Holographic zero sound at finite temperature in the Sakai-Sugimoto model

JHEP04(2014)149 Published for SISSA by Springer Received:March 14, 2014 Revised:April 3, 2014 Accepted:April 4, 2014 Published:April 24, 2014 Holographic zero sound at finite temperature in the Sakai-Sugimoto model Brandon S. DiNunno,aMatthias Ihl,bNiko Jokelacand Juan F. Pedrazaa aTheory Group, Department of Physics and Texas Cosmology Center, The University of Texas at Austin, Austin, TX, 78712, U.S.A. bCentro de F´ısica do Porto e Departamento de F´ısica e Astronomia, Faculdade de Ciˆencias da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal cDepartamento de F´ısica de Part´ıculas e Instituto Galego de F´ısica de Altas Enerx´ıas (IGFAE), Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain E-mail: [email protected],[email protected], [email protected],[email protected] Abstract: In this paper, we study the fate of the holographic zero sound mode at finite temperature and non-zero baryon density in the deconfined phase of the Sakai-Sugimoto model of holographic QCD. We establish the existence of such a mode for a wide range of temperatures and investigate the dispersion relation, quasi-normal modes, and spectral functions of the collective excitations in four different regimes, namely, the collisionless quantum, collisionless thermal, and two distinct hydrodynamic regimes. For sufficiently high temperatures, the zero sound completely disappears, and the low energy physics is dominated by an emergent diffusive mode. We compare our findings to Landau-Fermi liquid theory and to other holographic models. Keywords: Gauge-gravity correspondence, AdS-CFT Correspondence, Holography and condensed matter physics (AdS/CMT) ArXiv ePrint: 1403.1827 Open Access,c The Authors. Article funded by SCOAP3.doi:10.1007/JHEP04(2014)149 JHEP04(2014)149 Contents 1 Introduction 1 2 The Sakai-Sugimoto model at finite temperature 3 2.1 General setup 3 2.2 Chemical potential 4 3 Zero sound mode in the Sakai-Sugimoto model 5 3.1 DBI action and longitudinal fluctuations 5 3.2 Asymptotic solution 7 4 Numerical results 9 4.1 Quasi-normal modes and dispersion relations 10 4.1.1 Behavior of attenuation rate ΓTat non-zero temperature 12 4.2 Spectral functions 13 4.3 Absence of a Fermi surface 18 5 Conclusions and outlook 19 1 Introduction Understanding the phase structure of QCD at finite temperature and non-zero baryon density is a difficult task and a major focus of current research. At sufficiently large energies, QCD is weakly interacting and perturbation theory is adequate. However, at low energies QCD is strongly interacting and perturbative methods become unreliable, creating a demand for new theoretical tools. The discovery of the gauge/gravity correspondence [1– 3] has granted us access to the study of a large class of strongly-coupled non-abelian gauge theories. One of the first and most successful incarnations of holographic QCD is the SakaiSugimoto model [4,5], which is dual to a SU(Nc) gauge theory with Nfchiral flavors. This model is obtained by considering NfD8-D8-branes embedded in the background of NcD4-branes compactified on a circle (with antiperiodic boundary conditions for the fermions). If NfNc, the backreaction of the flavor branes on the geometry can be sensibly neglected and this, in the field theory, corresponds to working in a “quenched” approximation which disregards quark loops. Although many aspects of the Sakai-Sugimoto model closely resemble real-life QCD, this approach is ultimately troubled by the largeNclimit inherent in the holographic description. In particular, for large-Nc, the lowtemperature low-density phase of holographic QCD becomes a crystalline solid instead of the Fermi liquid for the Nc= 3 case. Moreover, considering the leading 1/Nccorrections is a difficult task because it requires doing loop calculations in the bulk. – 1 – JHEP04(2014)149 On the other hand, the deconfined phase of holographic QCD behaves as a diffusive conductor