Collective excitations of massive flavor branes

Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B 909 (2016) 677–724 www.elsevier.com/locate/nuclphysb Collective excitations of massive flavor branes Georgios Itsios a,d,e, Niko Jokela b,c,∗, Alfonso V. Ramallo d,e aDepartment of Physics, University of Oviedo, Avda. Calvo Sotelo 18, 33007 Oviedo, Spain bDepartment of Physics, FIN-00014 University of Helsinki, Finland cHelsinki Institute of Physics, P.O. Box 64, FIN-00014 University of Helsinki, Finland dDepartamento de Física de Partículas, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain eInstituto Galego de Física de Altas Enerxías (IGFAE), E-15782 Santiago de Compostela, Spain Received 25 February 2016; received in revised form 29 May 2016; accepted 6 June 2016 Available online 9 June 2016 Editor: Leonardo Rastelli Abstract We study the intersections of two sets of D-branes of different dimensionalities. This configuration is dual to a supersymmetric gauge theory with flavor hypermultiplets in the fundamental representation of the gauge group which live on the defect of the unflavored 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 adds flavor to the system. We work in the quenched approximation, i.e., where the flavor branes are considered as probes, and focus specifically on the case in which the quarks are massive. We study the thermodynamics and the speeds of first and zero sound at zero temperature and non-vanishing chemical potential. We show that the system undergoes a quantum phase transition when the chemical potential approaches its minimal value and we obtain the corresponding non-relativistic critical exponents that characterize its critical behavior. In the case of (2 +1)-dimensional intersections, we further study alternative quantization and the zero sound of the resulting anyonic fluid. We finally extend these results to non-zero temperature and magnetic field and compute the diffusion constant in the hydrodynamic regime. The numerical results we find match the predictions by the Einstein relation. ©2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. *Corresponding author. E-mail addresses: [email protected] (G. Itsios), [email protected] (N. Jokela), [email protected] (A.V. Ramallo). http://dx.doi.org/10.1016/j.nuclphysb.2016.06.008 0550-3213/©2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 678 G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 1. Introduction There is hope that the gauge/gravity holographic duality could serve to characterize new types of compressible states of matter, i.e., states with non-zero charge density which vary continuously with the chemical potential. Indeed, holography provides gravitational descriptions of strongly interacting systems without long-lived quasiparticles, situations which cannot be accommodated within the standard Landau’s Fermi liquid theory. Although the field theories with known holographic dual are very different from those found so far in Nature, there are good reasons to believe that these studies could reveal generic universal features of strongly interacting quantum systems [1]. In this paper we approach this problem in a top-down model of intersecting branes of different dimensionalities. We will consider a stack of Nccolor Dp-branes which intersect Nfflavor Dq-branes (q≥p) along ncommon directions. This configuration, which we will denote by (n | p⊥q), is dual to a (p +1)-dimensional SU(Nc)gauge theory with Nffundamental hypermultiplets (quarks) living on a (n +1)-dimensional defect [2]. In the context of holography, we will work in the large Nc’t Hooft limit with NfNc. In this limit the quarks are quenched and the Dq-branes can be treated as probes, whose action is the Dirac–Born–Infeld (DBI) action, in the gravitational background created by the Dp-branes. The embedding of the flavor branes is parameterized by a function which measures the distance between the two types of branes. The field theory dual of this distance is the mass of the hypermultiplet. Moreover, in order to engineer a system with non-zero baryonic charge density, we must switch on a suitable gauge field on the worldvolume of the flavor brane [3]. We will also study the influence of a magnetic field directed along two of the spatial directions of the worldvolume. In [4] we studied the collective excitations of generic brane intersections corresponding to massless quarks and we uncovered a certain universal structure. The purpose of this article is to extend the results of [4] to the case in which the quarks have a non-zero mass. We will study first the system at zero temperature and non-zero chemical potential. This is the so-called collisionless 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]. These results were generalized to non-zero temperature in [7,8] and to non-vanishing magnetic field in [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 speed of zero sound for massless quarks. This is intimately related with the fact that λdetermines the scaling dimension of the charge density or to put it slightly differently, λacts as the polytropic index in the equation of state for the holographic matter. In the case of massive quarks the embedding of the Dq-brane is non-trivial and must be determined in order to extract the different physical properties. When the charge density is nonvanishing, the brane reaches the horizon of the geometry, i.e., we have a black hole embedding. This embedding depends on a function which parameterizes the shape of the flavor brane in the background geometry and, in general, must be found by numerical integration of the equations of motion of the probe. However, in the case of intersections (n | p⊥q) which preserve some amount of supersymmetry at zero temperature Tand chemical potential μsome remarkable simplification occurs. Indeed, as shown in [29], in these intersections one can choose a system of coordinates such that the embedding function is a cyclic variable of the DBI Lagrangian when T=0 and μ =0. As a consequence, the embedding function and the physical properties of the configuration, can be found analytically. In particular, one can study the zero temperature ther- G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 679 modynamics of these systems and find the speed of first sound. This was done in refs. [12,13] for the D3–Dqintersections (3 | 3 ⊥7), (2 | 3 ⊥5), and (1 | 3 ⊥3). Moreover, by studying the quasinormal fluctuation modes of the probe, one can also compute analytically the speed of zero sound which, non-trivially, equals that of the first sound [8,11,30]. In this paper we generalize these results for any (n | p⊥q) intersection with #ND =4, i.e., when n =(p +q−4)/2. These cases correspond to those brane intersections which are supersymmetric in flat space at low energies as the gravitational and Ramond–Ramond forces cancel out. Here the index λcan only take three different values λ =2, 4, 6, corresponding to codimension 2 (Dp–Dp), codimension 1 (Dp–D(p +2)), and codimension 0 (Dp–D(p +4)) intersections, respectively. As in the conformal D3-background, the speeds of first and zero sounds coincide. Moreover, we find the same kind of universality as in the massless case: the speed is the same for those intersections which have the same λindex, or