Critical behaviour of hydrodynamic series
PPrepared for submission to JHEP
Critical behaviour of hydrodynamic series
M. Asadi H. Soltanpanahi , , F. Taghinavaz IPM, School of Particles and Accelerators, P.O. Box 19395-5531, Tehran, Iran Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter,South China Normal University, Guangzhou 510006, China Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Southern Nuclear ScienceComputing Center, South China Normal University, Guangzhou 510006, China Institute of Theoretical Physics, Jagiellonian University, S. Lojasiewicza 11, PL 30-348Krakow, Poland
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We investigate the time-dependent perturbations of strongly coupled N = 4SYM theory at finite temperature and finite chemical potential with a second order phasetransition. This theory is modelled by a top-down Einstein-Maxwell-dilaton descriptionwhich is a consistent truncation of the dimensional reduction of type IIB string theoryon AdS × S . We focus on spin-1 and spin-2 sectors of perturbations and computethe linearized hydrodynamic transport coefficients up to the third order in gradientexpansion. We also determine the radius of convergence of the hydrodynamic mode inspin-1 sector and the lowest non-hydrodynamic modes in spin-2 sector. Analytically,we find that all the hydrodynamic quantities have the same critical exponent near thecritical point θ = . Moreover, we establish a relation between symmetry enhancementof the underlying theory and vanishing the only third order hydrodynamic transportcoefficient θ , which appears in the shear dispersion relation of a conformal theory on aflat background. a r X i v : . [ h e p - t h ] F e b ontents The collisions of heavy ion at relativistic energies produce a hot and dense nuclear mattercomposed of deconfined quarks and gluons known as the strongly coupled quark-gluonplasma (QGP). The studies of the QGP at the Relativistic Heavy Ion Collider (RHIC)and the Large Hadron Collider (LHC) have led to the important results that explore awide variety of QGP-related phenomena [1–3]. To describe in and out of equilibrium– 1 –roperties of this new phase at extreme conditions, Relativistic Hydrodynamics (RH) isa powerful tool [4–6]. The RH approach has great triumphs to explain the experimentalevents of the collider labs. Furthermore, the condensed matter physics has also benefitedfrom the RH applications [7]. All of these evidences show the ”unreasonable effectiveness”of the RH irrespective of the energy scales [8].On the other hand, experimental observations imply that the QGP is a stronglyinteracting matter and the perturbative calculations cease to apply [2]. Therefore,non-perturbative methods such as the gauge/gravity duality may shed some lights onvarious properties of the QGP. The well-known example of the AdS/CFT correspondencestates that the type IIB supergravity on the AdS × S is dual to the 4 − dimensionalsuper Yang-Mills (SYM) theory living on the boundary of the AdS [9–11]. Indeed, thisduality is a strong-weak duality which maps a strongly-coupled quantum gauge fieldtheory to a weakly-coupled classical gravity in one higher dimension. One of the mainconsequences of the AdS/CFT application is the prediction of universal ratio of shearviscosity to entropy density, ηs = π , which agrees very well with the experimental data[12, 13].The RH approach is an effective theory for long wave-length regime in which eachconserved quantity can be expressed in terms of a gradient expansion [4]. The coefficientsof this series are the so called transport coefficients which contain the information ofthe underlying microscopic theory. One of the main concern regarding an infinite seriesexpansions is the convergence features. The divergences which exist in the RH series inthe real space [14] can be handled by using the Borel-Pad´e techniques [15–19]. Thisprovides good information about the origin of divergent points and even the regionin which the RH series might be convergent. On the other hand, the convergence ofthe gradient expansion in real space can be related to the radius of convergence of RHin momentum space [20, 21]. In the momentum space, the location of the singularpoints reflect the existence of non-hydro modes and the credit of the RH depends onthe strength of hydro modes over non-hydro modes [22, 23].Recently, the radius of convergence of hydrodynamics series have been investigatedin various holographic cases. The radius of convergence of the shear-mode series in4-dimensional AdS-Reissner-Nordstrom (AdS-RN) black brane has been studied in [24].It has been extended to full range of the charge all the way to the extremal value bothin 4 and 5-dimensional AdS-RN black branes in [25] (see also [26]) and different typesof pole collisions at different values of the charge has been found. Likewise, the analyticproperties of dispersion relations in all sectors have been studied for N = 4 SYM theoryat zero chemical potential dual to AdS black branes in [27, 28].– 2 –n this paper, we consider the 1-R charge black hole (1RCBH) model whichis an analytical top-down string theory construction [29–34]. It is obtained from 5-dimensional maximally supersymmetric gauged supergravity and is holographically dualto a 4-dimensional strongly coupled N = 4 SYM theory at finite chemical potentialunder a U (1) subgroup of the global SU (4) symmetry of R-charges. The interestingfeature of this model is the existence of a second order phase transition at a certainpoint of the parameter space called critical point. This phase transition belongs to themodel B dynamical universality class [35] and due to the large N c approximation, thisphase transition shares same critical exponents as a mean-field model [36, 37].Different aspects of this background have been investigated. Indeed, it was shownthat various quantities such as R-charge conductivity [38], complexity [39], mutualinformation [40] and entanglement of purification [41, 42] remain finite, while theirslopes diverge at the critical point with the same critical exponent θ = . The valueof the dynamical critical exponent was also confirmed in [43] and [44] by studying thenon-hydrodynamical quasi normal modes (QNMs) of the external fields and quantumquench in this background, respectively.Here, we would like to derive the transport coefficients for the dual theory of thisbackground and benefit from the standard recipe [45–48]. We perform this calculationup to the third order of gradient expansion. In spin-2 sector we compute the relaxationtime τ π and κ in the second order, and λ (3)17 and λ (3)1 − λ (3)16 in the third order of expansionfrom the Kubo formula. Furthermore, in the spin-1 sector we obtain another third ordertransport coefficient, θ ≡ − ( λ (3)1 + λ (3)2 + λ (3)4 ), by investigating the dispersion relationof shear hydro mode.According to the holographic dictionary, the hydrodynamic excitations correspondto the QNMs associated with poles of retarded Green’s function of the conservedcurrents. Additionally, there is an infinite series of non-hydrodynamic QNMs in eachsector resembling of the Christmas trees [49]. We investigate the radius of convergenceof hydrodynamic series in the spin-1 sector for the whole range of the parameter spaceby