with restored chiral symmetry and may be identified with a strongly coupled liquid [6]. Indeed, there are some indications that in this regime holographic QCD might be in a Fermi liquid phase. First, the density dependent part of the heat capacity at low temperature is linear in T[7].1This behavior is expected for systems with a Fermi surface, where only a fraction of quasiparticles is excited at small temperatures. Moreover, in the phase where chiral symmetry is restored, the energy density varies as n5/3 B, which is the expected power for non-relativistic fermions [8].2 It is well known that, at sufficiently low temperatures, most metals can be described by Landau-Fermi liquid theory, a phenomenological model where the behavior of interacting fermions are given in terms of quasiparticle excitations of a Fermi surface. In particular, in perturbative QCD the existence of a Fermi liquid phase at high enough chemical potential is well established [11–13], but it is an interesting question whether one can still find signatures of a sharp Fermi surface at strong coupling. This seems unlikely at first sight, but in the large ’t Hooft coupling limit, quark-quark interactions are suppressed and there is a chance of observing such behavior at least in some regions of the parameter space. One of the main features reminiscent of Fermi liquids is the existence of a gapless excitation in the longitudinal current-current correlators [14]. This collective excitation, known as the zero sound, arises from oscillations of the Fermi surface which change its shape but not its size. In the context of holography the existence and characterization of such mode has been studied extensively in the literature, starting with the seminal paper [15], generalized to non-zero temperature [16] and to magnetic field strength [17], with an ever-growing body of work that includes [18–34]. These studies have shed light on the question of whether a true Fermi liquid description can be recovered at low energies in a holographic setup.3,4 In the deconfined phase of holographic QCD, in particular, it was found that a zero sound mode exists at least in the zero temperature limit T/µ 1 [7]. However, it is well-known that for such high densities the model is unstable towards formation of baryon charge density waves [39]. The reason behind this is that there are Chern-Simons terms for the gauge fields which typically lead to spatially modulated phases, similar to other #ND=6 cousins, D3/D7’ [40] and D2/D8’ [41] brane intersection models. An interesting 1In the Sakai-Sugimoto model, the deconfined phase occurs for T > Tcr where Tcr is a critical temperature set by the Kaluza-Klein scale. Here, by low temperature we mean Tµ, where µis the baryonic chemical potential. 2In fact, this behavior holds for temperatures below Tcr and high enough baryon densities [9,10]. In this regime, the nuclear matter is still in a solid phase and is characterized by a condensate of instantons on the probe flavor branes. 3An interesting construction leading to a Fermi liquid behavior was recently proposed in [35]. In this paper, the authors considered a phenomenological model in which the bosonic sector is governed by the DBI action, and whose charged sector is purely fermionic. It would be interesting to study the longitudinal current-current correlators for this setup and determine if it supports the existence of a zero sound mode. 4There has also been a lot of interest in understanding non-Fermi liquids and marginal Fermi liquids in the context of holography [36–38]. These are situations with non-trivial IR fixed points where a Fermi surface still exists but the quasiparticle description breaks down, and could play an important role in understanding, e.g., the strange metal phase of high Tccuprates and thus high temperature superconductivity. – 2 – JHEP04(2014)149 question we may ask here is whether this mode survives at temperatures and densities that are stable against this striped phase