe action. It would be very interesting to develop a general formalism forthe collective excitations of the probe brane which could incorporate all the particular cases An interesting alternative viewpoint without fluxes is discussed in [46, 47]. In this context too, however,one would need to take other Wess-Zumino terms into account (together with modifying the UV asymptotics)and our results are not directly applicable.
Acknowledgments
We thank Yago Bea and Carlos Hoyos for discussions andcritical readings of the manuscript. N.J. is supported by the Academy of Finland grantno. 1268023. A. V. R. and G. I. are funded by the Spanish grant FPA2014-52218-P,by the Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042), by Xunta de Galicia(GRC2013-024), and by FEDER. G. I. is also funded by FPA2012-35043-C02-02.
A Fluctuation equations of motion
In this appendix we obtain the Lagrangian density, and the corresponding equations ofmotion, for the fluctuations of the embedding scalar and the gauge fields at non-vanishingcharge density d (cid:54) = 0 and magnetic field B (cid:54) = 0. As was the case for the background equations,it is useful to treat the analysis for T = 0 and T (cid:54) = 0 using different parametrization. A.1 Fluctuations at zero temperature
In this subsection we focus on T = 0 case. Let us consider a fluctuation of the gauge fieldand embedding as in (3.1) and (3.2). The induced metric g takes the form: g = ¯ g + ˆ g , (A.1)38here ¯ g is the zeroth-order metric and ˆ g is the perturbation. Let us split ˆ g in the form:ˆ g = ˆ g (1) + ˆ g (2) . (A.2)The non-zero elements of ˆ g (1) are:ˆ g (1) ρx µ = z (cid:48) r − p ∂ µ ξ , ˆ g (1) ρρ = 2 z (cid:48) r − p ∂ ρ ξ , (A.3)whereas ˆ g (2) has the form: ˆ g (2) ab = 1 r − p ∂ a ξ ∂ b ξ . (A.4)(we are taking the radius R = 1 in (2.14)). In order to expand the DBI D q -brane action wenotice that the Born-Infeld determinant can be written as: (cid:112) − det( g + F ) = (cid:113) − det (cid:0) ¯ g + F (0) (cid:1) (cid:112) det(1 + X ) , (A.5)where the matrix X is given by: X ≡ (cid:0) ¯ g + F (0) (cid:1) − (cid:0) ˆ g + f (cid:1) . (A.6)To evaluate the right-hand side of eq. (A.5), we shall use the expansion: (cid:112) det(1 + X ) = 1 + 12 Tr X −
14 Tr X + 18 (cid:0) Tr X (cid:1) + O ( X ) . (A.7)Moreover, in the inverse matrix (cid:0) ¯ g + F (0) (cid:1) − we will separate the symmetric and antisym-metric parts: (cid:16) ¯ g + F (0) (cid:17) − = G − + J , (A.8)where J is the antisymmetric component and the symmetric matrix G is the open stringmetric. The relevant components of G are: G tt = − ¯ g rr (1 + z (cid:48) )¯ g rr | ¯ g tt | (1 + z (cid:48) ) − A (0) (cid:48) t , G x i x j = δ ij ¯ g xx G ρρ = − ¯ g tt ¯ g rr | ¯ g tt | (1 + z (cid:48) ) − A (0) (cid:48) t . (A.9)Using the fact that ¯ g rr | ¯ g tt | = 1, and eliminating z (cid:48) and A (0) (cid:48) t , we get: G tt = − ρ λ + d | ¯ g tt | ρ λ = − ρ λ + d ( ρ + z ) − p ρ λ G ρρ = ρ λ + d − c ¯ g rr ρ λ = ( ρ + z ) − p ρ λ + d − c ρ λ G x i x j = δ ij ( ρ + z ) − p , (A.10)39hich are just the components written in (3.3). The elements of the antisymmetric matrix J are: J tρ = −J ρt = − A (0) (cid:48) t ¯ g rr | ¯ g tt | (1 + z (cid:48) ) − A (0) (cid:48) t = − d (cid:112) ρ λ + d − c ρ λ . (A.11)By explicit calculation one can verify that Tr X is given by:Tr X = 2 z (cid:48) r − p G ρρ ∂ ρ ξ + 2 J tρ f ρt + G ab r − p ∂ a ξ ∂ b ξ , (A.12)while Tr X is: Tr X = −G ac G bd f cd f ab + G ac G bd ˆ g (1) ab ˆ g (1) cd + 2( J tρ ) (cid:104) (ˆ g (1) tρ ) + ( f tρ ) (cid:105) − J tρ G ab ˆ g (1) ρa f tb . (A.13)This last expression can be written more explicitly as:Tr X = −G ac G bd f cd f ab + 2 ( z (cid:48) ) r − p G ρρ G ab ∂ a ξ ∂ b ξ + 2 ( z (cid:48) ) r − p ( G ρρ ) ( ∂ ρ ξ ) (A.14)+2( J tρ ) (cid:104) ( z (cid:48) ) r − p ( ∂ t ξ ) + ( f tρ ) (cid:105) − z (cid:48) r − p J tρ G ab ∂ a ξ f tb − z (cid:48) r − p J tρ G ρρ ∂ ρ ξ f tρ . From these expressions we get that:12 Tr X −
14 Tr X + 18 (cid:0) Tr X (cid:1) = z (cid:48) r − p G ρρ ∂ ρ ξ + J tρ f ρt + 14 G ac G bd f cd f ab + G ab r − p (cid:104) − ( z (cid:48) ) G ρρ r − p (cid:105) ∂ a ξ ∂ b ξ − ( z (cid:48) ) r − p ( J tρ ) ( ∂ t ξ ) + z (cid:48) r − p J tρ G ab ∂ a ξ f tb . (A.15)Let us now obtain the Lagrangian density from these results. First of all, we can check thatthe first-order terms do not contribute to the equations of motion and, therefore, we justdrop them. Moreover, in the second-order terms we can substitute r by r ( ρ ), given by: r ( ρ ) = (cid:112) ρ + z ( ρ ) . (A.16)Taking into account the zeroth-order Lagrangian and that:1 − ( z (cid:48) ) G ρρ r − p = 1 − A (0) (cid:48) t z (cid:48) ) − A (0) (cid:48) t , (A.17)we get: L = −N ρ λ (cid:113) z (cid:48) ) − A (0) (cid:48) t (cid:34) G ac G bd f cd f ab + 12 r − p − A (0) (cid:48) t z (cid:48) ) − A (0) (cid:48) t G ab ∂ a ξ ∂ b ξ − ( z (cid:48) ) r − p ( J tρ ) ( ∂ t ξ ) + z (cid:48) r − p J tρ G ab ∂ a ξ f tb (cid:35) . (A.18)Substituting the values of z (cid:48) and A (0) (cid:48) t (written in (2.20)), the Lagrangian density for thefluctuations at zero temperature can be written as in (3.4).40 .2 Fluctuations at non-zero temperature In this subsection we focus on T (cid:54) = 0 and B (cid:54) = 0, by fluctuating the scalar and the gaugefields (7.33). First we compute the variation of the induced metric. By using the expansions dθ = ˙ θ dr + 2 ˙ θ ∂ a ζ drdx a + ∂ a ζ ∂ b ζ dx a dx b + · · · cos θ = cos θ − sin(2 θ ) ζ − cos(2 θ ) ζ + · · · , (A.19)where x a = ( x µ , r ) = ( t, x i , r ), we can represent the induced metric g in the form: g = ¯ g + ˆ g , (A.20)where ¯ g is the zeroth-order metric and ˆ g is the perturbation. We will expand ˆ g up to secondorder in the fluctuations. Accordingly, let us split ˆ g in the form:ˆ g = ˆ g (1) + ˆ g (2) , (A.21)where ˆ g (1) (ˆ g (2) ) are the first (second) order terms of ˆ g . The non-zero elements of ˆ g (1) are:ˆ g (1) rr = 2 r p − ˙ θ ˙ ζ , ˆ g (1) rx µ = r p − ˙ θ ∂ µ ζ , ˆ g (1) mn = − r p − sin(2 θ ) ζ γ mn , (A.22)whereas those of ˆ g (2) are:ˆ g (2) ab = r p − ∂ a ζ ∂ b ζ , ˆ g (2) mn = − r p − cos(2 θ ) ζ γ mn , (A.23)where m, n are indices along the internal ( q − n − γ mn is the metric of a unit S q − n − . Let us now define the open string metric G and the antisymmetric tensor J as in(A.8), with F (0) being the gauge field strength (7.10). The components of the inverse of theopen string metric in this case are: G tt = − ¯ g rr ( f − p + r ˙ θ ) | ¯ g tt | ¯ g rr (1 + r f p ˙ θ ) − ˙ A (0)2 t , G rr = | ¯ g tt | f p | ¯ g tt | ¯ g rr (1 + r f p ˙ θ ) − ˙ A (0)2 t , G x x = G x x = ¯ g xx ¯ g xx + B , G x i x j = δ ij ¯ g xx , ( i, j = 3 , , . . . ) , G mn = γ mn r ¯ g rr cos θ , (A.24)where A (0) is the gauge potential for the field strength F (0) . Using these explicit equationsfor the metric and eliminating ˙ A (0) t , we get: G tt = − r − p f p (cid:104) d H (cos θ ) λ (cid:105) , G rr = r − p f p r f p ˙ θ (cid:104) d H (cos θ ) λ (cid:105) , G x x = G x x = ¯ g xx ¯ g xx + B ≡ G x x , G x i x j = δ ij ¯ g xx , ( i, j = 3 , , . . . ) , G mn = γ mn r ¯ g rr cos θ . (A.25)41he only non-zero elements of the antisymmetric matrix J are: J tr = −J rt = − ˙ A (0) t | ¯ g tt | ¯ g rr (1 + r f p ˙ θ ) − ˙ A (0)2 t J x x = −J x x = − B ¯ g xx + B . (A.26)More explicitly: J tr = −J rt = − dH (cos θ ) λ (cid:112) H (cos θ ) λ + d (cid:113) r f p ˙ θ J x x = −J x x = − B ¯ g xx + B ≡ J xy . (A.27)We next define the matrix X as in (A.6) and we perform the expansion (A.7) of the DBIdeterminant. The traces of X needed are:Tr X = G MN ˆ g MN − J MN f MN , (A.28)andTr X = (cid:0) G MN G P Q − J MN J P Q (cid:1) (ˆ g MP ˆ g NQ − f MP f NQ ) − G MN J P Q ˆ g MP f NQ . (A.29)In these formulas the indices M , N , P , and Q run over all worldvolume directions (includingthe angular ones). The Lagrangian density for the fluctuations is given by: L = L (cid:104) X −
14 Tr X + 18 (cid:0) Tr X (cid:1) + O ( X ) (cid:105) , (A.30)where L is the zeroth-order Lagrangian density, given by: L = −N H (cos θ ) λ (cid:113) r f p ˙ θ (cid:112) d + H (cos θ ) λ . (A.31)Notice that the equation for the embedding θ ( r ) can be written as: ∂ r (cid:104) L r ¯ g rr G rr ˙ θ (cid:105) = − λ θ L . (A.32)Let us now consider the first-order contributions to L . They originate from the Tr X termin (A.30). Therefore: L (1) = L (cid:104) G MN ˆ g (1) MN − J MN f MN (cid:105) . (A.33)By using the values of the first-order metric written in (A.22), we get that the first term in(A.33) can be written as: L G MN ˆ g (1) MN = L (cid:104) r ¯ g rr G rr ˙ θ ˙ ζ − λ θ ζ (cid:105) . (A.34)42ntegrating by parts the first term in (A.34) and using (A.32) one can easily check that(A.34) reduces to a total derivative and, therefore, can be dropped from the Lagrangian.Moreover, the second term in (A.33) can be written as: − L J MN f MN = N d f tr + L B ¯ g xx + B f x x , (A.35)and clearly does not contribute to the equations of