computing the corresponding QNMs. In particular, we want to explore whetherthere is a critical behaviour and if so what is the associated critical exponent. Weshow that, depend on the value of the ratio µT , different forms of mode collision canhappen. Near the critical point there is a level-crossing between the hydro-mode andlowest non-hydro mode which is originally from the gauge field perturbation. While nearthe zero chemical potential the radius of convergence is determined by a level-crossingbetween the hydro-mode and the lowest non-hydro mode from the gravity perturbation.To study the linearized equations of motion, enormous simplification will occur if Throughout this paper, by 1RCBH we mean black hole with flat horizon. – 3 –e use the master equations formalism [50–52] which has two great advantages. First,we can investigate the set of equations analytically to compute higher order transportcoefficients and also to find the radius of convergence of the shear hydrodynamic series.Second, it speeds up significantly the numerical computations of QNM in the spin-1sector.The organization of this paper is as follows. In section 2 we review the buildingblocks of hydrodynamics and gradient expansion and introduce the transport coefficientsup to the third order in gradient expansion for the conformal hydrodynamics. In section3 we give the preliminary ingredients about the Green’s function and how to derive thetransport coefficients from the two-point functions in the AdS/CFT formalism via theKubo formula. Section 4 is devoted to review the thermodynamics of the 1RCBH modeland fix our notation. In section 5 we study the hydrodynamics of the 1RCBH modeland illustrate the details for perturbations in spin-2 and spin-1 sectors. The second andthird order transport coefficients as well as the dispersion relation of the shear modeare derived. In section 6 we compute the QNM frequencies in spin-2 and spin-1 sectorsfor complex momenta. We find the radius of convergence of the hydrodynamic seriesin shear channel using both analytical and numerical approaches. Likewise, in spin-2sector the radius of convergence of lowest non-hydro modes is determined. In section 7we investigate the behaviour of the transport coefficients and the hydrodynamic radiusof convergence near the critical point of the phase transition and we show that all ofthem exhibit the same critical exponent θ = . We conclude with a summary and anoutlook to further directions in section 8. In this section we will review the basic principles of the relativistic hydrodynamicsof the boundary theory. The existence of the hydrodynamic equations is due to theconservation laws which are related to the continuous symmetries of the underlyingtheory. These symmetries yield the conserved quantities whose fluctuations are long-lived and long distance propagating modes, i.e. ω → k →
0. If one considers arelativistic hydrodynamic including a local U (1) symmetry, then the hydrodynamicequations can be read as ∇ µ T µν = F νµ J µ , ∇ µ J µ = 0 , (2.1)where T µν is the energy-momentum tensor corresponding to the space-time symmetriesand J µ is a conserved current corresponding to a local U (1) symmetry of the underlying We use the capital Latin letters (
M, N , . . . ) for the bulk coordinates and Greek letters ( µ, ν , . . . )for the boundary coordinates. We also adopt the convention (cid:126) = c = 1. – 4 –heory. In order to solve the Equations (2.1), one may use the fact that T µν and J µ canbe expressed as functions of the local temperature T ( x ν ), the local four velocity u µ ( x ν )and the local chemical potential µ ( x ν ) which are known as the hydrodynamic variables.Usually, the constitutive relations for the energy-momentum tensor and the currentdensity are written in the Landau frame as T µν = (cid:15)u µ u ν + p ∆ µν + Π µν ,J µ = nu µ + j µ , (2.2)where ( (cid:15), p, n ) are the equilibrium energy density, pressure and charge density, respec-tively. In flat space-time the operator ∆ µν ≡ η µν + u µ u ν is a projector perpendicular tothe fluid velocity u µ and the flat metric η µν = diag ( − , , , µν and j µ tensorsare dissipative contributions which can be expressed in terms of the derivatives of thehydrodynamic variables [4–6]. If the underlying theory possess the conformal symmetry,then the gradient expansion of the dissipative parts can be written as [53, 54]Π µν = − ησ (cid:104) µν (cid:105) + ητ π (cid:18) Dσ (cid:104) µν (cid:105) + 13 σ (cid:104) µν (cid:105) ∇ · u (cid:19) + κ (cid:18) R (cid:104) µν (cid:105) − u α R α (cid:104) µν (cid:105) β u β (cid:19) + λ (2)1 σ (cid:104) µ α σ ν (cid:105) α + λ (2)2 σ (cid:104) µ α Ω ν (cid:105) α + λ (2)3 Ω (cid:104) µ α Ω ν (cid:105) α + (cid:88) i =1 λ (3) i O (3) µνi + O ( ∂ ) ,j µ = − σT ∆ µν ∂ ν ( µT ) + O ( ∂ ) . (2.3)The structure of the operators are chosen such that they transform homogeneouslyunder the Weyl transformations [53]. Among the transport coefficients appeared in(2.3), some of them are more familiar such as the shear viscosity, η , the relaxation time, τ π and the conductivity, σ , while other ones are less recognized, like the non-linearizedsecond order transport coefficients λ (2) i and the third order transport coefficients λ (3) i .Due to the Landau matching condition, dissipative terms are transverse to the velocityprofile u µ Π µν = u µ j µ = 0 and because of the conformal symmetry we have g µν Π µν = 0.The shear stress tensor is defined as σ (cid:104) µν (cid:105) ≡ ∇ (cid:104) µ u ν (cid:105) and for a given second rank tensorwe use the following notation A (cid:104) µν (cid:105) ≡
12 ∆ µα ∆ νβ ( A αβ + A βα ) −
13 ∆ µν ∆ αβ A αβ . In addition, the vorticity field Ω µν and the convective derivative D are expressed asΩ µν = 12 ∆ µα ∆ νβ (cid:18) ∇ α u β − ∇ β u α (cid:19) ,D ≡ u µ ∇ µ . – 5 –he third order operators O (3) i in Equation (2.3) are complicated Weyl-covariant tensorsbuilt out of the third order gradients whose detailed forms can be found in Ref. [54].The corresponding third order transport coefficients λ (3) i are not uniquely defined butfor a given underlying theory there exist at most 20 of them.The transport coefficients ( η, σ ) of the 1RCBH model are calculated in the [38],while the other ones are still undetermined. It is shown that the shear viscosity and theconductivity exhibit a critical behavior close to the critical point of the second orderphase transition [38]. A natural question is whether other hydrodynamic properties (e.g.the transport coefficients, the radius of convergence) of the underlying theory wouldexhibit similar behaviour. In the following section we present the basic ingredients andmethods to derive the transport coefficients for N = 4 SYM theory at finite temperatureand finite chemical potential by using the gauge/gravity duality. In section 5 we willbenefit from our results to derive the corresponding second and third order transportcoefficients. For a strongly coupled field theory which has dual gravity interpretation, the AdS/CFTcorrespondence has a recipe to derive the transport coefficients [45–48]. In this section,our main focus is to review the building blocks of this recipe. This includes the formalismto obtain the retarded Green’s functions, the variational approach and the Kubo formula.