formation, and this paper is devoted to answering this question.5,6More specifically, we will investigate the fate of the holographic zero sound mode in the deconfined phase of the Sakai-Sugimoto model in the full space of parameters Tand µ. We will characterize the different transitions, starting from the expected quantum regime T/µ 1 [7] and ending with a high temperature regime T/µ 1 where the zero sound mode completely disappears and the physics is dominated by an emergent diffusive mode [6]. The paper is organized as follows: in section 2, we briefly review some important features of the Sakai-Sugimoto D4/D8/D8 model relevant to the investigation of the collective excitations in consideration. In section 3, we derive the equations of motion for the longitudinal fluctuations of the gauge field and we study in detail the asymptotic behavior of the solutions for both the near-boundary and near-horizon regions in the bulk. In section 4, we present our numerical results for the dispersion relation, quasi-normal modes, and spectral functions for the various regions of the parameter space. Finally, in section 5 we conclude with a discussion of the important similarities and differences of the zero sound mode in the Sakai-Sugimoto model compared to other holographic models and Landau-Fermi liquid theory, and comment briefly on possible future directions in the research on holographic quantum liquids. 2 The Sakai-Sugimoto model at finite temperature 2.1 General setup Let us consider the near horizon geometry generated by the NcD4-branes ds2=u R3 2−f(u)dt2+dx2 i+dx2 4+u R−3 2du2 f(u)+u2dΩ2 4, eΦ=gsu R3 4, F4=dC3=2πNc Ω4 ω4, where t,xi,i= 1,2,3,and x4represent the dimensions of the worldvolume of the D4branes, uis the radial (holographic) coordinate and dΩ2 4the metric of the unit four-sphere, while Ω4and ω4denote the volume and volume form of the unit four-sphere, respectively. The function f(u) := 1 −uH u3is the usual emblackening factor, where uH, the location of the horizon, is related to the temperature of the field theory via TH=3u1/2 H 4πR3/2.(2.1) 5The spatially modulated instabilities can be mitigated and effectively washed out by turning on a sufficiently large magnetic field and/or a mass for the quarks [16,17,30]. The former is an obvious generalization of the present work, albeit a very involved one due to a new phase opening up in the theory [42] that leads to very interesting physics [43]. We will leave this issue for a future work. 6The confining phase can also be unstable due to spatial inhomogeneities upon introducing chemical potentials [44–46]. – 3 – JHEP04(2014)149 This describes the high temperature, deconfined phase of the Sakai-Sugimoto model [4,5, 47] in Minkowski signature, appropriate for real time dynamics.7,8 The radius of curvature Ris given by R3=πgsNcl3 s=πλα0,(2.2) where λis the ’t Hooft coupling constant. The scale R4=2 3R3 uΛ1/2determines the critical temperature Tcr =1 2πR4at which a Hawking-Page (confinement/deconfinement) phase transition occurs [47].9In order to stay within the high temperature regime of the model, we demand TH> Tcr, or equivalently uH> uΛ. In the zero, or low temperature, confined phase the NfNcprobe D8/D8-branes describe a non-trivial profile in the x4(u) direction and the two branches merge at a radial position u0≥uΛ, which is the location of the tip of the cigar-shaped subspace {x4, u}. In the high temperature, deconfined phase that we will be interested in, the {x4, u}subspace will be cylinder-shaped and the straight embeddings, for which ∂ux4= 0, will be energetically favored. 2.2 Chemical potential Introducing a non-zero chemical potential amounts to introducing a flux Ftu 6= 0 along the brane worldvolume [9,10,51] (see also [52–55]). The corresponding DBI action reads SD8=−NTZdu u5 2p1−(∂uAt)2,(2.3) where NT=N Tand N=µ8 gsΩ4R3 2=√2 3(2π)−11 2NcNf √λ. Here we made a gauge choice to set Au=0 and also rescaled At→2πAt. As usual, the conserved charge associated with Atreads u5 2∂uAt p1−(∂uAt)2=d. (2.4) It follows that the electric field satisfies ∂uAt=d √d2+u5,(2.5) from which one can read off the chemical potential µas the asymptotic value of At, µ=At(u→ ∞) = Z∞ uH du ∂uAt=d 3πu3/2 H 2F11 2,3 10,13 10,−d2 u5 H.