codimension. However, in the massive case the speed of sound depends continuously on the chemical potential, i.e., on the charge density, and vanishes when the chemical potential reaches its minimal value, which corresponds to a vanishing charge density d. Actually, as argued in [30] for the D3–D7 and D3–D5 intersections, there is a quantum phase transition as d→0 which exhibits a non-relativistic scaling behavior with hyperscaling violation. At the transition point the black hole embeddings with d=0degenerate into a Minkowski embedding with zero charge density. Here we will find the critical exponents for the general #ND =4 intersections, generalizing the results of [30]. When the number nof common dimensions of the color and flavor branes is equal to two, the matter hypermultiplets live on a (2 +1)-dimensional theory. In this case one can perform an alternative quantization of the quasinormal modes, which consists in imposing mixed Dirichlet– Neumann boundary conditions at the UV. As shown in [31], this alternative quantization amounts to transforming the charged excitations into particles of fractional statistics, i.e., anyons (see also [28,32–34] for the analysis of different aspects of the holographic anyonic systems). In [4] we studied the zero sound mode as a function of the constant that measures the degree of mixing the UV boundary conditions. We found that the anyonic zero sound is generically gapped and that this gap can be fine-tuned to zero if a suitable magnetic field is switched on. This choice corresponds to the case, where the anyons experience no effective magnetic field. In this paper we generalize these results to the case in which the quarks are massive. In this article we also study the hydrodynamic regime that is reached when the temperature 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. When the temperature is non-zero the embedding function is no more a cyclic coordinate of the DBI action and cannot therefore be found analytically. Thus, we study this T= 0 case by 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, by taking the T→0limit. We also study numerically the system in the presence of a magnetic field B. We compare the results for the diffusion constant obtained from the fluctuation analysis at T=0 with the ones predicted by the Einstein relation, which gives Din terms of the DC conductivity σand the charge susceptibility χ. Both σand χcan be obtained from the embedding function. We find a very good agreement between the numerical results for Dand the value given by the Einstein relation. The rest of this paper is organized as follows. In section 2we formulate our top-down holographic model, solve the equations of motion of the probe at T=0 and μ =0, and study the thermodynamics at zero temperature. In particular, in this section we find the speed of first sound and compute the charge susceptibility at T=0. In section 3we write the equations of motion for the fluctuations of the probe at zero temperature. In section 4we analyze the zero sound and find 680 G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 analytically the dispersion relation of this collective mode. Section 5is devoted to the study of the scaling behavior near the quantum critical point. In section 6we study the zero sound mode in an anyonic fluid. Section 7contains our results at non-zero temperature and magnetic field. We summarize our results and discuss some possible future research directions in section 8. We complement and give further details of our analysis in several appendices. Appendix A.1 contains a detailed derivation of the Lagrangian of the fluctuations at zero temperature which is used in section 3. In Appendix A.2 we work out the equations of motion of the fluctuations at T=0. In Appendix B we find the correlator of two transverse currents and extract the DC conductivity in the absence of magnetic field. Finally, in Appendix C we provide an alternative derivation of the conductivity, valid also when B=0. 2. Massive Dp–Dqsystems with charge Let us begin our analysis by introducing our setup and studying its properties at zero temperature and magnetic field. We will consider a generic Dp-brane metric at zero temperature of the type: ds2 10 =gtt(r) dt2+gxx(r) (dx1)2+···+(dxp)2+grr(r) d y·dy, (2.1) where y=(y1, ..., y9−p)are the coordinates transverse to the Dp-brane and the functions gtt, gxx, and grr depend on the transverse radial direction r=y·y. We now embed NfDq-brane probes, with NfNc, extended along the directions (t, x1,...,xn,y1,...,yq−n). (2.2) We will refer to this configuration as a (n | p⊥q) intersection (nis the number of common spatial directions of the Dpand Dq). This intersection is represented by the array: x1··· xnxn+1··· xpy1··· yq−nyq−n+1··· y9−p Dp : × ··· × × ··· × − ··· − − ··· − Dq : × ··· × − ··· − × ··· × − ··· − We shall denote by zthe coordinates ytransverse to the Dq-brane: z=(z1,...,z9+n−p−q), (2.3) with zm=yq−n+mfor m =1, ..., 9 +n −p−q. Moreover, we define ρas the radial coordinate for the subspace spanned by (y1, ..., yq−n): ρ2=(y1)2+···+(yq−n)2.(2.4) Let us make a short comment on the global symmetries. The original Dp-background has a rotational symmetry in the yidirections, this corresponds to the SO(9 −p) R-symmetry. When we add Nfcoincident probe Dq-branes we introduce U(Nf)flavor symmetry. The Dp–Dq-intersection (n | p⊥q) breaks the original R-symmetry, which can be easily read off from the isometries. We end up with the global symmetry SO(n, 1) ×U(Nf) ×SO(p −n)p× SO(q −n)q×SO(9 +n −p−q). The last group will be further broken when we consider massive Dq-brane embeddings. Since, dy2=dρ2+ρ2d2 q−n−1+dz2,(2.5) G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 681 the background metric in these coordinates can be written as: ds2 10 =gtt(r) dt2+gxx(r) (dx1)2+···+(dxn)2+(dxn+1)2+···+(dxp)2 +grr(r) dρ2+ρ2d2 q−n−1+dz2.(2.6) Let us consider a stack of Dq-branes with a non-trivial profile in the transverse space. We will choose our transverse coordinates in such a way that this profile can be parameterized as z= (z1(ρ), 0, ..., 0). In what follows we just write z(ρ) instead of z1(ρ) and we will denote by r=r(ρ) the function: r(ρ) =ρ2+z(ρ)2.(2.7) The induced metric on the Dq-brane worldvolume at zero temperature is: ds2 q+1=gtt(ρ) dt2+gxx(ρ) (dx1)2+···+(dxn)2 +grr(ρ)(1+z2)dρ2+ρ2d2 q−n−1,(2.8) with z=dz/dρ. Let us compute the DBI action of the Dq-brane in the case in which there is a worldvolume gauge field Fwith components ρt. Thus, we will take Fto be given by: F=A tdρ ∧dt , (2.9) where A t=∂ρAtand we have chosen a gauge for Asuch that Aρ=0. This means that we aim to study holographic matter at non-zero baryon charge density by introducing a chemical potential for the diagonal U(1) ⊂U(Nf). The DBI action becomes: SDq =−NfTDq dq+1ξe −φ−det(g +2παF)=dt dnxdρLDBI ,(2.10) with the Lagrangian density LDBI given by: LDBI =−Ne−φρq−n−1g n 2 xx g q−n−1 2 rr grr|gtt|(1+z2)−(2πα)2A2 t,(2.11) where Nis the normalization factor N=NfTDq Vol(Sq−n−1), (2.12) and where the tension of the Dq-brane and the volume of the unit sphere are TDq =1 (2π)q√αq+1gs ,Vol (Sq−n−1)=2πq−n 2 q−n 2.(2.13) For a Dp-brane background at zero temperature, the metric and the dilaton are given by: −gtt =gxx =r R7−p 2,g rr =R r7−p 2,e −2φ=R r(7−p)(p−3) 2.