To compute the n -point functions one needs to know the generating functional Z ( J ) ofthe theory Z ( J ) = (cid:90) D Φ e − S [Φ ,J ] , where J is the source term. The n -point functions are nothing but the n ’th functionalderivatives of Z [ J ] with respect to JG ( n ) ( x , . . . , x n ) = i n δ n Z ( J ) δJ ( x ) . . . δJ ( x n ) . To our purpose, the two-point functions are of great importance. Consider a quantumfield theory which has some gauge invariant operators, say ˆ O i . Due to the causalstructure of the relativistic hydrodynamics, we will focus only on the retarded two-pointfunctions which are defined in momentum space as G Rij ( k ) = − i (cid:90) d x e − ik · x θ ( t ) (cid:104) (cid:104) ˆ O i ( x ) , ˆ O j (0) (cid:105) (cid:105) , – 6 –here k = ( ω, −→ q ). From the spectral representations, if one knows the retarded Green’sfunction, then the two-point functions (advanced, symmetric and Feynman) will be easyto find [45].In a holographic setup, there is a straightforward method to compute the retardedtwo-point functions of the boundary theory. Suppose that J is a source for an operatorˆ O in the boundary theory with an interaction Lagrangian density term J ˆ O . Accordingto the Euclidean description of the AdS/CFT correspondence, we have the followingrelation (cid:104) e (cid:82) ∂M J ˆ O (cid:105) = e − S cl (Φ) , (3.1)where the left-hand side is the expectation value of the generating functional for theoperator ˆ O living on the boundary ∂M , while the right-hand side is the exponent of theclassical on-shell action subject to the boundary condition Φ (cid:12)(cid:12) ∂M = J . Note that theEquation (3.1) is written in Euclidean signature while we are interested in Minkowskitwo-point functions. The procedure to make this connection is explained neatly insection 4 of Ref. [46] and we follow their recipe to compute G Rij ( k ). The transport coefficients can be determined via the variational approach and Kuboformula [4]. To quantify the hydrodynamics response, we use the conserved currents, T µν ( x ) and J µ ( x ), since they have microscopic definitions. One can use the variationalapproach to derive a specific two-point functions in the following steps [4]:1. Write down the constitutive relations and expand them up to the first order in thehydrodynamic fluctuations ( δT, δµ, δu µ ) and the source fluctuations ( δA µ , δg µν ).2. Use the Equation (2.1) to find the hydrodynamic fields ( δT, δµ, δu µ ) in terms ofthe source fields ( δA µ , δg µν ).3. Plug the solutions into the constitutive relations (2.2) and make them on-shell.4. Then, we use the on-shell currents to define the retarded Green’s functions as G RJ µ J ν ≡ − δ ( √− g (cid:104) J µ (cid:105) ) δA ν (cid:12)(cid:12)(cid:12)(cid:12) δg = δA =0 , G RJ µ T αβ ≡ − δ ( √− g (cid:104) J µ (cid:105) ) δh αβ (cid:12)(cid:12)(cid:12)(cid:12) δg = δA =0 ,G RT µν J α ≡ − δ ( √− g (cid:104) T µν (cid:105) ) δA α (cid:12)(cid:12)(cid:12)(cid:12) δg = δA =0 , G RT µν T αβ ≡ − δ ( √− g (cid:104) T µν (cid:105) ) δh αβ (cid:12)(cid:12)(cid:12)(cid:12) δg = δA =0 . Now we are in a position to obtain the transport coefficients. Each Green’s functioncorresponds to a specific set of transport coefficients [4]. We are interested in a 3+1– 7 –imensional hydrodynamic theory with a SO(3) symmetry in the spacial directions.Using this symmetry we can set the momentum along the z coordinate. To our purpose,we will focus on a so-called shear retarded Green’s function G Rxy,xy ≡ G RT xy T xy which hasthe following gradient expansion up to the third order[53, 54] G Rxy,xy ( ω, q ) = p − iηω + ητ π ω − κ (cid:0) ω + q (cid:1) − i λ (3)17 ω + i λ (3)1 − λ (3)16 − λ (3)17 ) ωq . (3.2)From this relation it is transparent that the transport coefficients can be found by takingspecial limits of G Rxy,xy which is the well-known Kubo formula [4, 53, 54]. Here is theexplicit form of the Kubo formula for each transport coefficient in this channel, η = − lim ω → lim q → ddω Im G Rxy,xy ,κ = − ω → lim q → ddq Re G
Rxy,xy ,ητ Π − κ ω → lim q → ddω Re G
Rxy,xy , (3.3) λ (3)17 = − ω → lim q → ddω Im G
Rxy,xy ,λ (3)1 − λ (3)16 − λ (3)17 = 2 lim ω → lim q → d dωdq Im G
Rxy,xy . These relations are universal which can be applied to any conformal field theory,including one with dual gravity description. For example, in the context of AdS/CFTcorrespondence and by using Equation (3.3) one can find the first and second ordertransport coefficients of the strongly interacting N = 4 SYM [45–48, 53]. Although,some difficulties may arise such as implementing the holographic renormalization toremove the divergent terms [55–57], but the calculation is straightforward. In section5, we will show how to utilize the holographic machinery as well as the Equation (3.3)to obtain the transport coefficients of a 4-dimensional CFT with second order phasetransition which is dual to the 1RCBH backgrounds. In this section, we present a short review on the gravity setup which mimics a secondorder phase transition of the boundary theory. The bulk theory is a top-down stringtheory construction which is a consistent truncation of the super-gravity on AdS × S geometry keeping only one scalar field and one gauge field coupled to the Einstein gravity.– 8 –he thermal solution is an asymptotically AdS black brane geometry with nontrivialprofile of the scalar and gauge fields, so-called 1RCBH [29–34]. This geometry is dualto a four dimensional strongly coupled gauge theory N = 4 SYM at finite temperatureand chemical potential. The 1RCBH model is described by the following Einstein-Maxwel-dilaton (EMD) action S bulk = 116 πG (cid:90) d x √− g (cid:20) R − f ( φ )4 F MN F MN − ∂ M φ∂ M φ − V ( φ ) (cid:21) . (4.1)where G is the five dimensional Newton’s constant. The self-interacting dilaton potential V ( φ ) and the Maxwell-dilaton coupling f ( φ ) are given by V ( φ ) = − L (cid:0) e φ √ + 4 e − √ φ (cid:1) ,f ( φ ) = e − √ φ . where L is the AdS radius and without loose of generality we set L = 1 now-on. Thecorresponding equations of motion are1 √− g ∂ M (cid:0) √− gg MN ∂ N φ (cid:1) − f (cid:48) ( φ )4 F MN F MN − V (cid:48) ( φ ) = 0 ,∂ M (cid:0) √− gf ( φ ) F MN (cid:1) = 0 , (4.2) R MN − g MN (cid:18) V ( φ ) − f ( φ )4 F (cid:19) − f ( φ )2 F MO F ON − ∂ M φ∂ N φ = 0 . The charged static stationary black brane solutions are given by ds = e A (˜ r ) (cid:0) − h (˜ r ) dt + d(cid:126)x (cid:1) + e B (˜ r ) ˜ r h (˜ r ) d ˜ r ,A (˜ r ) = − log ˜ r + 16 log (cid:16) (cid:101) Q ˜ r (cid:17) ,B (˜ r ) = log ˜ r −
13 log (cid:16) (cid:101) Q ˜ r (cid:17) ,h (˜ r ) = 1 − (cid:102) M ˜ r (cid:101) Q ˜ r ,φ (˜ r ) = − (cid:114)
23 log (cid:16) (cid:101) Q ˜ r (cid:17) , A (˜ r ) = (cid:102) M (cid:101) Q (cid:32) ˜ r h (cid:101) Q ˜ r h − ˜ r (cid:101) Q ˜ r (cid:33) dt, (4.3)– 9 –here ˜ r = 0 is the boundary of asymptotically AdS background and (cid:102) M and (cid:101) Q are themass and charge of the black brane, respectively. The black hole horizon is given by thelargest root of the blackening function, h (˜ r ) | ˜ r =˜ r h = 0,˜ r h = (cid:118)(cid:117)(cid:117)(cid:116) (cid:113) (cid:102) M + (cid:101) Q + (cid:101) Q (cid:102) M , and the gauge field A has only a time component which vanishes at the horizon due toregularity conditions. We use the following reparametrization M ≡ r h (cid:102) M , Q ≡ r h (cid:101) Q, r ≡ r h ˜ r, (4.4)to introduce the dimensionless parameters. In this manner the horizon radius can befixed to one which leads to a simple relation between the mass and charge, namely M = (cid:112) Q . (4.5)Using the above dimensionless parameters simplifies our calculation presented in thefollowing sections. We will use this parameterization now-on. The Hawking temperature T of the black brane is given by T = 2 + Q π ˜ r h (cid:112) Q , (4.6)which according to the gauge-gravity correspondence is equal to the temperature of theboundary theory. In addition, the chemical potential of the dual theory reads as µ = lim r → A t ( r ) = Q ˜ r h (cid:112) Q . (4.7)It is straightforward to see that once the charge parameter vanishes, we will recover thegeometry of the AdS -Schwarzschild background as one may expect.One can characterize the 1RCBH model either by two non-negative parameters( M, Q ) from gravity point of view or ( µ, T ) from the boundary point of view. Nevertheless,the reparameterization introduced in Equation (4.4) can manifest an extra scalingsymmetry in the system. Using Equations (4.6) and (4.7) one may make it moretransparent to find the charge as a ratio of the boundary theory parameters, Q = √ ± (cid:113) − ( µ/Tπ/ √ ) (cid:16) µ/Tπ/ √ (cid:17) . (4.8)– 10 –ince Q is real the above equation indicates that µT ∈ [0 , π/ √ µT ∈ (cid:2) , π/ √ (cid:1) there are two distinct solutionscorresponding to values of Q , while Q = √ µ = πT / √