motion of the fluctuations. Let us nowconcentrate on the second-order terms in L . After some work, we get: L = L (cid:104) (cid:16) G ab G cd − J ab J cd + 12 J ac J bd (cid:17) f ac f bd + r ¯ g rr (cid:0) − r ¯ g rr G rr ˙ θ (cid:1) G ab ∂ a ζ ∂ b ζ − λ (cid:16) (cid:0) − λ (cid:1) tan θ (cid:17) ζ − λ r ¯ g rr G rr tan θ ˙ θ ζ ˙ ζ − r ¯ g rr (cid:0) J tr (cid:1) ˙ θ ( ∂ t ζ ) + λ θ J ab ζ f ab + r ¯ g rr ˙ θ (cid:16) J tr G ab ∂ a ζf tb + J ab G rr ∂ a ζf rb − J ab G rr ∂ r ζf ab (cid:17)(cid:105) . (A.36)Let us integrate by parts the ζ ˙ ζ term on the second line of (A.36). In this process we generatethe following contribution to L : λ ∂ r (cid:2) L r ¯ g rr G rr tan θ ˙ θ (cid:3) ζ = − λ L (cid:0) tan θ (cid:1) ζ + λ L r ¯ g rr G rr ˙ θ cos θ ζ , (A.37)where we have used the embedding equation (A.32). Plugging this result into (A.36) we getthe final form of the Lagrangian for the fluctuations, which is given by: L = L (cid:104) (cid:16) G ab G cd − J ab J cd + 12 J ac J bd (cid:17) f ac f bd (A.38)+ (cid:16) − r ¯ g rr G rr ˙ θ (cid:17)(cid:16) r ¯ g rr G ab ∂ a ζ ∂ b ζ − λ θ ζ (cid:17) − r ¯ g rr (cid:0) J tr (cid:1) ˙ θ ( ∂ t ζ ) + λ θ J ab ζ f ab + r ¯ g rr ˙ θ (cid:16) J tr G ab ∂ a ζf tb + J ab G rr ∂ a ζf rb − J ab G rr ∂ r ζf ab (cid:17)(cid:105) . Let us now work out the equations of motion derived from this Lagrangian density. We willassume that all fields only depend on t , r and one of the Cartesian coordinates (say x ). Firstof all, we write the equation of a r in the a r = 0 gauge. We get the following Gauss’ law: G tt ∂ t ˙ a t + G xx ∂ i ˙ a i = r ¯ g rr J tr ˙ θ ∂ t ˙ ζ + ¯ λ J tr G rr tan θ ∂ t ζ . (A.39)The equation for a t becomes: ∂ r (cid:104) L G rr (cid:16) G tt ˙ a t − r ¯ g rr J tr ˙ θ ˙ ζ (cid:17) − L J tr (cid:16) λ θ ζ + J xy ∂ x a y (cid:17)(cid:105) + L G x x (cid:2) G tt ∂ x f xt − r ¯ g rr J tr ˙ θ ∂ x ζ (cid:3) + L J tr J xy ∂ x ˙ a y = 0 . (A.40)The equation of a x is: ∂ r (cid:104) L (cid:16) G rr G xx ˙ a x + J tr J xy ∂ t a y (cid:17)(cid:105) + L G tt G xx ∂ t f tx + L G xx r ¯ g rr J tr ˙ θ ∂ t ∂ x ζ − L J tr J xy ∂ t ˙ a y = 0 . (A.41)43aking into account that L J tr = constant, this last equation can be rewritten as: ∂ r (cid:2) L G rr G xx ˙ a x (cid:3) + L G tt G xx ∂ t f tx + L G xx r ¯ g rr J tr ˙ θ ∂ t ∂ x ζ = −L J tr ∂ r (cid:0) J xy (cid:1) ∂ t a y . (A.42)Moreover, after some simplifications, the equation of motion of a y can be written as: ∂ r (cid:2) L G rr G xx f ry (cid:3) + L G xx (cid:0) G tt ∂ t f ty + G xx ∂ x f xy (cid:1) = L ∂ r (cid:0) J xy (cid:1) (cid:16) J tr f tx − r ¯ g rr G rr ˙ θ ∂ x ζ (cid:17) . (A.43)Finally, let us write the equation of motion of the scalar fluctuations. We get: ∂ r (cid:104) L r ¯ g rr G rr (cid:16)(cid:0) − r ¯ g rr G rr ˙ θ (cid:1) ˙ ζ − J tr ˙ θ ˙ a t (cid:17)(cid:105) + λ θ L (cid:0) − r ¯ g rr G rr ˙ θ (cid:1) ζ + λ θ L J tr ˙ a t + L r ¯ g rr (cid:0) − r ¯ g rr G rr ˙ θ (cid:1)(cid:0) G tt ∂ t ζ + G xx ∂ x ζ (cid:1) −L r ¯ g