(2.14) This background satisfies grr|gtt| =1 and the Lagrangian density LDBI can be written as: LDBI =−Nρq−n−1r R(2n−p−q+4)(7−p) 41+z2−(2πα)2A2 t.(2.15) In the following we will scale out the constant R, i.e., we will take directly R=1. To avoid clutter, we also redefine the gauge field by absorbing the factors of the string length 682 G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 2παAμ→Aμ. Moreover, we will restrict ourselves to the case in which the embedding function z(ρ) is a cyclic variable, i.e., when LDBI depends on zand not on z. The only dependence on zin (2.15) is the one induced by the power of rmultiplying the DBI square root. Therefore z(ρ) is cyclic only when the following condition between n, p, and qis satisfied: n=p+q−4 2.(2.16) One can check that this happens only in the supersymmetric intersections with #ND =4: (p | p⊥ p+4), (p −1 | p⊥p+2), and (p −2 | p⊥p). In the following we will restrict ourselves to these cases. Let us define λas: λ=2(q −n−1)=q−p+2.(2.17) Notice that λ =6, 4, 2for the intersections Dp–D(p +4), Dp–D(p +2), and Dp–Dp, respectively. We can then write the Lagrangian density as: LDBI =−Nρλ 21+z2−A2 t.(2.18) The cyclic nature of zand Atimplies the following conservation laws: 1 N ∂LDBI ∂z=− ρλ 2z 1+z2−A2 t≡−c 1 N ∂LDBI ∂A t=ρλ 2A t 1+z2−A2 t≡d, (2.19) with cand dbeing constants of integration. These relations can be inverted as: z=c ρλ+d2−c2,A  t=d ρλ+d2−c2.(2.20) When c=d=0, both z(ρ) and At(ρ) are constant and we have a Minkowski embedding. Let us suppose that cdoes not vanish. Then, it follows from (2.20) that A tand zare related as: A t=d cz.(2.21) When c2=d2=0 both z(ρ) and At(ρ) diverge at ρ=0. Therefore, we discard this configuration and we will assume in the following that d2>c 2. In this case, from the expression of zand A twritten in (2.20) it is easy to conclude that the point ρ=0is reached. In what follows we will assume that this condition holds. We will integrate the equation for At(ρ) by imposing that At(0) =0. We have: At(ρ) =d ρ  0 d¯ρ ¯ρλ+d2−c2.(2.22) This integral can be computed analytically and expressed in terms of the hypergeometric function as: At(ρ) =d d2−c21 2−1 λ ρ ρλ+d2−c21 λ F1 λ,1 2+1 λ;1+1 λ;ρλ ρλ+d2−c2.(2.23) G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 683 Similarly, the embedding function z(ρ) can be written as: z(ρ) =c d2−c21 2−1 λ ρ ρλ+d2−c21 λ F1 λ,1 2+1 λ;1+1 λ;ρλ ρλ+d2−c2.(2.24) Notice that when d2>c 2the brane reaches the Poincaré horizon of the metric at ρ=z=0 and we have a black hole embedding. The two constants dand care related to the charge density and condensate of the dual theory, respectively. 2.1. Zero temperature thermodynamics Let us first consider the intersections with λ >2. We also restrict to T=0, as at non-zero temperature not much can be said analytically. In this case the functions At(ρ) and z(ρ) in (2.23) and (2.24) approach a constant value in the UV region ρ→∞. According to the standard AdS/CFT dictionary, the flavor chemical potential μis the UV value of At: μ=At(ρ →∞)=d d2−c21 2−1 λ F1 λ,1 2+1 λ;1+1 λ;1=d d2−c21 2−1 λ γ, (2.25) where γis the constant γ=1 √π1 2−1 λ1+1 λ(2.26) and we used the identity F(A, B; C; 1) =(C) (C−A−B) (C−A) (C−B). The mass parameter mof the embedding is defined as m =z(ρ →∞). It follows from (2.24) that: m=c d2−c21 2−1 λ γ. (2.27) Let us invert (2.25) and (2.27) and compute cand din terms of μand m. First, we notice that: μ2−m2=d2−c22 λγ2.(2.28) Since d2≥c2, eq. (2.28) implies that μ ≥mfor the embeddings we are considering. Moreover, from (2.28) we get d2−c2as a function of μand mand, using this result in (2.25) and (2.27), we obtain c=mγ−λ 2μ2−m2λ−2 4,d=μγ−λ 2μ2−m2λ−2 4.(2.29) When λ >2, μ =min (2.29) corresponds to c=d=0, i.e., to the Minkowski embeddings with vanishing density discussed above. Actually, as illustrated in Fig. 1, the topology of the embeddings changes when m →μ, where a quantum phase transition takes place. The order 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 1+z2−A2 ton-shell =ρλ 2 ρλ+d2−c2,(2.30) 684 G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 Fig. 1. In this figure we plot the different embeddings for λ =4and m/μ =0.1, 0.8, 0.999 (bottom-up). The Minkowski embeddings at zero density and m/μ =1 would correspond to the constant horizontal line z/m =1. we find Son-shell =−N ∞  0 ρλ ρλ+d2−c2dρ , (2.31) which is divergent and must be regulated. We will do it by subtracting the same integral with the integrand evaluated at the UV (ρ→∞). We arrive at Sreg on-shell =−N ∞  0 ρλ 2ρλ 2 ρλ+d2−c2−1dρ =2N λ+2d2−c21 λ+1 2γ. (2.32) The zero-temperature grand canonical potential is given by minus the regulated on-shell action: =−Sreg on-shell =−2N λ+2d2−c21 λ+1 2γ. (2.33) In terms of mand μthe grand canonical potential can be written as: =−2N λ+2γ−λ 2μ2−m2λ+2 4,(2.34) where we have used (2.28). Moreover, the charge density is: ρch =−∂ ∂μ =μNγ−λ 2μ2−m2λ−2 4=Nd, (2.35) which confirms our identification of the constant d. It is worth noting that the formulas that we will write down do not have the factor of the (infinite) volume of the gauge theory directions VRn, rather all thermodynamic quantities are densities per unit volume. Next, we compute the energy density as: =+μρ ch =N (λ +2)γ−λ 2μ2−m2λ−2 4(λμ2+2m2). (2.36) To calculate the speed of first sound uswe make use of the equation u2 s=∂P ∂ =∂P ∂μ∂ ∂μ−1 ,(2.37) G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 685 where Pis the pressure. Let us first compute the derivative appearing in the numerator. Since P=−, we get from (2.34): ∂P ∂μ =μNγ−λ 2μ2−m2λ−2 4.(2.38) Moreover, from (2.36) we have: ∂ ∂μ =μNγ−λ 2μ2−m2λ−6 4λ 2μ2−m2.(2.39) These yield u2 s=2μ2−m2 λμ 2−2m2,(2.40) which is the result we were looking for. As a check notice that (2.40) gives u2 s=2/λ for m =0, which is the universal result found in [4].1Moreover, the speed of sound (2.40) depends on the integers (n, p, q) through the combination λ, i.e., usis the same for conformal and nonconformal brane backgrounds with the same index λ. In particular, for the D3–D7 and D3–D5 supersymmetric intersections we have: u2 s=μ2−m2 3μ2−m2,for D3–D7 , u2 s=μ2−m2 2μ2−m2,for D3–D5 .(2.41) These results agree with the calculation in [12,13]. Notice that the speed of sound vanishes in 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 −2|p⊥p) intersections. In these systems At(ρ) and z(ρ) grow logarithmically when ρ→∞and the AdS/CFT dictionary must be adapted accordingly. Indeed, in this case the chemical potential and the mass are obtained from the subleading terms of Atand zin the UV. Moreover, the on-shell action has additional logarithmic divergences, which must be eliminated with new counterterms [35,36]. As the result of this analysis one gets that the grand canonical potential for black hole embeddings takes the form  =−a(μ2−m2), where ais a positive constant [37]. Repeating the calculation of us performed above, it is straightforward to verify that u2 s=1in this λ =2 case. Notice that this value is exactly the one obtained by taking λ =2in (2.40). 3. Fluctuations We now allow fluctuations of both the gauge field along the Minkowski directions of the intersection and of the scalar function in the form: Aν=A(0) ν+aν(ρ, xμ), z=z0(ρ) +ξ(ρ,xμ), (3.1) 1For the massless #ND =4 intersections one can rewrite the global symmetry in a suggestive form: SO(n, 1) × SU(Nf) ×U(1) ×SO(3 −λ/2)p×SO(1 +λ/2)q×SO(5 −n). The SO(1 +λ/2)qpart rotates a sphere of λ/2 dimensions, which curiously coincides with the value for the speed of sound (2.40) for m =0. 