2) is the merging point of thetwo branches. We will show that thermodynamic quantities of the 1RCBH backgrounddiverge at the merging point declaring that this is the critical point of a second orderphase transition point. In Ref. [38] it was shown that the solutions with − / + sign inEquation. (4.8) are thermodynamically stable/unstable black branes. In this paper weare only interested in stable geometries and choose the branch corresponding to the − sign in Equation. (4.8). It turns out that introducing a new variable y as y + 2 π (cid:16) µT (cid:17) = 1 , y ∈ [0 , , both simplifies our equations and makes the investigations close to the critical pointmore intelligible. Accordingly, one may express the charge Q , the Hawking temperature T and the chemical potential µ of the black brane in terms of this new dimensionlessparameter Q = (cid:114) − y y , T = 2 π ˜ r h (cid:112) (3 − y )(1 + y ) , µ = πT (cid:112) − y √ . (4.9)It is easy to see that the critical point ( Q = √
2) and the AdS -Schwarzschild background( Q = 0) correspond to y = 0 and y = 1, respectively.In the context of the AdS/CFT correspondence, we have L κ = N c π where κ is thefive dimensional gravitational constant given by κ = 8 πG . By using the Equation(4.9) the Bekenstein entropy density for the black brane geometry (4.3) can be computedas sN c T = π
16 (3 − y ) (1 + y ) . As stated by the holography principles, the entropy density of the boundary theoryequals to the entropy density of the black brane. Likewise, the R-charge density of theboundary model ρ = lim ˜ r → δSδ Φ (cid:48) is given by ρN c T = √ (cid:112) − y (3 − y ) . (4.10)Having obtained the entropy and charge density with the aid of the thermodynamicequations s = ( ∂p∂T ) µ and ρ = ( ∂p∂µ ) T , one can derive the pressure of the dual stronglycoupled theory pN c T = π
128 (3 − y ) (1 + y ) . (4.11)– 11 –he conformal symmetry of the underlying boundary theory leads to a relation betweenthe energy density ε and pressure, namely ε = 3 p which is nothing but the tracelessness ofthe stress tensor. On the other hand, the Hessian matrix of the boundary thermodynamicquantities is ∂ ( s, ρ ) ∂ ( T, µ ) = (cid:18) ∂s∂T ∂s∂µ∂ρ∂T ∂ρ∂µ (cid:19) , whose Jacobian J , the determinant of the above matrix, is given by J N c T = 3 π
256 (3 − y ) (1 + 1 y ) . Clearly, the Jacobian is divergent at y = 0 ( µT = π √ ) signaling that there is a second orderphase transition. The thermodynamics of the 1RCBH model is reviewed comprehensivelynear the critical point, y = 0, in the Ref. [43]. Moreover, it is easy to show that at y = 1 we recover the thermodynamics of the N = 4 SYM at zero chemical potential. We will extend the previous studies of the hydrodynamics dual to the 1RCBH model [38]to the higher orders of gradient expansion. There are two distinct computational waysto study the hydrodynamics of the current model which, in principle, at the end of theday should lead to the same results. The first method, is to use the gauge fixing process,following KS [49], and classify the perturbations according to the diffeomorphism andgauge transformations. On top of that, the little SO ( D −
2) group can be used tospecify various sectors of the perturbations. The second method, is to adopt the masterequations formalism [50–52]. Unlike the KS method, in master formalism one shouldwrite the gauge invariant combination of all perturbations before using any gauge fixingwhich leads to gauge invariant linearized equations. There exist particular combinationsof fields such that the equations of motion reduce to the Schrodinger type equations.While for spin-2 sector there is no difference between those methods, the latter methodhas great advantages and simplifies vastly the equations in other sectors. In this paper,we study the hydrodynamics in spin-2 and spin-1 sectors, utilizing the master formalism[50–52], and leave the spin-0 sector to the future works.We close this part by addressing the importance of total action. As stated in section3, the transport coefficients are derived from two-point functions which are nothing butthe derivatives of on-shell total action with respect to the sources. The total action is S tot = S bulk + S GH + S ct , We thank Andrzej Rostworowski for constructive discussion on master equations formalism. – 12 –here S bulk is the bulk action given in Equation (4.1), S GH is the Gibbons-Hawkingboundary term and S ct is the counter term [58], S GH = 18 πG (cid:90) d x √− γK, (5.1) S ct = − πG (cid:90) ∂M d x √− γ (cid:20) R r (cid:18) R µν R µν − R f ( φ ) F µν F µν (cid:19) + φ (cid:18) r − (cid:19) (cid:21) , where γ ij is the boundary induced metric, K is the corresponding extrinsic curvature, K = γ ij ∂ r γ ij . We would like to emphasize that the S ct contributes to the second ordertransport coefficients κ and τ Π , while it has no impact on the first and third order ones. We start by investigating the spin-2 sector which includes only one, namely pertur-bation h xy ( t, z, r ) = g xx ( r ) H xy ( r ) e − iωt + iqz in which we use the SO(3) symmetry of thebackground to fix the momentum along z direction. The linearized Einstein equationfor this perturbation in Schwarzschild coordinate is given by H (cid:48)(cid:48) xy ( r ) − (cid:18) r + 2 1 + M r rh ( r )(1 + Q r ) (cid:19) H (cid:48) xy ( r ) − q h ( r ) − w α h ( r )(1 + Q r ) H xy ( r ) = 0 ,h ( r ) = 1 − M r Q r , α ≡ (cid:112) Q Q , (5.2)where we use the dimensionless frequency and momentum defined as w ≡ ω πT , q ≡ q πT . The Equation (5.2) is a subtle linear second order differential equation which is very hardto find the exact analytical solutions. However, we are interested in the hydrodynamiclimit, i.e. w (cid:28) q (cid:28)
1. The solution representing the ingoing wave at the horizon r = 1 can be written as (for more details see Appendix A), H xy ( r ) = (1 − r ) − i w H ( r ) H (0) , where H ( r ) is regular at the horizon. In hydrodynamic limit we use the following ansatz H ( r ) = ∞ (cid:88) i =0 (cid:15) i H i ( r ) , ( w , q ) → ( (cid:15) w , (cid:15) q ) , (5.3)– 13 –o solve Equation (5.2) order by order in (cid:15) . The explicit form of the solutions up tothird order is given in Equation (A.11).The next step is to compute the on-shell action and keep all the terms quadratic in H xy following the recipe explained in Ref. [46] which leads to S tot = π N c T α (cid:20) (1 − r )(1 + M r ) r H (cid:48) xy ( r ) H xy ( r )+ (cid:32) ( Q + 2) ( q − w )4 M r − M (cid:33) H xy ( r ) (cid:21) r → . (5.4)We would like to highlight a practical point at this stage in our computation. Thecontribution of the counter term in (5.4) starts from the second order which is given inthe second line of Equation (5.4). On the other hand, each term in the counter term(5.1) is composed of even number of derivatives. Therefore, the general form of S tot isthe same in 2 k and 2 k + 1 order of expansion. Since we are interested in computing thetransport coefficients up to the third order, we only need to plug the solution of H xy to the corresponding order in (5.4). The retarded Green’s function can be obtained bytaking the second derivative of this total action with respect to the source [46], G Rxy,xy = N c π T (cid:20) (3 − y ) (1 + y )32 − i w α y − α q (1 + y )(3 − y )+ 4 w α (1 − y ))(3 − y )(1 + y ) − i w λ (3)17 + i wq ( λ (3)1 − λ (3)16 − λ (3)17 ) (cid:21) . Now we are in the position to apply the Kubo formula (3.3) and compute the transportcoefficients up to the third order of gradient expansion, η = πN c T (3 − y ) (1 + y )64 ,κ = N c T (3 − y )(1 + y )32 ,τ π = 2 − y ) πT (3 − y ) , (5.5) λ (3)17 = N c T (1 + y )128 π (cid:18) log( 41 + y ) (cid:18) y ) log( 1 + y (cid:19) + 2(3 − y )Li ( 3 − y (cid:19) ,λ (3)1 − λ (3)16 = N c T (1 + y )128 π (cid:18) log( 41 + y ) (cid:18)