(2.6) 7In the D3/D7 model at finite baryon density, the probe brane embeddings are “black hole” embeddings that fall into the horizon [50]. This is the analogue of the high-temperature, deconfined phase we are interested in here. Note, however, that in the deconfined phase of the Sakai-Sugimoto, there exists a region of parameter space where the (“short cusp”) U-shaped embeddings are at least meta-stable, even at non-zero baryon density/chemical potential [9]. 8For some recent criticism on the standard mechanism of the deconfinement/confinent phase transition in the Sakai-Sugimoto model, see [48,49]. 9The parameter uΛsets the position of the tip of the cigar in the zero, or low temperature, confined phase of the Sakai-Sugimoto model and also determines the critical temperature at which the deconfinement transition happens. – 4 – JHEP04(2014)149 3 Zero sound mode in the Sakai-Sugimoto model 3.1 DBI action and longitudinal fluctuations We want to study the massless excitation coupled to the density operator in the D4/D8/D8-system. This requires analyzing the linearized equations of motions that follow from the quadratic action describing the fluctuations of the gauge fields living on the D8/D8-branes.10 The DBI action for the D8-brane reads SDBI,D8 =−T8ZdΩ4Zd4xZuB uH du e−Φp−det [Gmn +Fmn],(3.1) where Gmn and Fmn are the induced metric and gauge field strength, respectively. For the investigation of the zero sound mode, we want to study small (longitudinal) fluctuations of the gauge field. We are only interested in fluctuations which are independent of the S4coordinates. Therefore we work in a gauge where Au≡0 and will set Ay(xµ, u) = Az(xµ, u) = 0.11 Let us introduce the following useful functions: g(u) := pd2+u5, f1(u) := g(u)f(u), f2(u) := g3(u) u5, f3(u) := R3g(u) u3f(u),(3.2) in terms of which the DBI action for the longitudinal modes reduces to SDBI,D8 =−1 2NTZd4xZuB uH du f1(u) (∂uAx)2−f2(u) (∂uAt)2−f3(u) (∂xAt−∂tAx)2. (3.3) The gauge field is expanded as follows At(xµ, u) = At(u) + Zd4k (2π)4eikµxµat(kµ, u), Ax(xµ, u) = Zd4k (2π)4eikµxµax(kµ, u),(3.4) where we choose kµ= (−ω, k, 0,0) and sometimes suppress the dependence of aµon kµ. Expanding the DBI action to second order in the fluctuations yields the following equations of motion for the longitudinal modes, ∂u[f2(u)∂uat(kµ, u)] −k2f3(u)at(kµ, u) + ω kax(kµ, u)= 0,(3.5a) ∂u[f1(u)∂uax(kµ, u)] + ω2f3(u)ax(kµ, u) + k ωat(kµ, u)= 0.(3.5b) Let us introduce the gauge invariant quantity E(kµ, u) := kat(kµ, u) + ωax(kµ, u),(3.6) 10The complete D8 action consists of a Dirac-Born-Infeld (DBI) and a Chern-Simons (CS) term. However, for the longitudinal part of the gauge field fluctuations, it suffices to study the DBI part of the action. Note also that the gauge field and metric perturbations decouple. 11This can be done consistently, since the longitudinal fluctuations decouple from the transversal ones, cf. e.g., [6]. – 5 – JHEP04(2014)149 and moreover impose Gauss’ law, which is a constraint equation following from the equation of motion for au(k, u), ka0 x(u) + ωG(u)a0 t(u) = 0,(3.7) where the prime denotes derivation with respect to uand G(u) := f2(u) f1(u)=g2(u) u5f(u). Using Gauss’ law it is possible to express a0 t(u) = k k2−ω2G(u)E0(u), a00 t(u) = k k2−ω2G(u)E00(u) + kω2G0(u) (k2−ω2G(u))2E0(u),(3.8) a0 x(u) = ωG(u) ω2G(u)−k2E0(u), a00 x(u) = ωG(u) ω2G(u)−k2E00(u)−ωk2G0(u) (ω2G(u)−k2)2E0(u).