692 G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 At first-order in δω, we get: δω =−2 p−3 2(5−p) cp λd γ (d2−c2) 6−p 5−p−1 λ (λ d2−2c2) 7−p 2(5−p) +1k 7−p 5−p.(4.32) In terms of mthis expression becomes: δω =−2 p−3 2(5−p) cp λ μ (1−m2) 6−p 5−p−1 2 (λ −2m2) 7−p 2(5−p) +1k 7−p 5−p,(4.33) where we used the following relation of μ, d, and m: μ=γd2 λ(1−m2)1 λ−1 2.(4.34) Let us use the expression of cpin (4.9) and separate the imaginary and real parts: Im δω =−πλ μ (5−p) p−3 5−p 1 5−p22 p−3 2(5−p) (1−m2) 6−p 5−p−1 2 (λ −2m2) 7−p 2(5−p) +1k 7−p 5−p Re δω =πλ μ (5−p) p−3 5−p 1 5−p2cot π 5−p2 p−3 2(5−p) (1−m2) 6−p 5−p−1 2 (λ −2m2) 7−p 2(5−p) +1k 7−p 5−p.(4.35) In particular, for p=3the real part of Re δω vanishes at the order we are working in (4.35) and the complete dispersion relation is given by: ωp=3=±√21−m2 λ−2m21 2 k−iλ μ 1−m2 (λ −2m2)2k2.(4.36) In order to compare with the results in [8,11], let us substitute μby its expression in terms of the density d(eq. (4.34)). We find ωp=3=±√21−m2 λ−2m21 2 k−iλ2 d2 λ 1 2 1 2−1 λ1 λ (1−m2)3 2−1 λ (λ −2m2)2k2.(4.37) In particular, for the D3–D5 system we take λ =4 and arrive at the following dispersion relation: ωD3–D5=±1−m2 2−m21 2 k−i4 d1 2 1 2 1 42 (1−m2)5 4 (2−m2)2k2.(4.38) In Fig. 2 we check (4.38) by comparing it with the results obtained by numerical integration at non-zero (but small) temperature. As it can be appreciated in this figure, the agreement is very good, both for the speed of zero sound csand for the attenuation (i.e., the imaginary part of ω). G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 693 Fig. 2. We depict the speed of zero sound (left) and the attenuation divided by momentum squared (right) for the D3–D5 intersection. The dots have been obtained by integrating numerically the fluctuation equations (A.48) and (A.49) at extremely small temperature (we use ˆ d=106; ˆ d, ˆω, and ˆ kare defined in (7.17)). The continuous curve corresponds to the analytic expression (4.38). 4.4. The p=4case 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, which implies that E(ρ) and ξ(ρ) behave near the horizon at low frequency as: E(ρ) =Aρ+Ac 42+A 2log ρ 2+··· ξ(ρ) =Bρ+Bc 42+B 2log ρ 2+··· ,(4.39) where c4is the constant: c4=iπ+1−2γE.(4.40) In (4.40) γE=0.577 ···is the Euler–Mascheroni constant. Let us now try to obtain the expansion (4.39) by performing the limits in the opposite order. As in [12], we have to compute the next correction to (4.17) and (4.19) near the horizon. First we notice that the equations satisfied by E(ρ) and ξ(ρ) near ρ=0a re just obtained by taking p=4in (4.5): E =−2 ρ3E, ξ  =−2 ρ3ξ. (4.41) Neglecting the right-hand side in (4.41) and integrating twice, we arrive at a linear solution as in (4.17) and (4.19). To go beyond this approximation we plug the values of Eand ξinto the right-hand side of (4.41) and perform the integration. In the low-frequency limit ω2ρ, we have: E(ρ) =E(0)+b1C1+b2C2+(a1C1+a2C2)ρ +(a1C1+a2C2)2log ρ+··· ξ(ρ) =ξ(0)+˜ b1C1+˜ b2C2+(˜a1C1+˜a2C2)ρ +(˜a1C1+˜a2C2)2log ρ+··· .(4.42) Let us now match (4.39) and (4.42). By comparing the linear and logarithmic terms of these equations we arrive at the same values of Aand Bas those written in (4.21). Moreover, using these values of Aand Band identifying the constant terms, we find the following matrix relation between (E(0), ξ(0))and (C1, C2): 694 G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 E(0) ξ(0)=2(c4−log 2)a 1−b12(c4−log 2)a 2−b2 2(c4−log 2)˜a1−˜ b12(c4−log 2)˜a2−˜ b2C1 C2.(4.43) As in the p<4 case, the sources vanish non-trivially when the determinant of the matrix written in (4.43) is zero, namely: (a 1˜a2+a2 2) 2(c4−log 2)2−(a1˜ b2+˜a2b1+2a2b2) 2(c4−log 2) +b1˜ b2+b2 2=0.(4.44) Notice that (4.44) is obtained from (4.23) by taking p=4 and changing cp→c4−log 2on the latter. Using this observation it is straightforward to find the dispersion relation encoded in (4.44). At leading order in ω∼k(4.44) reduces to (4.25), which means that the leading dispersion relation is just given by (4.29) and (4.30). Moreover, the next-to-leading contribution δω is: δω =−√2λ μ(c4−log 2)(1−m2)3 2 (λ −2m2)5 2 k3.(4.45) The imaginary part of δω is easily deduced from (4.45): Im δω =−π√2λ μ (1−m2)3 2 (λ −2m2)5 2 k3.(4.46) Notice that (4.46) is the same as in the first equation in (4.35) for p=4. Similarly, the real part of δω can be written as: Re δω =√2λ μ2γE−1+log 2(1−m2)3 2 λ−2m2k2 (1−m2)3 2 (λ −2m2)5 2 k3.(4.47) 4.5. The λ =2case For λ =2the integral J1(ρ), defined in (4.13), is not convergent and, therefore, the expressions written in (4.15) for E(ρ) and ξ(ρ) at low frequency are not correct. In order to obtain the solution of (4.12) for λ =2, let us define the integral ¯ J1(ρ) as: ¯ J1(ρ) ≡ ∞  ρ d¯ρ¯ρ2 (¯ρ2+d2−c2)3 2−1 ¯ρ=ρ ρ2+d2−c2−1+log 2ρ ρ2+d2−c2+ρ . (4.48) Then, (4.12) for λ =2 can be integrated as: E(ρ) =E(0)−(ω2−k2)C 1[¯ J1(ρ) −log ρ] −[(k2−ω2)c2+ω2d2]C1−cd k C2J2(ρ) ξ(ρ)=ξ(0)+C2[¯ J1(ρ) −log ρ]+ddC 2−ckC 1J2(ρ) , (4.49) where E(0)and ξ(0)are constants. When ρis very large the integrals ¯ J1(ρ) and J2(ρ) vanish by construction and thus E(ρ) and ξ(ρ) behave at the UV as: G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 695 E(ρ) =E(0)+(ω2−k2)C 1log ρ+··· ξ(ρ) =ξ(0)−C2log ρ+··· ,(ρ→∞). (4.50) As argued in [35], when the logarithmic behavior displayed in (4.50) is present, the sources are identified with the coefficients of the logarithms, which should vanish. It is clear from the behavior of ξ(ρ) in (4.50) that we must require that C2=0. Moreover, the logarithmic term in E(ρ) is absent either when C1=0or when: ω=±k. (4.51) If C1=C2=0it follows from (4.49) that the functions E(ρ) and ξ(ρ) are constant and also the matching with the near-horizon results in (4.8) imply that both Eand ξmust vanish. 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 c2 s=1. Notice that this result coincides with the value of the speed of first sound in (2.40) for λ =2. Moreover, when C2=0 and ω2=k2, eq. (4.49) reduces to: E(ρ) =E(0)−ω2d2C1J2(ρ) , ξ(ρ) =ξ(0)−cdkC 1J2(ρ) . (4.52) Taking ρ→0in (4.52) we can match this result with (4.8) and, as a consequence, we can show that E(0)and ξ(0)are related to the constant C1as: E(0)=C1 d2 d2−c2ω2+cpC1 d d2−c2ω 2(6−p) 5−p ξ(0)=C1 cd d2−c2k+cpC1 c d2−c2kω 2 5−p.