16 + (1 + y ) log( 1 + y (cid:19) + 2(3 − y )Li ( 3 − y (cid:19) . It is easy to check that the shear viscosity satisfies the universal relation ηs = 14 π , – 14 –hich is in complete agreement with [13, 38]. Moreover the y = 1 limit of those transportcoefficients coincide with the counterparts of the N = 4 SYM theory dual to AdS blackbrane background [53, 54]. One of the remarkable features of these transport coefficientsis their behaviour near the critical point of the phase transition. In the section 7 wewill show that all the transport coefficients reach to their critical value with the samecritical exponent θ = . In this section we study the hydrodynamic limit corresponding to the spin-1 sector. Inour setup, by employing the master equations formalism [52] the coupled equations ofmotion of the gauge invariant perturbations can be written in a decoupled form. Indeedthe chief advantage of using this approach is twofold. First, it authorizes the analyticalinvestigations of this sector, as we will discuss in this section. Second, the numericalcomputation of the QNM frequencies will be less costly. We will come back to this pointin section 6.Let us summarize the key steps of the master equations formalism [50–52] in thefollowing. At the first step, we should find the gauge-invariant combinations of theperturbation in spin-1 sector with plane wave ansatz Φ( r ) e − iωt + iqz for all the fields Z ( r ) ≡ h tx ( r ) + iωh zx ( r ) , Z ( r ) ≡ h rx ( r ) − h (cid:48) xz ( r ) + 2 A (cid:48) ( r ) h xz ( r ) , Z ( r ) ≡ a x ( r ) . The next step is to write the linearized equations of motion for the gauge invariantperturbations Z i ( i = 1 , ,
3) which are coupled equations. At the final step we shouldrewrite the gauge invariant combinations as linear combinations of the master scalarsand their derivatives such that they satisfy master equations. The master scalarscorresponding to the spin-1 sector are defined by Z ( r ) ≡ r h ( r ) e A ( r ) − B ( r ) q (3 A (cid:48) ( r )Ψ ( r ) + Ψ (cid:48) ( r )) , Z ( r ) ≡ − iω e A ( r )+ B ( r ) q r h ( r ) Ψ ( r ) , Z ( r ) ≡ e A ( r ) q (cid:112) f ( φ ( r )) Ψ ( r ) , and the master equations are given by (cid:3) (cid:18) (cid:101) Ψ (cid:101) Ψ (cid:19) − (cid:18) W , W , W , W , (cid:19)(cid:18) (cid:101) Ψ (cid:101) Ψ (cid:19) = 0 , where (cid:101) Ψ i ≡ Ψ i ( r ) e − iωt + iqz , the d’Alembert operator (cid:3) is the wave operator on thebackground and the potential matrix W is given in terms of the background functions.– 15 –hile in general the final equations are still coupled they have a simple form interms of the master scalars. Nevertheless, in our case, since the following conditionholds W , − W , W , = − α (cid:112) Q + 1 qQ = constant . The master equations (5.6) can be further simplified to decoupled equations as (cid:3) Ψ ± ± W ± Ψ ± = 0 , (5.6) W ± = r (4 Q r ( M r + 1) + Q (6 M r + 7) + 5 M r ) + 3( Q r + 1) / ± M r (cid:113) M + q Q α ( Q r + 1) / . This form of master equations make our analytical studies more intelligible. It turnsout that the equation with ” + ( − )” sign describes the shear channel perturbation(transverse perturbation of the gauge field) . In the rest of this section we solve Equation(5.6) with ” + ” sign perturbatively to find the hydrodynamic dispersion relation of theshear mode following the prescription given in Ref. [46]. We solve the relevant equationin Eddington-Finklestein (EF) coordinate by using the following ansatzΨ + ( r ) = r / (cid:88) n =0 (cid:15) n ψ n ( r ) , and we scale the frequency and momentum as ( w , q ) → ( (cid:15) w , (cid:15) q ). Again we utilizethe variation of parameters method reviewed in Appendix A to find the ψ n ’s. At thezeroth order in (cid:15) one can solve a second order differential equation for ψ with regularitycondition at the horizon ψ ( r ) = M r (cid:112) Q r + 1 , where the second integration constant is fixed by imposing ψ (1) = 1. In higher orders,without loose of generality, we impose ψ i (1) = 0 for i >
0. We find the solutionsanalytically up to the second order and the source-less boundary condition leads to thefollowing form of the spectral curve [28] in shear channel F shear ( q , w ) = w + i q ( Q + 2)4 ( Q + 1) − i q ( Q + 2)
64 ( Q + 1) − i w (cid:104) Q log (4 Q + 4) + 2 log (2 Q + 4) + 2 (cid:112) Q + 1 cot − ( (cid:112) Q + 1) − − (2 Q + 3) (cid:105) Q + 1)+ wq ( Q + 2) (cid:104)(cid:112) Q + 1 log (cid:16) Q +1 Q +2 (cid:17) + 2 cot − (cid:16)(cid:112) Q + 1 (cid:17)(cid:105) Q + 1) / + O ( w , w q , wq , q ) = 0 . – 16 –y solving this equation perturbatively in q , one can find the dispersion relation ofthe shear hydrodynamic mode w = − i − y q − i y + 1) log(2) − y + 1) log( y + 1) − y − ( y + 1) q + O ( q ) , (5.7)where we use Q = (cid:113) − y )1+ y . By comparing our results with general form of the dispersionrelation in the shear channel for conformal theories (see for example [54]), ω = − i ηε + p q + (cid:18) η τ π ( ε + p ) − θ ε + p (cid:19) q + O (cid:0) q (cid:1) , one can find a relevant third order transport coefficient θ as θ ≡ − ( λ (3)1 + λ (3)2 + λ (3)4 ) = N c T π y. (5.8)Note that at the critical point of the phase transition the θ vanishes. This could be ahint to the symmetry enhancement of the underlying theory at the critical point. Onemay compare this phenomenon with vanishing bulk viscosity for theories with conformalsymmetry. We will elaborate on this point in the discussion section. Let us close thissection by emphasizing that without using the master equations formalism [52] theabove computation could not be achieved in this straightforward manner (as pointedout in a simpler case such as AdS-RN black holes [26]). In principle, one can continueand find the higher order terms in the dispersion relation by solving ordinary secondorder differential equations for higher ψ n using the variation of parameters method. In this section we compute the QNM frequencies in spin-2 and spin-1 sectors associatedwith the poles of the corresponding retarded Green’s function of the boundary theory. Weconsider the complex momentum square q = | q | e iϕ and compute the QNM frequencies.That is simply because of the symmetries of the static background which leads to thefact that the linearized equations are functions of even powers of momentum.To compute the QNMs in our background we used the pseudo spectral Chebyshevdiscretization along the radial coordinate in EF parameterization. The boundaryconditions we should impose are ingoing wave at the horizon which translates to theregularity condition in EF coordinate, and sourceless Dirichlet boundary condition atthe asymptotic region which comes from the holographic dictionary. In all the caseswe consider in this work, the 30 number of grid points along the radial coordinate are– 17 – ⊗⊗ - � - � � � � - � - � - � - �� ⊗⊗ - � - � � � � - � - � - � - �� ⊗ ⊗ - � - � � � � - � - � - � - �� ⊗ ⊗ - � - � � � � - � - � - � - �� Figure 1 . The trajectory of the lowest QNMs in spin-2 sector at the critical point ( y = 0). Inthe first row we show the trajectories before and after the first collision between the lowestmodes at | q ∗ | = 1 .