(3.9) Any one of the eqs. (3.5), together with the constraint eq. (3.7), implies the remaining one. Defining F(u) := k2−ω2G(u), we thus arrive at a single second order equation for E(u) (similar equations, albeit in a different context, were discussed in [56]), E00(u) + f0 2(u) f2(u)−F0(u) F(u)E0(u)−f3(u)F(u) f2(u)E(u)=0,(3.10) or equivalently, E00(u) + −5 u+15u4 2g2(u)+ω2G0(u) k2−ω2G(u)E0(u)− R3u2k2−ω2G(u) g2(u)f(u)!E(u) = 0. (3.11) It turns out to be convenient to perform a change of variables y= 2rR3 u,(3.12) and define bµ:= d1/5 2R3/2,12 which can be expressed in terms of µtaking into account (2.6). In order to facilitate the translation between the yand ucoordinates, we define functions similar to the fiabove (by a slight abuse of notation, we will continue to use f(y) = 1 −y yH6and g(y) = p1 + bµ10y10), h1=f(y)g(y) y2, h2=g3(y) y2, h3=g(y) y2f(y),(3.14) and note the following transformation rules: u↔y, f1↔h1, f2↔h2, f3↔h3, 12Note that the critical density, above which an instability occurs in the transversal sector coupled to the Chern-Simons term, was determined in [39], eq. (18). For the Chern-Simons coupling in the Sakai-Sugimoto model, this yields bµcrit ≈3.7142π 3TH⇒eµcrit.≈3.714.(3.13) – 6 – JHEP04(2014)149 Moreover, in order to render eq. (3.11) dimensionless, we perform the following rescalings, y→yHey, ω →eω yH , k →ek yH ,bµ→eµ yH ,(3.15) so that13 ˜ω=3 2π ω TH ,˜ k=3 2π k TH ,˜µ=3 2π ˆµ TH .(3.17) Eq. (3.10) then becomes (in dimensionless variables) ¨ E(ey) + ˙ h2(ey) h2(ey)−˙ F(ey) F(ey)!˙ E(ey)−h1(ey)F(ey) h2(ey)E(ey) = 0.(3.18) Here the dot ˙ indicates derivation with respect to ey. 3.2 Asymptotic solution We proceed to solve eq. (3.18) by applying a standard Frobenius series expansion. First note that eq. (3.18) has non-essential (regular) singular points at ey=−1,0,+1,∞,and at the real roots of ey6ek2+eω2eµ10 ey4=ek2−eω2. The characteristic exponents at the horizon (ey= 1) are ±i˜ω 6, corresponding to solutions near the boundary (ey= 0) with incoming-wave (−i˜ω 6) and outgoing-wave (+i˜ω 6) boundary conditions, respectively. On general grounds, the solution to eq. (3.18) that satisfies an incoming-wave boundary condition near the horizon can be written as a linear combination of two local solutions near the boundary (with exponents 0 and 3) [57–59]: E(ey) = AZI(ey) + BZII (ey),(3.19) where ZI(ey) = 1 + bI 1ey+bI 2ey2+. . . , (3.20) ZII (ey) = ey31 + bII 1ey+bII 2ey2+. . ..(3.21) The coefficients bI,II iare determined by recursion relations following from eq. (3.18). For the case at hand, we find bI 1,3,5,... = 0, bII 1,3,5,... = 0, bI 2=−1 2ek2−eω2, bII 2=1 10 ek2−eω2, bI 4=−1 8ek2−eω22, bII 4=1 280 ek2−eω22, . . . . (3.22) 13For the presentation of the numerical computations reported in section 4, it is sometimes advantageous to work with the alternative dimensionless variables ω=ω ˆµ, k =k ˆµ.(3.16) – 7 – JHEP04(2014)149 Close to the horizon at ey= 1, we can find an incoming-wave solution to eq. (3.18) of the form E(ey) = f(˜y)−i˜ω 6R(ey),(3.23) where R(ey) is regular at the horizon. In order to find the retarded two-point correlation functions Gtt R(ω, k), Gxx R(ω, k), and Gtx R(ω, k), we merely need to compute the polarization function Π(ω, k), which can be easily achieved using the method outlined in [57,58] (for a more general, and very useful, prescription, cf. [60]): Essentially, one needs to functionally differentiate the on-shell boundary action with respect to the sources. In terms of Atand Ax, the boundary term takes the following form SB= lim u→uBNZd4xf(u)g(u)Ax∂uAx−g(u)f(u)G(u)At∂uAt,(3.24) where uBis some radial cut-off. In terms of E(kµ, u) (and E(kµ, y), resp.), we obtain SB= lim u→uBNZd4k (2π)4f(u)g(u)G(u) k2−ω2G(u)E(−kµ, u)∂uE(kµ, u),(3.25) = lim y→02NR9/2Zd4k (2π)4h2(y) k2−ω2G(y)E(−kµ, y)∂yE(kµ, y).(3.26) Now, close to the boundary, we can expand lim y→0E(−kµ, y)∂yE(kµ, y) = A(−kµ)A(kµ)2bI 2y+ 3 B A y2 y3 H +. . .,(3.27) where we have written the source of E(±kµ, y →0) as A(±kµ), in accordance with eq. (3.19). As usual, one needs to holographically renormalize the action, i.e., remove the divergent terms from the action by adding appropriate counter terms: SB,ren.=SB+Sct, Sct = 2NR9/2Zd4k (2π)4A(−kµ)A(kµ) y.(3.28) The matrix of (thermal) retarded correlators is given by Gtt R(ω, k) = k2Π(ω, k), Gxx R(ω, k) = ω2Π(ω, k), Gtx R(ω, k) = ω k Π(ω, k),(3.29) with Π(ω, k)≡δ2SB,ren. δE(−k, 0)δE(k, 0) .