(4.53) Notice that (4.53) coincides with (4.22) when λ =2, C2=0 and ω2=k2. In particular, these relations imply that the ratio of E(0)and ξ(0)is given by: E(0) ξ(0)=d ck. (4.54) The analysis performed so far in this section is valid for p<4. When p=4we have to go beyond the leading term in ω, as in section 4.4, in order to match the logarithmic terms in the near-horizon expansion. It is easy to check that the λ =2 solution written above can be corrected to match the ρ→0expansion in (4.39). The dispersion relation is still given by (4.51) and (4.53) continues to hold in this case. 5. Hyperscaling violation near the critical point As already mentioned, the probe D-brane systems analyzed above undergo a quantum phase transition as μ →mand the density dvanishes. It was shown in [30] that the critical points of the D3–D7 and D3–D5 intersections are described by a non-relativistic scale invariant field theory exhibiting hyperscaling violation. In this section we extend these results to the case of non-conformal backgrounds (i.e., for p= 3) and we compute the corresponding critical exponents. Let us thus follow the approach of [30] and study the behavior of the system near the quantum critical point at μ =m. Accordingly, we consider a chemical potential of the form: μ=m+¯μ, (5.1) 696 G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 where ¯μis considered to be small. At leading order in ¯μwe can expand the different thermodynamic functions of (2.34), (2.36), and (2.29) as: =−P≈−2λ+6 4 λ+2γ−λ 2Nm¯μ)λ+2 4 =f≈2λ−2 4γ−λ 2Nmλ+6 4¯μλ−2 4 d≈2λ−2 4γ−λ 2mλ+2 4¯μλ−2 4,(5.2) where fis the free energy density. The non-relativistic energy density eis defined as in [30]: e=−ρch m=−Ndm, (5.3) where ρch =Ndis the physical charge density. By using (2.36) and (2.29) we get: e=Nγ−λ 2μ2−m2λ−2 4λμ2+2m2 λ+2−μm.(5.4) Expanding at leading order in ¯μ, we arrive at: e≈2λ−2 4λ−2 λ+2Nγ−λ 2m¯μλ+2 4.(5.5) Comparing this result with the one for the pressure in (5.2), we obtain the following relation between eand P: e=λ−2 4P. (5.6) According to the analysis in [30], the relation between eand Pat zero temperature near the quantum critical point is: e=n−θ zP, (5.7) where θis the hyperscaling violation exponent and zis 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 θand z: θ=n−λ−2 4z. (5.8) Notice that the relation (5.8) between θand zcoincides with the ones found in [30] for the D3–D7 system (taking n =3 and λ =6) and for the D3–D5 intersection (taking n =2 and λ =4). In order to determine zwe look at the speed of sound (2.40) for μ ≈m. At first-order in ¯μit is given by: u2 s≈4 λ−2¯μ m,(5.9) and the corresponding dispersion relation is: ω≈4 λ−2¯μ mk. (5.10) Matching the scaling dimensions of both sides of (5.10) as in [30], using that [ω] =zand [k] =1, we conclude that: z=2.(5.11) G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 697 Therefore θtakes the value: θ=n−λ 2+1.(5.12) Taking into account that for the SUSY Dp–Dqintersections we are considering n=p+q−4 2,λ=q−p+2,(5.13) we can rewrite the expression of θsimply as: θ=p−2.(5.14) Notice that for a D3–Dqintersection 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 Tis small the free energy density can be approximated as: f(μ,m,T)=f(μ,m,T =0)+πρ ch T+O(T 2). (5.15) Then, the non-relativistic free energy density is given by: fnon-rel(μ,m,T)=f(μ,m,T)−ρch m=e+πρ ch T+O(T 2). (5.16) At leading order in ¯μwe have: fnon-rel(μ,m,T)=2λ−2 4λ−2 λ+2Nγ−λ 2m¯μλ+2 41+πλ+2 λ−2 T ¯μ+OT ¯μ2.(5.17) In the quantum critical region the non-relativistic free energy density should scale as: fnon-rel ∼¯μ2−αgT ¯μνz ,(5.18) where αis the exponent which characterizes the scaling of the specific heat capacity Cand νis the exponent corresponding to the correlation length ξ(i.e., C∼(T −Tc)−αand ξ∼(T −Tc)−ν near a phase transition at T=Tc). Comparing (5.18) and (5.17) it follows that, in our case, we have: 2−α=λ+2 4,νz=1.(5.19) Since z=2for our system, the exponents αand νare: α=6−λ 4,ν=1 2.(5.20) Using the expression of λin terms of pand qwritten in (5.13), we can recast αsimply as: α=1−q−p 4.(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) 698 G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 6. Zero sound in alternative quantization 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 to the fluctuation modes, i.e., one can adopt an alternative quantization scheme [39,40]. The equations of motion are the same for different quantizations, only the boundary conditions in the UV are different. On the dual field theory side this corresponds to having an anyonic fluid [31–34]. Let us impose the following boundary condition at the UV: lim ρ→∞ nρλ 2fρμ−1 2μαβ fαβ =0,(6.1) where nis a constant that characterizes the boundary condition (the normal quantization condition considered so far corresponds to n =0). As in [4], it is straightforward to prove that (6.1) is equivalent to require: lim ρ→∞ E=−inlim ρ→∞ ρλ 2a y,lim ρ→∞ ay=in ω2−k2lim ρ→∞ ρλ 2E.(6.2) Notice that, even if the equations of motion (3.13) and (3.14) for Eand ayare decoupled, the mixed boundary conditions (6.2) introduce a coupling between them. Therefore, to implement (6.2) we have to study the equation of motion of ay, written in (3.14). Near the horizon ρ≈0 this equation reduces to: a y+2 ρ7−pay=0,(6.3) which is just the same as (4.5). For p<5the solution of (6.3) is given by the right-hand-side of (4.7). Moreover, for p<4 this solution behaves for low frequencies as: ay(ρ) =Cρ+Cc p2 5−p+··· ,(p<4), (6.4) with Cbeing a constant. We now perform the two limits in the opposite order. For low frequencies (3.14) reduces to: ∂ρρλ+d2−c2a y=0,(6.5) whose integration is straightforward: ay(ρ) =a(0) y−C3J3(ρ) , (6.6) where a(0) y=ay(ρ →∞), C3is a constant of integration, and J3(ρ) is the following integral (for λ >2): J3(ρ) = ∞  ρ d¯ρ (¯ρλ+d2−c2)1 2=2 λ−2ρ1−λ 2F1 2,1 2−1 λ;3 2−1 λ;−d2−c2 ρλ.(6.7) Let us now expand ay(ρ) in powers of ρ. First, one can check that, for small ρ, the integral J3(ρ) can be approximated as: J3(ρ) ≈μ d−ρ √d2−c2,(6.8) G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 699 where μis the chemical potential (2.25). Therefore, for small ρ, aycan be approximated as: ay(ρ) ≈a(0) y−C3 μ d+C3 ρ √d2−c2.(6.9) Let us now match (6.4) and (6.9). From the linear terms, we get the following relation between the constants Cand C3: C=C3 √d2−c2.(6.10) Using this relation, and identifying the constant terms in (6.4) and (6.9), we get the following relation between a(0) yand C3: a(0) y=μ d+cp √d2−c22 5−pC3.(6.11) Let us now rewrite the boundary conditions (6.2) at low frequency and momentum. From the expressions of Eand ayin this regime (eqs. (4.12) and (6.6)), we conclude that they behave in the UV as: Eρ→∞ ≈(ω2−k2)ρ−λ 2C1,a  yρ→∞ ≈ρ−λ 2C3.(6.12) Taking this into account, we can recast the boundary conditions for the alternative quantization as a relation between the constants E(0), a(0) y, C2, and C3. Indeed, let us define E(0) nand a(0) y,nas: E(0) n≡E(0)+inC3,a (0) y,n=a(0) y−inC1.(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,ncan be related to the constants C1, C2, and C3. In matrix form this relation becomes: ⎛ ⎜ ⎝ E(0) n ξ(0) a(0) y,n ⎞ ⎟ ⎠=⎛ ⎜ ⎜ ⎝ 2 5−pcpa1−b12 5−pcpa2−b2in 2 5−pcp˜a1−˜ b12 5−pcp˜a2−˜ b20 −in0μ d+cp d2−c22 5−p ⎞ ⎟ ⎟ ⎠⎛ ⎝ C1 C2 C3⎞ ⎠,(6.15) where a1, a2, b1, b2and ˜a1, ˜a2, ˜ b1, ˜ b2are 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=0we must require that the determinant of the matrix in (6.15) be zero. This leads to: 2 5−pcpa1−b12 5−pcp˜a2−˜ b2−2 5−pcpa2−b22 5−pcp˜a1−˜ b1 ×μ d+cp √d2−c22 5−p+n2˜ b2−2 5−pcp˜a2=0.(6.16) At leading order in frequency and momentum this equation simplifies as: b1˜ b2+b2 2+dn2 μ˜ b2=0.(6.17) 700 G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 Fig. 3. We plot the dispersions in the D3–D5 model (p =3, λ =4). In both plots the red points stand for numerical results (the numerics were performed at extremely small temperature, that is for values of ˆ d=106and ˆ B=3 ·103introduced later in (7.17)) whereas the blue curves are the analytic from (6.23); we emphasize that the analytic result (6.23) is an educated guess, but reproduces the numerics precisely. (left) We vary the quantization parameter n =0, 1 2ncrit , ncrit (top-down) at fixed m μ=0.5. (right) The quantization parameter is chosen to be critical n =ncrit . Different lines correspond to varying m μ=0.1, 0.5, 0.8 (top-down). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Since: b1˜ b2+b2 2=γ2 λd2−c22 λ−22(d2−c2)k2−(λd2−2c2)ω2 0,(6.18) then (6.17) implies the following gapped dispersion relation: ω2=ω2 0=2(d2−c2) λd2−2c2k2+dn μ2 .(6.19) In terms of the reduced mass parameter m, defined in (4.27), we have ω2 0=21−m2 λ−2m2k2+dn μ2 .(6.20) One can also calculate the next order term in the dispersion relation. Indeed, one can check that ω=ω0+δω, where δω is given by: δω =−2 p−3 2(5−p) cp λ μ (1−m2) 6−p 5−p−1 2 (λ −2m2) 7−p 2(5−p) +1k 7−p 5−p−n2 λ−2 cp kμ3γ μλ 2ω p−3 5−p 0 (1−m2)1 2+λ 4 .(6.21) It was noticed in [4] for the massless embeddings that the effect of the alternative quantization is equivalent to switching on a magnetic field traversing the x1x2plane. Actually, it was found in [4] that the effect of a magnetic field Beffectively changes the parameter nas n→n−B d.(6.22) In the present massive case we cannot verify analytically the substitution rule (6.22) since the embedding function z(ρ) is not a cyclic variable in the presence of a Bfield. Therefore, we conjecture that the dispersion relation of the zero sound with general anyonic boundary conditions and magnetic field is given (at leading order) by: ω2 0=21−m2 λ−2m2k2+1 μ2dn−B2.(6.23) G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 701 Thus, the spectrum is generically gapped for non-vanishing Band n. However, it can be made gapless by adjusting the alternative quantization parameter nto the critical value: ncrit ≡B d.(6.24) This particular case corresponds to one, where the anyonic fluid experiences zero net effective magnetic field, thus the resulting spectrum is also gapless. In Fig. 3 we compare the results obtained 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 that the spectrum becomes gapless at n =ncrit . 7. Finite temperature Let us now consider the Dp–Dqintersections (n | p⊥q) at non-zero temperature and magnetic field. First, we introduce a more convenient system of coordinates. Let us represent the different components of the Cartesian coordinates ytransverse to the Dp-brane as: ym=rcos θη m,m=1,···,q−n, yl=rsin θξl,l=q−n+1,···,9−p, (7.1) where ηmand ξlsatisfy: q−n  m=1ηm2= 9−p  l=q−n+1ξl2=1.(7.2) Clearly, the ηm(ξl) are the coordinates of a (q −n −1)-sphere ((8 +n −p−q)-sphere). As: 9−p  l=q−n+1yl2=r2sin2θ, q−n  m=1ym2=r2cos2θ, (7.3) we identify the coordinates zand ρused so far with: z=rsin θ, ρ=rcos θ. (7.4) It is straightforward to check that dy·dy=dr2+r2dθ2+cos2θd 2 +sin2θd 2 ⊥,(7.5) where d2 =d2 q−n−1is the line element of the (q −n −1)-sphere of the Dq-brane worldvolume and d2 ⊥=d2 8+n−p−qis the metric of the (8 +n −p−q)-sphere transverse to the Dq-brane. The ten-dimensional metric of a black Dp-brane in these coordinates is: ds2 10 =r R7−p 2−fp(r) dt2+dx2 +R r7−p 2dr2 fp(r) +r2dθ2+cos2θd 2 +sin2θd 2 ⊥,(7.6) where Ris a constant radius and the blackening factor fpis: fp(r) =1−rh r7−p ,(7.7) 708 G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 fluctuation equations couple the transverse and longitudinal modes when B=0 and it is not clear to us how to deal with this coupling. For this reason we have computed σby applying the method of ref. [41]. The details of this calculation are explained in Appendix C. The final result for σis: σ=Nrλ h(1+rp−7 hB2)(cos θh)λ+d2 r7−p h+B2r 7−p 2 h.(7.40) It is now straightforward to write down the expression of Dwhich follows from (7.30). Indeed, let us define Bas: B=H(cos θ)λ(1+r2fp˙ θ2) d2+H(cos θ)λ,(7.41) where His the quantity defined in (7.12). Then, the Einstein relation gives the following value of the diffusion constant: D=rλ h(1+rp−7 hB2)(cos θh)λ+d2 r 7−p 2 h+r p−7 2 hB2 ∞  rh dr B Hcos θλ 2 d2+Hcos θλ ×1+dλ 2tan θ∂θ ∂d +r2fp˙ θ B ∂˙ θ ∂d.(7.42) In Fig. 4 we compare the predictions of (7.42) for the D2–D6 model and the numerical results obtained by direct integration of the coupled fluctuation equations (A.48)–(A.50). As can be appreciated in this figure, the agreement between the two methods is very good. 8. Summary and conclusions In this paper we studied the collective excitations of flavor Dq-branes in the supergravity background generated by color Dp-branes. The two set of branes are separated in their transverse directions, which corresponds to adding massive flavors in the dual field theory. We first studied this Dp–Dqmodel at T=0and μ =0in the quenched approximation. The non-zero chemical potential is generated by a suitable worldvolume gauge field on the probe. We then generalized these results for T=0 and non-vanishing magnetic field. At zero temperature and non-vanishing chemical potential the supersymmetric Dp–Dqintersections with #ND =4 can be studied analytically. We obtained their thermodynamics and first and zero sound, generalizing previous results in the literature for the conformal cases with p=3. These results allow to characterize the quantum phase transition that occurs when μ =m and d=0. In this point several thermodynamic quantities vanish and the system displays a non-relativistic scaling behavior with hyperscaling violation. We have been able to compute the corresponding critical exponents. We also analyzed the massive flavor brane systems at non-zero temperature and magnetic field. We verified numerically that, when the magnetic field is non-vanishing, the zero sound spectrum becomes gapped, with the gap given by B/μ. Moreover, when Tis large enough the system enters into a hydrodynamic regime, which is dominated by a diffusion mode. We determined numerically the corresponding diffusion constant and verified the validity of the Einstein relation. G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 709 When the intersection is (2 +1)-dimensional we performed an alternative quantization of 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 gapless if the magnetic field is chosen appropriately. In fact, this choice corresponds to a fluid of anyons experiencing zero effective magnetic field, thus the occurrence of gapless mode was expected. Our understanding of the anyonic fluid is still lacking, though. In order to describe its properties better one would need to make a definite choice for the SL(2, Z)transformation as 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 to investigate the transport properties and collective phenomena of the anyon fluid in terms of the statistics, proportional to the quantization parameter n. There are several other open topics which deserve further investigation. The Dp-brane metrics with p= 3 violate hyperscaling [42] with θ=−(p −3)2/(5 −p). It would be worth to explore the relation between this scaling of the background and the one found above for the probe. Another interesting problem for the future would be the analysis of more general Dp–Dq intersections. Contrary to the supersymmetric cases studied here, the massive embeddings of a general Dp–Dqmodel are generically unstable and one must turn on fluxes on the worldvolume of the probe to stabilize them (see, for example [43–45]). These additional worldvolume gauge fields give an important contribution to the Wess–Zumino term of the probe action.3It would be very