505 and w ∗ = − . i . In the second row we show the trajectories beforeand after the second collision at | q ∗ | = 2 .
621 and w ∗ = ± . − . i . enough to find ∼ − relative accuracy. To cross-cheekour numerical findings we compute the dispersion relation of the hydro mode in smallreal momenta in shear channel and compare it with the analytical result presented inEquation (5.7) which shows perfect agreement.Finding the radius of convergence for series is an interesting topic in complexanalysis [59]. Suppose we have an analytic (spectral) curve T ( u, v ) = 0 of complexvariables ( u, v ) in the complex plane and we want to find v (cid:63) = v ( u ). These solutionsare classified as regular points and critical points. Regular points are the zeros ofcurve which ∂ i T ∂v i | v = v (cid:63) ( u ) (cid:54) = 0 for i ≥
1, while critical points of order j are the zeroeswhich ∂ i T ∂v i | v = v (cid:63) ( u ) = 0 for 1 ≤ i ≤ j . By definition, the radius of convergence for v (cid:63) isdetermined by the location of the nearest critical point to the origin. Therefore, at thecritical point there is a degeneracy of solutions which specifies the radius of convergence.As explained in details in [23], if the spectral curve is non-analytic at some points thenthere may be other sources of singularities. In all the cases studied in this paper we did– 18 – �� ��� ��� ��� ��� ������������������������������� ��� ��� ��� ��� ��� ��� - ���� - ���� - ���� - ���� - ���� - ���� Figure 2 . Left panel: the y dependency of the convergence radius of the lowest non-hydromodes in spin-2 sector computed numerically. Right panel: the corresponding frequency atwhich the lowest non-hydro modes collide. In both panels the solid blue lines are the resultsand the red-dotted lines are the fits close to the critical point given in Equation (6.1). not find any sign of non-analyticity in the spectral curves for momenta smaller or equalto the critical momentum. In this section we consider the perturbations in the spin-2 sector and compute thecorresponding QNMs numerically. Since there is no hydro-mode in this sector we canonly find the radius of convergence of the non-hydro modes and we will focus only onthe lowest modes. In small values of | q | each mode has a closed trajectory for ϕ from 0to 2 π while for larger values they may collide and share their trajectories. Due to thesymmetries the collision between the pair modes is always on the imaginary axes in thecomplex frequency plane with a purely imaginary momentum. The trajectories of theQNMs are qualitatively the same for the whole range of y . In Figure 1, as an example,we illustrate the trajectories of the modes before and after the first collision for y = 0.The general features of the lowest collision in this sector does not change in thewhole range of y . In Figure 2 we show the radius of convergence of the lowest modein the spin-2 sector and also the corresponding frequency as functions of y . Close tothe critical point the radius of convergence and the corresponding frequency are linearfunctions in y which can be fitted by | q ∗ | (cid:39) . − . y, i w ∗ (cid:39) .
637 + 0 . y. (6.1)In section 7 we will discuss on these relation. We thank Michal Heller for pointing this out. – 19 – �� ��� ��� ��� ��� ������������������ ��� ��� ��� ��� ��� ��� ������������������
Figure 3 . Left panel: the y dependency of the convergence radius of shear mode. The red andblue dotted lines are our numerical computation and the green-solid line is analytical resultsgiven in Equation (6.2). Right panel: The corresponding frequency for 0 ≤ y ≤ .
596 (solidgreen line) and the linear fit near critical point given in Equation (6.4).
In this section we consider the perturbations in the spin-1 sector, including the shearand the transverse gauge field channels, and compute the lowest QNMs using numericaltechniques.In small values of | q | each mode has a closed trajectory for ϕ from 0 to 2 π . In otherwords, in this regime each mode can be found uniquely in complex momentum squareplane. On the other hand for larger values of | q | this may change due to level-crossingor collision of the modes [27]. In fact, depends on how far the medium is from thecritical point either of phenomena may happen. In the rest of this section we presentvarious plots of the trajectory of the QNM frequencies for the complex momentumsquare and discuss the main features at different regimes. One of the purposes is toinvestigate the convergence radius of the hydrodynamic series in the whole regime ofour model. In particular its behaviour near the second order phase transition will beexplored in the next section.To compute the QNMs of the system in complex momentum square, we employthe master equations formalism [52] explained in section 5.2. The decoupled linearizedequations are given in Equation (5.6) and as pointed out in [24, 25] there is a collisionof the modes due to the appearance of a radical in the last term of the potentials W ± .If the lowest collision of the hydrodynamic mode occurs at this | q | , then the radius ofconvergence can be related to this phenomena, | q ∗ | = − α M Q = (3 − y ) ( y + 1)8( y − . (6.2)– 20 – ⊗⊗⊗⊗ - � - � � � � - � - � - ��� ⊗⊗⊗⊗ - � - � � � � - � - � - ��� ⊗⊗⊗⊗ - � - � � � � - � - � - ��� ⊗⊗⊗⊗ - � - � � � � - � - � - ��� Figure 4 . The trajectory of the lowest QNMs in shear channel (solid lines) and in the transversechannel of the gauge field (dashed lines) for y = 1 (AdS-Schwarzschild black brane) in the firstrow and for y = 0 . | q ∗ | = 2 . | q ∗ | = 2 . It is easy to show that the critical momentum given in Equation (6.2) has a simpleexpression in terms of the thermodynamic variables q ∗ = i ε + pπρ . (6.3)This branch point is of square-root type and at this value of the momentum the shearand transverse gauge modes satisfy the same equations of motion. Therefore, for eachmode in the former there is a cousin in the latter which they meet at the criticalmomentum. In other words, the convergence radius of the hydrodynamics is constrainedby a crossing between the hydrodynamic mode and one of the modes in the transversegauge channel. – 21 – ⊗⊗ ⊗ ⊗⊗ - � - � � � � - ��� - ��� - ��� - ��� - ������ ⊗⊗ ⊗ ⊗⊗ - � - � � � � � - ��� - ��� - ��� - ��� - ��������� ⊗ ⊗⊗⊗ ⊗ - � - � � � � - � - � - �� ⊗ ⊗⊗⊗ ⊗ - � - � - � � � � � - � - � - �� Figure 5 . The trajectory of the lowest QNMs in shear channel for various values of theamplitude of the complex momenta square | q | for y = 0 , .
2. We use the same colors asFigure 4 to keep track of the shear modes and transverse gauge field modes in the coupledsystem. The joining happens at w = 0 . i and w = 0 . i respectively. Our main results can be summarized in Figure 3 in which the y dependency of theconvergence radius is demonstrated, where it indicates that there exist at least twodifferent structures for the first mode collision of the hydrodynamic QNM. In the leftpanel, the red and blue points are what we found by studying the QNMs of the systemin q numerically while the solid green line is the analytical formula for the convergenceradius (6.2) computed by employing the master equations formalism [52]. In the rightpanel we show the corresponding frequency for 0 ≤ y ≤ .
596 (solid green line) and thelinear fit near critical point given by − i w = 0 .
244 + 0 . y. (6.4)There are couple of intriguing regimes of the y parameter that we will investigateseparately in the following. Let’s start by recomputing the modes for AdS-Schwarzschildblack brane corresponding to y = 1. In this case we can compare our results with the onewhich is already known [28]. We also introduce our conventions for the presentations ofthe trajectories of the modes. In the first row of Figure 4, we show the trajectories of the– 22 –owest modes in spin-1 sector as functions of complex momenta for y = 1. In this case, forreal momenta the QNM frequencies are two Christmas trees including one for the shearmodes and one for the transverse gauge modes [49]. Interestingly, the first level-crossingoccurs between the latter modes at | q | = 0 . | q | = 2 .