(3.30) Therefore, we arrive at Π(ω, k) = 6NR9/2 y3 H 1 k2−ω2B A=16π3µ8Ω4 9gs R6T31 k2−ω2B A.(3.31) – 8 – JHEP04(2014)149 0.300 0.305 0.310 0.315 0.320 0.325 0.330 0.335 0 500 000 1.0 ´106 1.5 ´106 Ω ΧΩ 2 k =0.5 2 4 6 8 10 2 4 6 8 10 Ω ΧΩ 2 k =0.5 Figure 7. Spectral function χxx(˜ω) (normalized by 6NR9/2 y3 H˜ω2) in region II, collisionless thermal regime, for ek= 0.5 and ˜µ= 11 (red), ˜µ= 13 (blue), ˜µ= 15 (orange). Note that the normalization factor enhances the height of the peak for ˜ω < 1. Left panel: Zoomed in closer to the proximity of the peaks. Right panel: At slightly higher ˜ω, there is a second peak developing as the temperature increases and thermal excitations become important. 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0 2´107 4´107 6´107 8´107 Ω ΧΩ 2 k =0.01 5 10 15 20 25 0 5 10 15 20 25 Ω ΧΩ 2 k =0.01 Figure 8. Spectral function χxx(˜ω) (normalized by 6NR9/2 y3 H˜ω2) in region III, hydrodynamic regime, for ek= 0.01 and ˜µ= 5 (red), ˜µ= 10 (blue), ˜µ= 15 (orange). Left panel: The zero sound mode decays further and moves closer to the origin. Right panel: The hydrodynamic diffusive contribution becomes dominant compared to the remnants of the zero sound mode. – 15 – JHEP04(2014)149 0.0 0.5 1.0 1.5 2.0 0 500 000 1.0 ´106 1.5 ´106 2.0 ´106 Ω ΧΩ2 Μ =15 2 3 4 5 6 7 4 6 8 10 12 Ω ΧΩ2 Μ =15 Figure 9. Spectral function χxx(ω) (normalized by 6NR9/2 y3 Hω2) in region I, collisionless quantum regime, for eµ= 15 and k= 0.4 (red), k= 0.6 (blue), k= 1.0 (orange), k= 1.5 (pink), k= 2.0 (grey), and k= 3.0 (purple). Left panel: The sharp, distinct peaks correspond to the zero sound mode. Right panel: Zoomed in closer to the proximity of the peaks. 0.0 0.1 0.2 0.3 0.4 0.5 0 5000 10 000 15 000 20 000 25 000 Ω ΧΩ2 Μ =3 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0 2000 4000 6000 8000 Ω ΧΩ2 Μ =3 Figure 10. Spectral function χxx(ω) (normalized by 6NR9/2 y3 Hω2) in region II, collisionless thermal regime, for eµ= 3 and k= 1/30 (red), k= 1/15 (blue), k= 1/10 (orange), k= 2/15 (pink), k= 1/6 (grey), and k= 1/5 (purple). Left panel: The zero sound mode decays and broadens. Right panel: Zoomed in closer to the proximity of the peaks. – 16 – JHEP04(2014)149 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0 2´107 4´107 6´107 8´107 Ω ΧΩ2 k=0.01 0.000 0.005 0.010 0.015 0.020 0 2.0´106 4.0´106 6.0´106 8.0´106 1.0´107 1.2´107 1.4´107 Ω ΧΩ2 k=0.01 0.000 0.005 0.010 0.015 0 200 000 400 000 600 000 800 000 1.0 ´106 1.2 ´106 1.4 ´106 Ω ΧΩ2 k=0.01 Figure 11. Spectral function χxx(ω) (normalized by 6NR9/2 y3 Hω2) in region III, hydrodynamic regime, for k= 0.01 and eµ= 15 (red), eµ= 10 (blue), eµ= 5 (orange), eµ= 2.5 (pink), eµ= 1.25 (grey), eµ= 1 (purple). Going from the largest to the smallest peak corresponds to increasing the temperature. Note the different scales on the vertical axis. Each subsequent figure zooms into an area of the previous figure (cf. [25], figure 10). The zero sound mode destabilizes as we move deeper into the hydrodynamic regime. – 17 – JHEP04(2014)149 0 1 2 3 4 5 0 10 20 30 40 k  Re@GRD Ω =0 Figure 12. The real part of GR(eω= 0,ek) for various choices of eµ, from bottom to top, eµ= 0 (blue), eµ= 1 (purple), eµ= 5 (orange), eµ= 10 (green), eµ= 20 (red), eµ= 100 (black). Note that the GR(eω= 0,ek= 0) intercepts have been shifted to improve presentability. 4.3 Absence of a Fermi surface This question was originally posed in ref. [7] in the context of the zero temperature limit of the Sakai-Sugimoto model. Here we will review their arguments and extend the investigation to non-zero temperature. From the theory of Fermi liquids, we know that the existence of a sharp Fermi surface is associated with a discontinuity in the distribution function which in turn is directly observable as a singularity at k= 2kFin the retarded current-current Green’s functions in the small frequency limit, GR(ω→0, k)∼k 2kF−1ln k 2kF−1.