interesting to develop a general formalism for the collective excitations of the probe brane which could incorporate all the particular cases studied in the literature. It would also be interesting to analyze the systems in which the backgrounds are not generated by branes in flat space. Let us mention the cases of branes on the conifold (as in the Klebanov– Witten model [48]) and the ABJM model [49]. Since the massive embeddings depend on the particular model, it is expected that the results will not be completely universal. It is interesting, however, to determine the features common to all the cases. The collective excitations of brane intersections analyzed so far in the literature have been carried out in the probe approximation. Therefore, it is quite natural to explore the effects on the results of having dynamical quarks. In order to provide an answer to this problem we need to have supergravity backgrounds which include the backreaction of the flavor branes. By employing different approximations, these backgrounds can be found for some systems. Let us mention the case of ABJM with smeared flavor branes [50–53], which are geometries free of pathologies, although they do not incorporate the effect of non-zero density. This effect is included in the geometry recently found in [54], which is dual to three-dimensional super Yang–Mills theory with compressible matter. In the near future we intend to study the collective excitations of the flavor branes for some of these systems. Acknowledgements We thank Yago Bea and Carlos Hoyos for discussions and critical readings of the manuscript. N.J. is supported by the Academy of Finland Grant No. 1268023. A.V.R. and G.I. are funded 3An 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. 710 G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 by the MINECO and FEDER grant FPA2014-52218-P, by the Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042), by Xunta de Galicia (GRC2013-024). G.I. is also funded by FPA2012-35043-C02-02. Appendix A. Fluctuation equations of motion In this appendix we obtain the Lagrangian density, and the corresponding equations of motion, for the fluctuations of the embedding scalar and the gauge fields at non-vanishing charge density d=0 and magnetic field B=0. As was the case for the background equations, it is useful to treat the analysis for T=0and T=0using 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 field and embedding as in (3.1) and (3.2). The induced metric gtakes the form: g=¯g+ˆg, (A.1) where ¯gis the zeroth-order metric and ˆgis the perturbation. Let us split ˆgin the form: ˆg=ˆg(1)+ˆg(2).(A.2) The non-zero elements of ˆg(1)are: ˆg(1) ρxμ=z 0 r7−p 2 ∂μξ, ˆg(1) ρρ =2z 0 r7−p 2 ∂ρξ, (A.3) whereas ˆg(2)has the form: ˆg(2) ab =1 r7−p 2 ∂aξ∂ bξ(A.4) (we are taking the radius R=1in (2.14)). In order to expand the DBI Dq-brane action we notice that the Born–Infeld determinant can be written as: −det(g +F)=−det ¯g+F(0)det(1+X) , (A.5) where the matrix Xis given by: X≡¯g+F(0)−1ˆg+f.(A.6) To evaluate the right-hand side of eq. (A.5), we shall use the expansion: det(1+X) =1+1 2Tr X−1 4Tr X2+1 8Tr X2+O(X3). (A.7) Moreover, in the inverse matrix ¯g+F(0)−1we will separate the symmetric and antisymmetric parts: ¯g+F(0)−1=G−1+J,(A.8) where Jis the antisymmetric component and the symmetric matrix Gis the open string metric. The relevant components of Gare: G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 711 Gtt =− ¯grr (1+z2 0) ¯grr |¯gtt|(1+z2 0)−A(0)2 t ,Gxixj=δij ¯gxx Gρρ =− ¯gtt ¯grr |¯gtt|(1+z2 0)−A(0)2 t .(A.9) Using the fact that ¯grr |¯gtt| =1, and eliminating z 0and A(0) t, we get: Gtt =−ρλ+d2 |¯gtt|ρλ=− ρλ+d2 (ρ2+z2 0)7−p 4ρλ Gρρ =ρλ+d2−c2 ¯grr ρλ=(ρ2+z2 0)7−p 4ρλ+d2−c2 ρλ Gxixj=δij (ρ2+z2 0)7−p 4 ,(A.10) which are just the components written in (3.3). The elements of the antisymmetric matrix Jare: Jtρ =−Jρt =− A(0) t ¯grr |¯gtt|(1+z2 0)−A(0)2 t=−dρλ+d2−c2 ρλ.(A.11) By explicit calculation one can verify that Tr Xis given by: Tr X=2z 0 r7−p 2 Gρρ ∂ρξ+2Jtρ fρt +Gab r7−p 2 ∂aξ∂ bξ, (A.12) while Tr X2is: Tr X2=−Gac Gbd fcd fab +Gac Gbd ˆg(1) ab ˆg(1) cd +2(Jtρ)2(ˆg(1) tρ )2+(ftρ)2−4Jtρ Gab ˆg(1) ρa ftb .(A.13) This last expression can be written more explicitly as: Tr X2=−Gac Gbd fcd fab +2(z 0)2 r7−pGρρ Gab ∂aξ∂ bξ+2(z 0)2 r7−p(Gρρ )2(∂ρξ)2(A.14) +2(Jtρ)2(z 0)2 r7−p(∂tξ)2+(ftρ)2−4z 0 r7−p 2 JtρGab∂aξf tb −4z 0 r7−p 2 JtρGρρ ∂ρξf tρ . From these expressions we get that: 1 2Tr X−1 4Tr X2+1 8Tr X2=z 0 r7−p 2 Gρρ ∂ρξ+Jtρ fρt +1 4Gac Gbd fcd fab +Gab 2r7−p 21−(z 0)2Gρρ r7−p 2∂aξ∂ bξ−(z 0)2 2r7−p(Jtρ)2(∂tξ)2+z 0 r7−p 2 JtρGab ∂aξf tb . (A.15) Let us now obtain the Lagrangian density from these results. First of all, we can check that the first-order terms do not contribute to the equations of motion and, therefore, we just drop them. Moreover, in the second-order terms we can substitute rby r0(ρ), given by: r0(ρ) =ρ2+z0(ρ)2.(A.16) 712 G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 Taking into account the zeroth-order Lagrangian and that: 1−(z 0)2Gρρ r 7−p 2 0 =1−A(0)2 t 1+(z 0)2−A(0)2 t ,(A.17) we get: L=−Nρλ 21+(z 0)2−A(0)2 t ×1 4Gac Gbd fcd fab +1 2r 7−p 2 0 1−A(0)2 t 1+(z 0)2−A(0)2 t Gab ∂aξ∂ bξ −(z 0)2 2r7−p 0 (Jtρ)2(∂tξ)2+z 0 r 7−p 2 0 JtρGab ∂aξf tb .(A.18) Substituting the values of z 0and A(0) t(written in (2.20)), the Lagrangian density for the fluctuations at zero temperature can be written as in (3.4). A.2. Fluctuations at non-zero temperature In this subsection we focus on T=0 and B=0, by fluctuating the scalar and the gauge fields (7.33). First we compute the variation of the induced metric. By using the expansions dθ2=˙ θ2 0dr2+2˙ θ0∂aζdrdx a+∂aζ∂ bζdx adxb+··· cos2θ=cos2θ0−sin(2θ0)ζ −cos(2θ0)ζ2+··· ,(A.19) where xa=(xμ, r) =(t, xi, r), we can represent the induced metric gin the form: g=¯g+ˆg, (A.20) where ¯gis the zeroth-order metric and ˆgis the perturbation. We will expand ˆgup to second order in the fluctuations. Accordingly, let us split ˆgin the form: ˆg=ˆg(1)+ˆg(2),(A.21) where ˆg(1)(ˆg(2)) are the first (second) order terms of ˆg. The non-zero elements of ˆg(1)are: ˆg(1) rr =2rp−3 2˙ θ0˙ ζ, ˆg(1) rxμ=rp−3 2˙ θ0∂μζ, ˆg(1) mn =−rp−3 2sin(2θ0)ζγ mn ,(A.22) whereas those of ˆg(2)are: ˆg(2) ab =rp−3 2∂aζ∂ bζ, ˆg(2) mn =−rp−3 2cos(2θ0)ζ2γmn ,(A.23) where m, nare indices along the internal (q −n −1)-sphere and γmn is the metric of a unit Sq−n−1. Let us now define the open string metric Gand the antisymmetric tensor Jas in (A.8), with F(0)being the gauge field strength (7.10). The components of the inverse of the open string metric in this case are: G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 713 Gtt =− ¯grr (f −1 p+r2˙ θ2 0) |¯gtt|¯grr(1+r2fp˙ θ2 0)−˙ A(0)2 t ,Grr =|¯gtt|fp |¯gtt|¯grr(1+r2fp˙ θ2 0)−˙ A(0)2 t , Gx1x1=Gx2x2=¯gxx ¯g2 xx +B2,Gxixj=δij ¯gxx ,(i,j=3,4,...) , Gmn =γmn r2¯grr cos2θ0 ,(A.24) where A(0)is the gauge potential for the field strength F(0). Using these explicit equations for the metric and eliminating ˙ A(0) t, we get: Gtt =− 1 r7−p 2fp1+d2 H(cos θ0)λ,Grr =r7−p 2fp 1+r2fp˙ θ2 01+d2 H(cos θ0)λ, Gx1x1=Gx2x2=¯gxx ¯g2 xx +B2≡Gxx ,Gxixj=δij ¯gxx ,(i,j=3,4,...) , Gmn =γmn r2¯grr cos2θ0 .(A.25) The only non-zero elements of the antisymmetric matrix Jare: Jtr =−Jrt =− ˙ A(0) t |¯gtt|¯grr(1+r2fp˙ θ2 0)−˙ A(0)2 t Jx1x2=−Jx2x1=− B ¯g2 xx +B2.