224 which are in agreementwith [28]. Therefore, by definition the radius of convergence of the hydrodynamic is | q ∗ | = 2 . y = 0. In all plots presented in thissection, we use the same colors for the modes and their trajectories. The correspondingmodes, for the real momenta are shown by dots and the positions of mode collisions areshown by crossed-circles.While close to y = 1 the convergence radius of the hydrodynamic series is due toa level-crossing between the hydro mode and the lowest non-hydro shear mode, for0 . < y < .
89 it’s not the case anymore. The level-crossing will be replaced by acollision and the hydro-mode will still have a closed trajectory. This is demonstrated inthe second row of the Figure 4 for y = 0 .
8. For this background the radius of convergenceof the hydrodynamic series is | q ∗ | = 2 .
553 which is associated with a mode collision at q ∗ = 2 . ± . i and w = ± . − . i .As we show in Figure 3 the radius of convergence has a maximum value at y = 0 . y = 0 .
596 the convergence radius of the hydrodynamic series is | q ∗ | = 2 .
854 wheretwo lines are crossed. Associated to this point, there is a collision between the hydromode and non-hydro shear modes at q ∗ = 2 . ± . i and w = ± . − . i as well as a level-crossing with transverse gauge field non-hydro mode at q ∗ = − . w = 0 . i . The most important point here is that the convergence radius ofhydrodynamic series is found due to multi-phenomena: a level-crossing and a collisionbetween the QNMs simultaneously.Finally, in Figure 5 we show the trajectory of the least damped modes in spin-1sector as functions of complex momenta at the critical point of the second order phasetransition y = 0 in the first row and y = 0 . ≤ y ≤
1. The radius of convergence of the hydrodynamicseries is | q ∗ | = for y = 0 and | q ∗ | = 1 .
47 for y = 0 .
2. Note that in this regime there are By collision of the modes, we mean they swap part of their trajectories such that they still havedisjoint trajectories. – 23 –lways infinite number of level-crossing between the shear and gauge field modes at the | q ∗ | , since the linearized equations are the same at this critical value of the momenta,see the Equation (5.6). The radius of convergence of the hydrodynamic series at thecritical point, given by y = 0, is almost half of its counterpart N = 4 SYM withoutchemical potential, corresponding to y = 1.Last but not least, in Figure 5 there is a second damping mode which can be seenin small y cases, as one of the lowest QNMs. This observation may give us a hint aboutthe QNM structure in the whole range of parameter space, in particular for y = 1. Aswe already discussed at the critical value of momentum given in Equation (6.2) twoequations of motion in spin-1 sector coincide with each other. In other words, for eachshear mode there should be a cousin in the transverse gauge field mode. On the otherhand, due to the symmetries of the background and QNM equations in each sector,either the modes are purely imaginary or they are in pair as ω ± = ± ω R − iω I . Havingsaid that, we can conclude that there should be odd numbers of purely imaginary modesin the transverse gauge field channel to accomplish the fact that there are odd numbersof modes in the shear channel (one hydro and infinite number of pair non-hydro modes).Based on this argument we infer that in AdS-Schwarzschild black brane geometry thetransverse perturbation of a gauge field should have at least one purely imaginary QNMwhich has not been addressed in the literature to our best knowledge.To summarize our results, for the convergence radius of the hydrodynamic seriesin shear channel we would like to point out that there is a competition between thelowest non-hydro modes in the shear channel and the ones in the transverse channel tojoin to the hydro-mode. In the range 0 ≤ y < .
596 which ends to the phase transitionpoint the latter is responsible, while in 0 . < y ≤ As mentioned before, the theory we study in this paper enjoys a critical point at µ = πT / √
2. The behavior of different observables near this critical point has beenstudied in [38–44]. In all cases it was shown that the critical exponent is θ = . Allquantities remain finite (except the diffusion constant which vanishes), while their slopesdiverge at the critical point. By using the chain rule, it is easy to show that if a quantity P has the following expression near the critical point in terms of the dimensionlessparameter y = (cid:113) − µ π T P − P c ∼ y m , (7.1)– 24 –hen, the behaviour of the associated quantity near critical point at either fixed temper-ature or fixed chemical potential will be given by, P − P c ∼ | µ − µ c | m/ , T = fixed , (7.2) P − P c ∼ | T − T c | m/ , µ = fixed , (7.3)where P c is the critical value of the quantity. That means for m < θ = m in both schemes. In particular, if a quantity is linear in y close to thecritical point, then the corresponding critical exponent will be θ = .Here, we would like to study the behavior of transport coefficients as well as theradius of convergence of the hydrodynamic series | q ∗ | near the critical point and wechoose to present the results for fixed temperature. By using Equations (5.5), (5.8),(6.1), (6.2), (6.4) and expanding close to the critical point up to the linear term in y wereach the following expressions η (cid:39) πN c T c (cid:18)
98 + 38 y (cid:19) , κ (cid:39) N c T c y ) ,τ π (cid:39) − log(2)2 πT c (cid:18) −
43 log(2) − −
109 log(2) − y (cid:19) , θ (cid:39) N c T c π y,λ (3)17 (cid:39) N c T c ( π − −
2) log(2))192 π × (cid:32) (cid:0) (cid:1) − −
4) log(2) π − −
2) log(2) + 3 (cid:0) (cid:0) (cid:1) − − (2) + 8 log(2) (cid:1) π − −
2) log(2) y (cid:33) ,λ (3)1 − λ (3)16 (cid:39) N c T c ( π − −
4) log(2))192 π × (cid:32) (cid:0) (cid:1) − −
8) log(2) π − −
2) log(2) + 3 (cid:0) (cid:0) (cid:1) − − (2) + 8 log(2) (cid:1) π − −
2) log(2) y (cid:33) , | q ∗ | hydro spin-1 (cid:39)
98 + 32 y, | q ∗ | non-hydro spin-2 (cid:39) . − . y, − i ( w ∗ ) hydro spin-1 (cid:39) .
244 + 0 . y, i ( w ∗ ) non-hydro spin-2 (cid:39) .