rr ( J tr ) ˙ θ ∂ t ζ + L r ¯ g rr J tr G xx ˙ θ ∂ x f tx = L ∂ r (cid:0) J xy (cid:1) r ¯ g rr ˙ θ G rr f xy . (A.44)Let us next Fourier transform the gauge field and the scalar to momentum space as in (7.34)and let us define the electric field E as the gauge-invariant combination: E = k a t + ω a x . (A.45)In momentum space the Gauss law (A.39) becomes: ω G tt ˙ a t − k G xx ˙ a x = ω r ¯ g rr J tr ˙ θ ˙ ζ + ¯ λ ω J tr G rr tan θ ζ . (A.46)We can combine (A.46) and (A.45) to get ˙ a t and ˙ a x in terms of the gauge-invariant combi-nation E and the scalar field ζ :˙ a t = G xx k ˙ E + ω r ¯ g rr J tr ˙ θ ˙ ζ + ω λ J tr G rr tan θ ζ G tt ω + G xx k ˙ a x = G tt ω ˙ E − k ω r ¯ g rr J tr ˙ θ ˙ ζ − k ω ¯ λ J tr G rr tan θ ζ G tt ω + G xx k . (A.47)Moreover, using (A.47) one can demonstrate that (A.40) and (A.42) are equivalent to thefollowing equation for the electric field E : ∂ r (cid:34) L G rr G xx G tt ω + G xx k (cid:16) G tt ˙ E − k r ¯ g rr J tr ˙ θ ˙ ζ − k λ J tr G rr tan θ ζ (cid:17)(cid:35) −L G tt G xx E + k L G xx r ¯ g rr J tr ˙ θ ζ = i L J tr ∂ r ( J xy ) a y , (A.48)where G xx has been defined in (A.25). Similarly, we can work out the equation for the scalar ζ in terms of E . In momentum space this equation becomes: ∂ r (cid:104) L r ¯ g rr G rr (cid:16)(cid:0) − r ¯ g rr G rr ˙ θ (cid:1) ˙ ζ − J tr ˙ θ ˙ a t (cid:17)(cid:105) + λ θ L (cid:0) − r ¯ g rr G rr ˙ θ (cid:1) ζ + λ θ L J tr ˙ a t − L r ¯ g rr (cid:0) − r ¯ g rr G rr ˙ θ (cid:1)(cid:0) G tt ω + G xx k (cid:1) ζ + L r ¯ g rr ( J tr ) ˙ θ ω ζ + L r ¯ g rr J tr G xx ˙ θ kE = ik L ˙ θ r ¯ g rr G rr ∂ r ( J xy ) a y , (A.49)44here it should be understood that ˙ a t is given by the first equation in (A.47). Finally, theequation of motion of the transverse fluctuation a y is: ∂ r (cid:104) L G rr G xx ˙ a y (cid:105) − L G xx (cid:0) G tt ω + G xx k (cid:1) a y = − i L J tr ∂ r ( J xy ) E − ik L ˙ θ r ¯ g rr G rr ∂ r ( J xy ) ζ . (A.50) B Transverse correlators and the conductivity
Let us consider the case in which the magnetic field vanishes, B = 0. In this case, theequation of motion (A.50) for the transverse fluctuation a y is: ∂ r (cid:104) L G rr G xx ˙ a y (cid:105) − L G xx (cid:0) G tt ω + G xx k (cid:1) a y = 0 . (B.1)This equation can be rewritten as:¨ a y + ∂ r log (cid:104) L G rr G xx (cid:105) ˙ a y − G tt ω + G xx k G rr a y = 0 . (B.2)More explicitly, the equation of motion for a y is:¨ a y + ∂ r log (cid:34) (cid:113) d + r λ (cid:0) cos θ (cid:1) λ (cid:113) r f p ˙ θ f p (cid:35) ˙ a y + 1 + r f p ˙ θ r − p f p ( ω − f p k ) r λ (cos θ ) λ + ω d d + r λ (cid:0) cos θ (cid:1) λ a y = 0 . (B.3)We now study the equation of motion (B.3) for a y in the low frequency regime in which k ∼ O ( (cid:15) ) and ω ∼ O ( (cid:15) ). Let us first study (B.3) near the horizon r = r h . With thispurpose we expand θ ( r ) near r = r h : θ ( r ) ≈ θ h − λ − p ) r λ − h (cid:0) cos θ h (cid:1) λ tan θ h d + r λh (cid:0) cos θ h (cid:1) λ ( r − r h ) + · · · . (B.4)We also expand the coefficients of the equation of the transverse fluctuations: ∂ r log (cid:34) (cid:113) d + r λ (cid:0) cos θ (cid:1) λ (cid:113) r f p ˙ θ f p (cid:35) = 1 r − r h + d + · · · r f p ˙ θ r − p f p ( ω − f p k ) r λ (cos θ ) λ + ω d d + r λ (cid:0) cos θ (cid:1) λ = A ( r − r h ) + c r − r h + · · · , (B.5)45here A , d , and c are given by: A = ω (7 − p ) r − ph d = 12 r h ( p − d + ( p + λ − r λh (cid:0) cos θ h (cid:1) λ d + r λh (cid:0) cos θ h (cid:1) λ + λ − p ) r λ − h (cid:0) cos θ h (cid:1) λ (cid:2) d + r λh (cid:0) cos θ h (cid:1) λ (cid:3) tan θ h c = − − p r p + λ − h (cid:0) cos θ h (cid:1) λ d + r λh (cid:0) cos θ h (cid:1) λ k + 1(7 − p ) r − ph ω + λ − p ) r p +2 λ − h (cid:0) cos θ h (cid:1) λ tan θ h (cid:2) d + r λh (cid:0) cos θ h (cid:1) λ (cid:3) ω . (B.6)Let us now solve for a y in Frobenius series around r = r h : a y ( r ) = ( r − r h ) α (1 + β ( r − r h ) + . . . ) , (B.7)where the exponents α and β , at order (cid:15) , are given by: α = − iω (7 − p ) r − p h , β ≈ − ( α d + c ) . (B.8)From the expressions of d and c written in (B.6) we find that β is given by: β = i (cid:34) − p ) r − p h ( p − d + ( p + λ − r λh (cid:0) cos θ h (cid:1) λ d + r λh (cid:0) cos θ h (cid:1) λ + λ − p ) r − p h r λ − h (cid:0) cos θ h (cid:1) λ (cid:2) d + r λh (cid:0) cos θ h (cid:1) λ (cid:3) tan θ h (cid:35) ω + 17 − p r p + λ − h (cid:0) cos θ h (cid:1) λ d + r λh (cid:0) cos θ h (cid:1) λ k . (B.9)Let us now take the near-horizon and low frequency limits in opposite order. First, we write(B.3) as: ¨ a y + ˙ GG ˙ a y + Q a y = 0 , (B.10)where G ( r ) is given by: G ( r ) = (cid:113) d + r λ (cid:0) cos θ (cid:1) λ (cid:113) r f p ˙ θ f p . (B.11)Moreover, the expression of Q ( r ) at order (cid:15) is: Q ( r ) ≈ − r f p ˙ θ f p r λ + p − (cid:0) cos θ (cid:1) λ d + r λ (cid:0) cos θ (cid:1) λ k . (B.12)46et us redefine a y ( r ) as: a y ( r ) = F ( r ) α y ( r ) , (B.13)where α y ( r ) should be regular at r = r h and F ( r ) is given by: F ( r ) = ( r − r h ) α . (B.14)The resulting equation for α y is:¨ α y + (cid:16) ˙ GG + 2 ˙ FF (cid:15) (cid:17) ˙ α y + (cid:15) ( P + Q ) α y = 0 , (B.15)where we have explicitly introduced the powers of (cid:15) to keep track of the low frequencyexpansion and we have defined the new function P ( ρ ) as: P ( r ) ≡ ¨ FF + ˙ GG ˙ FF . (B.16)We will solve (B.15) order by order in a series expansion in (cid:15) of the form: α y = α + (cid:15) α + . . . . (B.17)As in the massless case, ˙ α = 0 if we impose regularity at r = r h . Furthermore, without lossof generality we can take: α = 1 . (B.18)The equation for α is ¨ α + ˙ GG ˙ α = − P − Q . (B.19)This equation can be solved by variation of constants. We put:˙ α ( r ) = A ( r ) G ( r ) , (B.20)where A ( r ) is a function to be determined. By direct substitution into (B.19) we get that A ( r ) must satisfy: ˙ A = − G ( P + Q ) . (B.21)The solution of this equation for A at leading order in (cid:15) is: A ( r ) = − G ˙ FF − c − (cid:90) rr h G (¯ r ) Q (¯ r ) d ¯ r , (B.22)where c is a constant to be determined. Let us next define the integral I ( r ) as: k I ( r ) ≡ − (cid:90) rr h G (¯ r ) Q (¯ r ) d ¯ r , (B.23)or, more explicitly: I ( r ) = (cid:90) rr h (cid:113) r f p (¯ r ) ˙ θ (¯ r ) (cid:113) d + ¯ r λ (cid:0) cos θ (¯ r ) (cid:1) λ ¯ r λ + p − (cos θ (¯ r )) λ d ¯ r . (B.24)47herefore, ˙ α can be written as:˙ α = − (cid:34) c (cid:113) r f p ˙ θ (cid:113) d + r λ (cid:0) cos θ (cid:1) λ f p + αr − r h (cid:35) + I ( r ) k G ( r ) . (B.25)The constant c is determined by requiring that ˙ α be regular at r = r h . We get: c = i (cid:112) r λh (cos θ h ) λ + d r − p h ω . (B.26)As ˙ α y = ˙ α + O ( (cid:15) ), we get:˙ α y = − i (7 − p ) r − p h (cid:34) − pr h (cid:113) r f p ˙ θ (cid:112) r λh (cos θ h ) λ + d (cid:112) r λ (cos θ ) λ + d f p − r − r h (cid:35) ω + I ( r ) k G ( r ) . (B.27)This solution should match (B.7). One can check that this is indeed the case since ˙ α y ( r = r h ) = β . Moreover:˙ a y = ˙ α + αr − r h + O ( (cid:15) ) = ˙ α y + αr − r h + O ( (cid:15) ) . (B.28)Thus, we can write: ˙ a y = − G ( r ) (cid:104) i (cid:112) r λh (cos θ h ) λ + d r − p h ω − I ( r ) k (cid:105) . (B.29)In order to obtain the (cid:104) J y J y (cid:105) correlator from these results, let us point out that the termdepending on a y of the Lagrangian density is of the form: L ( a y ) = F ( f y r ) = F ( ˙ a y ) , (B.30)where F is given by: F = −N r λ (cid:0) cos θ (cid:1) λ √ ∆ G yy G rr = −N G . (B.31)Then, the (cid:104) J y J y (cid:105) correlator takes the form: (cid:104) J y ( p ) J y ( − p ) (cid:105) = N (cid:104) Γ ω iω + Γ k k (cid:105) , (B.32)where the coefficients Γ ω and Γ k are:Γ ω = (cid:112) r λh (cos θ h ) λ + d r − p h Γ k = − (cid:90) ∞ r h d ¯ r (cid:113) r f p (¯ r ) ˙ θ (¯ r ) (cid:113) d + ¯ r λ (cid:0) cos θ (¯ r ) (cid:1) λ ¯ r λ + p − (cid:0) cos θ (¯ r ) (cid:1) λ . (B.33)48otice that the DC conductivity σ is given by σ = N Γ ω . Therefore: σ = N (cid:112) r λh (cos θ h ) λ + d r − p h . (B.34)In terms of ˆ d = d/r λ h , the conductivity can be written as: σ = N r p + λ − h (cid:113) (cos θ h ) λ + ˆ d . (B.35) C Conductivity by the Karch-O’Bannon method
In this appendix we evaluate the conductivity of the probe by using the method developedin [41]. Let us work in the ( r, θ ) variables of section 7 and consider a D q -brane probe withthe following worldvolume gauge field: A = A t dt + ( Et + a x ( r )) dx + ( Bx + a y ( r )) dy , (C.1)where x and y are two spatial Minkowski directions along the brane. The field strengthcorresponding to (C.1) is: F = ˙ A t dr ∧ dt + Bdx ∧ dy + E dt ∧ dx + ˙ a x dr ∧ dx + ˙ a y dr ∧ dy . (C.2)Notice that, as in the main text, the field A t is dual to the charge density, whereas a x and a y are dual to the components of the current along the directions x and y , respectively. Noticealso that we have switched on an electric field E in the x direction and a