(A.26) More explicitly: Jtr =−Jrt =− d H(cos θ0)λH(cos θ0)λ+d2 1+r2fp˙ θ2 0 Jx1x2=−Jx2x1=− B ¯g2 xx +B2≡Jxy .(A.27) We next define the matrix Xas in (A.6) and we perform the expansion (A.7) of the DBI determinant. The traces of Xneeded are: Tr X=GMN ˆgMN −JMN fMN ,(A.28) and Tr X2=GMN GPQ−JMN JPQ(ˆgMP ˆgNQ −fMP fNQ)−4GMN JPQ ˆgMP fNQ . (A.29) In these formulas the indices M, N, P, and Qrun over all worldvolume directions (including the angular ones). The Lagrangian density for the fluctuations is given by: L=L01+1 2Tr X−1 4Tr X2+1 8Tr X2+O(X3),(A.30) 714 G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 where L0is the zeroth-order Lagrangian density, given by: L0=−NH(cos θ0)λ1+r2fp˙ θ2 0 d2+H(cos θ0)λ.(A.31) Notice that the equation for the embedding θ0(r) can be written as: ∂rL0r2¯grr Grr ˙ θ0=−λ 2tan θ0L0.(A.32) Let us now consider the first-order contributions to L. They originate from the Tr Xterm in (A.30). Therefore: L(1)=L01 2GMN ˆg(1) MN −1 2JMN fMN .(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: L0 2GMN ˆg(1) MN =L0r2¯grr Grr ˙ θ0˙ ζ−λ 2tan θ0ζ.(A.34) Integrating 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: −1 2L0JMNfMN =Ndf tr +L0 B ¯g2 xx +B2fx1x2,(A.35) and clearly does not contribute to the equations of motion of the fluctuations. Let us now concentrate on the second-order terms in L. After some work, we get: L=L01 4GabGcd −JabJcd +1 2JacJbd fac fbd +r2¯grr 21−r2¯grrGrr ˙ θ2 0Gab∂aζ∂ bζ−λ 41+1−λ 2tan2θ0ζ2 −λ 2r2¯grr Grr tan θ0˙ θ0ζ˙ ζ−r4¯g2 rr 2Jtr2˙ θ2 0(∂tζ)2+λ 4tan θ0Jab ζf ab +r2¯grr ˙ θ0JtrGab∂aζftb +JabGrr∂aζfrb −1 2JabGrr∂rζfab.(A.36) Let us integrate by parts the ζ˙ ζterm on the second line of (A.36). In this process we generate the following contribution to L: λ 4∂rL0r2¯grr Grr tan θ0˙ θ0ζ2=−λ2 8L0tan θ02ζ2+λ 4L0r2¯grrGrr ˙ θ2 0 cos2θ0 ζ2,(A.37) where we have used the embedding equation (A.32). Plugging this result into (A.36) we get the final form of the Lagrangian for the fluctuations, which is given by: L=L01 4GabGcd −JabJcd +1 2JacJbd fac fbd (A.38) +1−r2¯grrGrr ˙ θ2 0r2¯grr 2Gab∂aζ∂ bζ−λ 4 cos2θ0 ζ2−r4¯g2 rr 2Jtr2˙ θ2 0(∂tζ)2 +λ 4tan θ0Jab ζf ab +r2¯grr ˙ θ0JtrGab∂aζftb +JabGrr∂aζfrb −1 2JabGrr∂rζfab. G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 715 Let us now work out the equations of motion derived from this Lagrangian density. We will assume that all fields only depend on t, rand one of the Cartesian coordinates (say x). First of all, we write the equation of arin the ar=0 gauge. We get the following Gauss’ law: Gtt ∂t˙at+Gxx ∂i˙ai=r2¯grr Jtr ˙ θ0∂t˙ ζ+¯ λ 2 Jtr Grr tan θ0∂tζ. (A.39) The equation for atbecomes: ∂rL0GrrGtt ˙at−r2¯grr Jtr ˙ θ0˙ ζ−L0Jtr λ 2tan θ0ζ+Jxy ∂xay +L0GxxGtt ∂xfxt −r2¯grr Jtr ˙ θ0∂2 xζ+L0Jtr Jxy ∂x˙ay=0.(A.40) The equation of axis: ∂rL0Grr Gxx ˙ax+Jtr Jxy ∂tay+L0Gtt Gxx ∂tftx +L0Gxx r2¯grr Jtr ˙ θ0∂t∂xζ−L0Jtr Jxy ∂t˙ay=0.(A.41) Taking into account that L0Jtr =constant, this last equation can be rewritten as: ∂rL0GrrGxx ˙ax+L0GttGxx∂tftx +L0Gxxr2¯grrJtr ˙ θ0∂t∂xζ=−L0Jtr∂rJxy∂tay. (A.42) Moreover, after some simplifications, the equation of motion of aycan be written as: ∂rL0Grr Gxx fry+L0Gxx Gtt ∂tfty +Gxx ∂xfxy  =L0∂rJxyJtr ftx −r2¯grr Grr ˙ θ0∂xζ.(A.43) Finally, let us write the equation of motion of the scalar fluctuations. We get: ∂rL0r2¯grrGrr1−r2¯grrGrr ˙ θ2 0˙ ζ−Jtr ˙ θ0˙at+λ 2 cos2θ0 L01−r2¯grrGrr ˙ θ2 0ζ +λ 2tan θ0L0Jtr ˙at+L0r2¯grr1−r2¯grrGrr ˙ θ2 0Gtt∂2 tζ+Gxx∂2 xζ −L0r4¯g2 rr(Jtr)2˙ θ2 0∂2 tζ+L0r2¯grrJtrGxx ˙ θ0∂xftx =L0∂rJxyr2¯grr ˙ θ0Grrfxy .(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 Eas the gauge-invariant combination: E=ka t+ωa x.(A.45) In momentum space the Gauss law (A.39) becomes: ωGtt ˙at−kGxx ˙ax=ωr2¯grrJtr ˙ θ0˙ ζ+¯ λ 2ωJtr Grr tan θ0ζ. (A.46) We can combine (A.46) and (A.45) to get ˙atand ˙axin terms of the gauge-invariant combination Eand the scalar field ζ: ˙at=Gxx k˙ E+ω2r2¯grrJtr ˙ θ0˙ ζ+ω2¯ λ 2 Jtr Grr tan θ0ζ Gttω2+Gxxk2 ˙ax=Gtt ω˙ E−kωr2¯grrJtr ˙ θ0˙ ζ−kω ¯ λ 2 Jtr Grr tan θ0ζ Gttω2+Gxxk2.(A.47) 716 G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 Moreover, using (A.47) one can demonstrate that (A.40) and (A.42) are equivalent to the following equation for the electric field E: ∂rL0Grr Gxx Gttω2+Gxxk2Gtt ˙ E−kr2¯grrJtr ˙ θ0˙ ζ−kλ 2 Jtr Grr tan θ0ζ −L0Gtt Gxx E+kL0Gxx r2¯grrJtr ˙ θ0ζ=iL0Jtr ∂r(Jxy)a y,(A.48) where Gxx 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L0r2¯grrGrr1−r2¯grrGrr ˙ θ2 0˙ ζ−Jtr ˙ θ0˙at+λ 2 cos2θ0 L01−r2¯grrGrr ˙ θ2 0ζ +λ 2tan θ0L0Jtr ˙at−L0r2¯grr1−r2¯grrGrr ˙ θ2 0Gttω2+Gxxk2ζ +L0r4¯g2 rr(Jtr)2˙ θ2 0ω2ζ+L0r2¯grrJtrGxx ˙ θ0kE =ikL0˙ θ0r2¯grrGrr∂r(Jxy)ay,(A.49) where it should be understood that ˙atis given by the first equation in (A.47). Finally, the equation of motion of the transverse fluctuation ayis: ∂rL0GrrGxx ˙ay−L0GxxGttω2+Gxxk2ay= −iL0Jtr∂r(Jxy)E −ikL0˙ θ0r2¯grrGrr∂r(Jxy)ζ . (A.50) Appendix B. Transverse correlators and the conductivity Let us consider the case in which the magnetic field vanishes, B=0. In this case, the equation of motion (A.50) for the transverse fluctuation ayis: ∂rL0Grr Gxx ˙ay−L0Gxx Gttω2+Gxx k2ay=0.(B.1) This equation can be rewritten as: ¨ay+∂rlog L0Grr Gxx˙ay−Gttω2+Gxx k2 Grr ay=0.(B.2) More explicitly, the equation of motion for ayis: ¨ay+∂rlog d2+rλcos θ0λ 1+r2fp˙ θ2 0 fp˙ay +1+r2fp˙ θ2 0 r7−pf2 p (ω2−fpk2)rλ(cos θ0)λ+ω2d2 d2+rλcos θ0λay=0.(B.3) We now study the equation of motion (B.3) for ayin the low frequency regime in which k∼O() and ω∼O(2). Let us first study (B.3) near the horizon r=rh. With this purpose we expand θ0(r) near r=rh: θ0(r) ≈θh−λ 2(7−p) rλ−1 hcos θhλtan θh d2+rλ hcos θhλ(r −rh)+··· .(B.4) G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 717 We also expand the coefficients of the equation of the transverse fluctuations: ∂rlog d2+rλcos θ0λ 1+r2fp˙ θ2 0 fp=1 r−rh+d1+··· 1+r2fp˙ θ2 0 r7−pf2 p (ω2−fpk2)rλ(cos θ0)λ+ω2d2 d2+rλcos θ0λ=A (r −rh)2+c2 r−rh+··· ,(B.5) where A, d1, and c2are given by: A=ω2 (7−p)2r5−p h d1=1 2rh (p −8)d2+(p +λ−8)rλ hcos θhλ d2+rλ hcos θhλ+λ2 8(7−p) r2λ−1 hcos θh2λ d2+rλ hcos θhλ2tan2θh c2=− 1 7−p rp+λ−6 hcos θhλ d2+rλ hcos θhλk2+1 (7−p)2r6−p h ω2 +λ2 4(7−p)3 rp+2λ−6 hcos θh2λtan2θh d2+rλ hcos θhλ2ω2.(B.6) Let us now solve for ayin Frobenius series around r=rh: ay(r) =(r −rh)α(1+β(r−rh)+...) , (B.7) where the exponents αand β, at order 2, are given by: α=− iω (7−p)r 5−p 2 h ,β≈−(α d1+c2). (B.8) From the expressions of d1and c2written in (B.6) we find that βis given by: β=i1 2(7−p)r 7−p 2 h (p −8)d2+(p +λ−8)rλ hcos θhλ d2+rλ hcos θhλ +λ2 8(7−p)2r 5−p 2 h r2λ−1 hcos θh2λ d2+rλ hcos θhλ2tan2θhω +1 7−p rp+λ−6 hcos θhλ d2+rλ hcos θhλk2.(B.9) Let us now take the near-horizon and low frequency limits in opposite order. First, we write (B.3) as: ¨ay+˙ G G˙ay+Qa y=0,(B.10) 724 G. Itsios et al. / Nuclear Physics B 909 (2016) 677–724 [49] O. Aharony, O. Bergman, D.L. Jafferis, J. Maldacena, N=6 superconformal Chern–Simons-matter theories, M2-branes and their gravity duals, J. High Energy Phys. 0810 (2008) 091, http://dx.doi.org/10.1088/1126-6708/ 2008/10/091, arXiv:0806.1218 [hep-th]. [50] E. Conde, A.V. Ramallo, On the gravity dual of Chern–Simons-matter theories with unquenched flavor, J. High Energy Phys. 1107 (2011) 099, http://dx.doi.org/10.1007/JHEP07(2011)099, arXiv:1105.6045 [hep-th]. [51] N. Jokela, J. Mas, A.V. Ramallo, D. 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