637 + 0 . y. (7.4)Interestingly, all the quantities are linear in y which means that they have the samedynamical critical exponent θ = . It is noticed that in Equation (7.4) we organizethe expressions such that the prefactors are the corresponding value for the AdS-Schwarzschild black brane which is a typical critical exponent for the mean fieldtheories [36], related to the Large N c limit [37]. It is intriguing to expect that otherdynamical quantities such as higher transport coefficients may share the same criticalbehaviour close to the transition point. In the case of | q ∗ | and w (cid:63) this form can not be applied. – 25 –et us note that, although the background (4.3) has linear expansion close to anyvalue of y , namely g MN = g (0) MN + g (1) MN ( y − y ) + O (cid:0) ( y − y ) (cid:1) ,A t = A (0) t + A (1) t ( y − y ) + O (cid:0) ( y − y ) (cid:1) ,φ = φ (0) + φ (1) ( y − y ) + O (cid:0) ( y − y ) (cid:1) , the physical quantities associated to the boundary theory do not inherit this behaviournecessarily. After reviewing general aspects of hydrodynamics as a gradient expansion in section2, Green’s function and linear response theory in section 3, we focused on a specificexample from section 4 onwards, namely the 4-dimensional N = 4 SYM theory atfinite temperature and finite chemical potential which poses a second order phasetransition at µ = πT / √
2. This theory is dual to a Einstein-Maxwell-dilaton theorywhich is constructed by a consistent reduction and truncation of 10-dimensional typeIIB supergravity on AdS × S , keeping only the metric, one scalar field and one gaugefield [29–34]. The black hole solutions with planar horizon and with sourceless scalarfield boundary condition in this theory are so-called 1RCBHs which are asymptoticallyAdS. In sections 5-7 we computed the linear hydrodynamic transport coefficients up tothe third order in spin-1 and spin-2 sectors as well as the radius of convergence of thehydrodynamic series in shear channel for arbitrary temperature and chemical potential.Surprisingly we found that close to the critical point of the phase transition all thequantities share the same critical exponent θ = which might be related to the large N c limit of our setup [36, 37]. We speculate that the other hydrodynamic transportcoefficients, including the non-linear ones, exhibit the same critical behaviour. Tosupport our proposal let us recall a universal relation among the second order transportcoefficients [61, 62], 4 λ (2)1 + λ (2)2 = 2 ητ π . which we expect to be hold in our setup too.As one of our main results we found that the only third order transport coefficientwhich appears in shear hydrodynamic dispersion relation dies out at the critical point, Although the Einstein-dilaton theory with second order phase transition exhibit a universal criticalexponent θ = in thermodynamic quantities in various dimensions [60]. – 26 – = 0. This peculiar observation might be related to the symmetry enhancement of theunderlying theory at the critical point, in a same spirit that the bulk viscosity vanishesdue to conformal symmetry.Let us emphasize that both analytical and numerical studies in the spin-1 sectorhave been carried out by employing the master equations approach [50–52] to solvethe linearized equations of motion. To our best knowledge this is the first non-trivialexample of Einstein-Maxwell-dilaton theory which benefits enormously from masterequations formalism. We analyzed the QNM structure in the spin-1 sector with complexmomentum to compute the convergence radius of the hydrodynamic series. We foundthat in different regime different mode collisions and/or level-crossing is responsiblefor radius of convergence. In a range of parameters which ends to the critical point(0 ≤ y ≤ . . ≤ y ≤
1) we found that a mode collision or a level-crossing between the shearhydro mode and lowest shear non-hydro modes lead to the radius of convergence of thehydrodynamic series. For completeness, we have also studied the spectrum of QNMs inspin-2 sector. Due to the fact that there is no hydro mode in this sector, the radius ofconvergence is set by the collision of non-hydro modes and we have analyzed it in theentire range of the parameter space.Now we would like to address some generalizations of our results. As we alreadymentioned, the only third order transport coefficient which shows up in shear dispersionrelation, θ , vanishes at the critical point. This suggest that the full hydrodynamicexpansion may contain less number of transport coefficients at the critical points, evenat the higher order of expansion. In this manner, further investigations for other casesdeserve more attention in future studies. To complete our results, the next step isto study the spin-0 sector of the same model which includes three coupled linearizedequations. With some efforts we found numerically that the other third order transportcoefficient θ ≡ − ( λ (3)3 + λ (3)5 + λ (3)6 ) which appears in the sound hydro mode dispersionrelation [54] does not vanish at the critical point of the phase transition. It turns outthat a complete investigation of this sector with complex momentum is much moreinvolved and has some subtleties that we leave this study for future work.It could also be of interest to venture other models with second order phase transition,e.g. bottom-up holographic cases [60, 63–66], to study the radius of convergence of the– 27 –ydro-mode and to check our proposal of a relation between the symmetry enhancementof the underlying theory and vanishing higher order transport coefficients at the criticalpoint.Finally, it will be interesting to consider the large but finite coupling impacts to ourfindings with an ultimate goal of interpolating between weak and strong coupling results.This can be achieved for example by adding the Gauss-Bonnet term to the action. TheQNMs and some of the transport coefficients of such a theory at zero chemical potentialhave been already computed in [67]. Acknowledgement
We would like to thank Michal Heller, Romuald Janik, Jakub Jankowski, AndrzejRostworowski, Michal Spalinski and Andrei Starinets for valuable discussions and theircomments on the first version of this manuscript. This work is partly supported byGuangdong Major Project of Basic and Applied Basic Research No. 2020B0301030008,the National Natural Science Foundation of China with Grant No.12035007, Scienceand Technology Program of Guangzhou No. 2019050001.
A Spin-2 perturbation
Here we will address the details of calculations of section 5.1. We shall show that howto derive H ( r ) generally from variation of parameters method. This is a necessary stepto obtain the transport coefficients.Generally speaking, the Equation (5.2) is classified as the Heun differential equation[68] and it has four types of different singular points. To our purpose, only two of thesingular points are important which are at the horizon and at boundary, r (cid:63) = 1 , . (A.1)The singular points in the Heun equation are regular-singular points and near thesepoints we can solve the differential equation using the Frobenius ansatz H xy ( r ) → ( r − r (cid:63) ) a (cid:101) H ( r ) , (A.2)where (cid:101) H ( r ) is a regular function at r (cid:63) . The index ” a ” can be obtained by the regularitycondition. By plugging the ansatz (A.2) in linearized equation of motion (5.2) one cansolve the corresponding indicial equation [68]. Near our singular points (A.1) the resultsare (cid:101) H ( r ) → (1 − r ) − i w (cid:101) H ( h )1 ( r ) + (1 − r ) i w (cid:101) H ( h )2 ( r ) + · · · , (cid:101) H ( r ) → (cid:101) H ( b )1 + (cid:101) H ( b )2 r + · · · . (A.3)– 28 –hysical conditions dictate which term has to be picked up. For ingoing wave solutionnear the horizon we should take H xy ( r ) = (1 − r ) − i w H ( r ) . (A.4)Likewise, the Dirichlet boundary condition rules that (cid:101) H ( b )1 ( r →
0) = 0. This boundarycondition will give the spectrum of QNMs [49]. It is worthwhile to mention that valuesof index ” a ” depend on the chosen coordinates. For example in the EF coordinateswhere g tt = 0, the indices are derived as a = 0 , i w , As r → ,a = 0 , , As r → . (A.5)As we stated above, solving exactly the Equation (5.2) is a tedious job. But in thehydrodynamics limit ( w , q ) (cid:28)
1, we can solve it by the method of variation of parameters[28]. By plugging the ansatz (A.4) into the Equation (5.2) and using the followingexpansion H ( r ) = ∞ (cid:88) n =0 (cid:15) n H n ( r ) , ( w , q ) → ( (cid:15) w , (cid:15) q ) , (A.6)for each H n ( r ) the following equation would appear H (cid:48)(cid:48) n ( r ) − r (3 − y ) + 2 r (1 − y ) + 3(1 + y ) r (1 − r )(1 + y + r (3 − y )) H (cid:48) n ( r )= − ir − r w H (cid:48) n − ( r ) + C w H n − ( r ) + (cid:0) C w + C q (cid:1) H n − ( r ) . (A.7)The C , , are defined in below C ≡ i (1 + y )(1 − r )(1 + y + r (3 − y )) , C ≡ − r (3 − y ) + 16(1 + y ) + r (3 − y ) (5 + y )(1 − r )(3 − y )(1 + y + r (3 − y )) , C ≡ − r )(3 − y )(1 + y + r (3 − y )) . (A.8)Homogeneous solutions for the Equation (A.7) are given by H ( r ) = c h ( r ) + c h ( r )where h ( r ) = 1 , h ( r ) = 1 + y − y log(1 + y + r (3 − y )) + log(1 − r ) . (A.9)– 29 –ethod of variation of parameters states that having the homogeneous solutions isenough to find the general solution of the Equation (A.7) H n ( r ) = c h ( r ) + c h ( r ) + h ( r ) (cid:90) r du h ( u ) F n ( u ) W ( h , h ) − h ( r ) (cid:90) r du h ( u ) F n ( u ) W ( h , h ) . (A.10)Here, W ( h , h ) is the Wronskian of the two homogeneous solutions and F n ( u ) is righthand side of the Equation (A.7). The boundary condition at r = 1 demands that theintegration constants should vanish, c = 0 , c = 0, for n >
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