(4.3) We numerically evaluate GR(ω, k)∼B Ain this limit. Figure 12 shows a plot of GR(0,ek) for a wide range of ekand various temperatures. Of course, since [7] did not find any evidence for a sharp Fermi surface at zero temperature, we do not expect to find traces of a Fermi surface in the small frequency limit of the retarded Green’s functions at nonzero temperature either, and merely intend to corroborate their result. Indeed, we do not observe any characteristic structure consistent with a singularity in the Green’s functions. In [42], the authors explored if the model exhibits quantum oscillations associated with a series of crossings of the Landau levels with the Fermi surface. They concluded that the model does not possess de Haas-van Alphen oscillations, even at zero temperature. Similarly, our analysis does not seem to lead to further evidence for an observable Fermi surface, despite the existence of a collective excitation corresponding to zero sound. Some possible implications are listed in the conclusions. – 18 – JHEP04(2014)149 5 Conclusions and outlook The principal line of research followed in the present work was to clarify the nature of holographic quantum liquids, in particular, their low energy effective description at finite density, and to compare them with properties of real materials such as liquid Helium-3, which at sufficiently low temperatures are well understood in terms of Landau’s theory of Fermi liquids. We focused on the deconfined phase of the Sakai-Sugimoto model of holographic QCD, expanding on previous results regarding the existence of a zero sound mode in the longitudinal channel of the current-current correlators. More specifically, following [16,25], we generalized the zero-temperature (or infinite baryonic density) results of [7] for the zero sound mode to arbitrary values of Tand µand we studied in detail its behavior as T/µ was increased. Let us summarize some of the important findings of the Sakai-Sugimoto model at finite temperature and baryon density/chemical potential: •Some thermodynamic properties of the deconfined phase of the Sakai-Sugimoto model appear to be consistent with a strongly-coupled Fermi liquid interpretation. We must emphasize that the existence of such a phase at high enough baryon density is well established in perturbative QCD [11–13], which means that the existence of a Fermi surface seems to naively extrapolate to the strong coupling regime. In particular, the specific heat at low temperatures scales like Cv∼T[7], in agreement with the predictions of Landau-Fermi theory, while in the standard D3/D7 case Cv∼T6[25]. On the other hand, the energy density varies as n5/3 B, which is the expected equation of state for non-relativistic fermions [8]. •The model features a stable zero sound mode with a non-standard k3dependence of the attenuation rate at low and moderate temperatures (collisionless quantum and collisionless thermal regimes). •For large enough temperatures (hydrodynamic regime), the zero sound mode is effectively damped out by thermal fluctuations. The low energy physics in this case is dominated by an emergent diffusive mode, in agreement with the results of [6]. •The behavior of the attenuation rate at finite temperature in the three regimes discussed above is reminiscent of Landau-Fermi liquid theory, albeit with different temperature dependences. In particular, we found that ΓT∼constant in the collisionless quantum regime and ΓT∼T3in the collisionless thermal regime. In the hydrodynamic regime, we can distinguish between a cold and a hot phase where ΓT∼T−3 and ΓT∼T−1, respectively. In the latter regime, the attenuation rate refers to the diffusion mode, while in the first two cases, to the zero sound mode. •We did not discover any evidence for a direct signature of a sharp Fermi surface at zero or non-zero temperature. One possible explanation is that the Fermi surface structure expected in the spectral functions at low frequencies might only appear at – 19 – JHEP04(2014)149 order