magnetic field B across the xy plane. The DBI Lagrangian density for this configuration is given by: L = −N (cos θ ) λ √ Σ , (C.3)where N is the normalization constant (2.12) and Σ is defined as:Σ = (1 + r f p ˙ θ ) ( H − r λ + p − f − p E ) − r λ ˙ A t + r λ f p ( ˙ a x + ˙ a y ) − r λ + p − ( E ˙ a y + B ˙ A t ) , (C.4)with H = H ( r ) being the function introduced in (7.12). It follows from (C.3) and (C.4)that A t , a x , and a y are cyclic variables. Let d , j x , and j y be the corresponding conservedcanonical momenta. They are given by: d = (cos θ ) λ H ˙ A t + r λ + p − E B ˙ a y √ Σ − j x = (cos θ ) λ r λ f p ˙ a x √ Σ − j y = (cos θ ) λ ( r λ f p − r λ + p − E ) ˙ a y − r λ + p − EB ˙ A t √ Σ . (C.5)49otice that we have absorbed the normalization constant N in the definitions (C.5). Let usnow solve for ˙ A t , ˙ a x , and ˙ a y . First of all, we define the quantity X as: X ≡ (cid:104) d f p + r λ f p (cos θ ) λ − j x − j y (cid:105) (cid:2) f p ( r − p + B ) − E ) (cid:3) − (cid:0) d B f p − E j y ) . (C.6)Then, after some algebra, one can verify that:˙ A t = (cid:113) r f p ˙ θ r − p √ X (cid:104)(cid:0) E − r − p f p ) d − E B j y (cid:105) ˙ a x = (cid:113) r f p ˙ θ r − p f p √ X (cid:104) E − f p ( r − p + B ) (cid:105) j x ˙ a y = − (cid:113) r f p ˙ θ r − p √ X (cid:104) E B d − ( r − p + B ) j y (cid:105) . (C.7)Following closely the arguments in [41], let us determine the position r = r ∗ of the pseudo-horizon by imposing the three conditions at r = r ∗ : f ∗ p ( r − p ∗ + B ) = E j x + j y = f ∗ p (cid:2) r λ ∗ (cos θ ∗ ) λ + d (cid:3) E j y = B f ∗ p d , (C.8)where we have denoted θ ∗ ≡ θ ( r = r ∗ ) and f ∗ p ≡ f p ( r = r ∗ ). From the first equation in (C.8)we can determine r ∗ in terms of r h , E , and B . Indeed, we have: r − p ∗ = 12 (cid:104) E − B + r − ph + (cid:113) ( E − B + r − ph ) + 4 r − ph B (cid:105) . (C.9)Notice that r ∗ = r h if the electric field E vanishes. From now on we will assume that E issmall. Then, it follows from (C.9) that: r ∗ = r h + 17 − p r h r − ph + B E + O ( E ) , f ∗ p = E r − ph + B + O ( E ) . (C.10)We can use these expressions in the last two equations in (C.8) to get j x and j y . At leadingorder in E , we get: j x = r − p h (cid:113) r λh (1 + r p − h B ) (cos θ h ) λ + d r − ph + B Ej y = B dr − ph + B E . (C.11)Therefore, the longitudinal and transverse conductivities are given by: σ xx = N j x E = N r − p h (cid:113) r λh (1 + r p − h B ) (cos θ h ) λ + d r − ph + B σ xy = N j y E = N B dr − ph + B , (C.12)50here we have reintroduced the normalization factor N . Notice that σ xx in (C.12) coincideswith the value written in (7.40). The same value of σ xx can be found from the analysis ofthe transverse fluctuations (as the one in appendix B for B = 0) if we neglect the couplingbetween the fluctuation equations. References [1] For reviews see: J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal andU. A. 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