Nf/Nc, in which case we would need to include backreaction to have a visible effect.14 If this is the case, the first step would be to construct the backreacted background at finite chemical potential with the fluctuations turned off. On top of this background we would need to consider small fluctuations. Gauge fluctuations will generically mix with metric fluctuations and we would have to consider an appropriate gauge invariant combination for the sound channel as in [20]. The second possibility is that the zero sound mode that we observe is not actually due to the presence of a Fermi surface, but rather, should be interpreted as a Goldstone boson arising from the spontaneous breaking of a specific symmetry, see e.g. [22,33]. Both options would be interesting in their own right. Either we confirm the existence of a Fermi liquid phase at strong coupling or we discover a new low energy description of strongly-correlated electrons with no Fermi surfaces. •At large chemical potential to temperature ratio µ/T the model is expected to decay to a spatially inhomogeneous phase [39]. The critical value ˜µ∼3.7 for the on-set of the instability is of the same order as the threshold for the zero sound to be completely damped by thermal effects and ceasing to propagate, thus leaving only a fairly small range of parameter space for the zero sound. However, to precisely address this question one should first construct the striped solution out of the equations of motion following from the DBI+CS action, and compare the free energies to decipher which phase dominates. Thus, in conclusion, although some of the features of the model are compatible with the Landau theory of Fermi liquids, further research is necessary in order to determine whether this model of holographic QCD has a true low energy description in terms of quasiparticle excitations of a Fermi surface or if it constitutes a novel type of strongly-coupled quantum liquid phase without a Fermi surface at all. As mentioned in the introduction, the inclusion of a large magnetic field in the worldvolume of the flavor branes would be an interesting extension of the present work because such a case may not be unstable to the formation of a spatially modulated phase. The zero sound mode is expected to develop a gap in the dispersion relation, and would lead to interesting physics regarding, e.g, the fate of Kohn’s theorem for nonrelativistic fermions with pairwise interactions. Another interesting possibility for future research would be to investigate the case of non-supersymmetric flavor D7/D7and D5/D5-brane setups in the Klebanov-Witten and Klebanov-Strassler backgrounds [62–70]. Both cases are currently under consideration. Acknowledgments The authors would like to acknowledge very useful conversations and correspondence with Alfonso Ballon Bayona, Richard Davison, Mohammad Edalati, Matthias Kaminski, Matthew Lippert, Alfonso Ramallo, Gordon Semenoff, and Dimitrios Zoakos. M.I. is grateful to the Theory Group of the Department of Physics at the University of Texas at Austin, 14Some backreacted configurations in the Sakai-Sugimoto model were studied in [61]. – 20 – JHEP04(2014)149 the Department of Physics at the University of Washington, the Department of Physics at the University of British Columbia, and the Theory Group, Instituto Galego de F´ısica de Altas Enerx´ıas, Universidade de Santiago de Compostela for hospitality during intermediate stages of this work; moreover, M.I. would like to express his gratitude to Estibalitz Ukar and Christoph Sachse for hospitality during his extended stay in Austin, TX. The work of B.D and J.P is partially supported by the National Science Foundation under grant Grant No. PHY-1316033 and by the Texas Cosmology Center. M.I. is funded by the FCT fellowship SFRH/BI/52188/2013. The Centro de F´ısica do Porto is partially funded by FCT through the projects PTDC/FIS/099293/2008 and CERN/FP/116358/2010. 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