aa r X i v : . [ h e p - t h ] D ec UWO-TH-09/17
Transport at criticality
Alex Buchel , and Chris Pagnutti Department of Applied MathematicsUniversity of Western OntarioLondon, Ontario N6A 5B7, Canada Perimeter Institute for Theoretical PhysicsWaterloo, Ontario N2J 2W9, Canada
Abstract
We study second order phase transitions in non-conformal holographic models ofgauge theory/string theory correspondence at finite temperature and zero chemicalpotential. We compute critical exponents of the bulk viscosity near the transitionand interpret our results in the framework of available models of dynamical criticalphenomena. Intriguingly, although some of the models we discuss belong to differentstatic universality classes, they appear to share the same dynamical critical exponent.December 2009
Introduction
Gauge theory/string theory correspondence of Maldacena [1] presents a useful theo-retical framework to study strong coupling dynamics of gauge theories. In this paperwe use it to study bulk viscosity of non-conformal gauge theory plasma at criticalityof a second order phase transition. We focus on finite temperature phase transitionsat zero chemical potential . Furthermore, since the models we consider do not havelong-range (Goldstone boson) modes at the criticality, their hydrodynamics is that ofa standard viscous relativistic fluid , i.e, the local stress-energy tensor is given by T µν = ǫu µ u ν + P ( ǫ )∆ µν − η ( ǫ ) σ µν − ζ ( ǫ )∆ µν ( ∇ · u ) , ∆ µν = g µν + u µ u ν , σ µν = ∆ µα ∆ νβ ( ∇ α u β + ∇ β u α ) − d − µν ∆ αβ ∇ α u β , (1.1)where ǫ and P ( ǫ ) are the local energy density and pressure, u µ is the local d -velocity ofthe plasma, and η ( ǫ ) and ζ ( ǫ ) are the shear and the bulk viscosities correspondingly. Aplasma with the stress-energy tensor (1.1) allows for a propagation of the hydrodynamicsound waves with the following dispersion relation w = c s q − i Γ q + O (cid:0) q (cid:1) , (1.2)where c s is the speed of sound and Γ is the sound wave attenuation, c s = (cid:18) ∂P∂ǫ (cid:19) T = sc v , Γ = 2 π ηs (cid:18) d − d − ζ η (cid:19) , (1.3)and w = ω/ (2 πT ) and q = | ~q | / (2 πT ). Thus, if the dispersion relation for the soundwaves (1.2) in plasma can be determined from first principles, the bulk viscosity can becomputed from (1.3) . A holographic correspondence of Maldacena provides a frame-work for such first principle computations [3]: the dispersion relation of the soundwaves in strongly coupled plasma is identified with the dispersion relation of a hy-drodynamic graviton quasinormal mode (of a certain polarization) in gravitationalbackground holographically dual to the equilibrium thermal state of the plasma.In a conformal plasma the trace of the stress-energy tensor vanishes. As a result, c s (cid:12)(cid:12)(cid:12)(cid:12) CF T = 1 d − , ζη (cid:12)(cid:12)(cid:12)(cid:12) CF T = 0 . (1.4) Critical phenomena in strongly coupled plasma at finite density will be discussed in [2]. One also needs a ratio of shear viscosity to the entropy density. In models we analyze this ratiois universal [5, 6, 7]. He in the vicinity of the critical liquid-vaporpoint exhibits the ratio of the bulk-to-shear viscosities in the excess of a million [4].Below, we present the first example of a holographic model with divergent ratio of ζη in the vicinity of a second order phase transition.Techniques for the holographic computations of the bulk viscosity in non-conformalplasma were developed in [8]. In what follows we omit all the technical details of theanalysis. In the next section we mention three different proposals for the scaling ofbulk viscosity at the criticality of a second order phase transition [9, 10, 11]. In section3, building up on the previous work [12, 13, 14], we identify the second order phasetransitions (and the static universality classes) in holographic models of gauge-gravitycorrespondence: the N = 2 ∗ gauge theory plasma [15, 16, 17], the finite temperatureKlebanov-Tseytlin (cascading) gauge theory plasma [18, 19, 20, 21, 22, 23], and thephase transition in ”exotic black holes” we proposed earlier [24]. In section 4 we presentresults for the scaling of bulk viscosity at the criticality in these models. While thecomputations of bulk viscosity in N = 2 ∗ and the cascading gauge theories were donepreviously (see [8, 13, 25, 26, 14] ), our result for the bulk viscosity of plasma dual to”exotic black holes” of [24] is new. We conclude in section 5 by interpreting the scalingof the bulk viscosity in holographic models in the framework of models of dynamicalcritical phenomena [9, 10, 11]. We end up mapping the second order phase transitions in holographic models ofgauge/gravity correspondence to the ferromagnetic phase transition in p = d − W = W ( T, H ) which is the difference be-tween the free energy densities of the ordered and the disordered phase W ( T, H ) = Ω o ( T, H ) − Ω d ( T, H ) (2.1)as a function of temperature T and the (generalized) external magnetic field H . The(generalized) spontaneous magnetization M determines the response of the free energy3o the changes in the external control parameter, H , as M = − (cid:18) ∂ W ∂ H (cid:19) T . (2.2)The energy density ǫ , the entropy density s , and the spontaneous magnetization M satisfy the basic thermodynamic relation W = ǫ − s T − M H , (2.3)with the first law of thermodynamics d W = − s dT − M d H . (2.4)For convenience, we introduce a reduced temperature tt ≡ T − T c T c , (2.5)where T c is the critical temperature of the second order phase transition. For a standardferromagnetic phase transition t < W ( t, H ) < , M ( t, H = 0) = 0 ,t > W ( t, H ) > , M ( t, H = 0) = 0 . (2.6)At a second order phase transition the first derivatives of W are continuous while thehigher derivatives are not. Under the static scaling hypothesis we have W ( t, H ) = λ − p W ( λ y T t, λ y H H ) , (2.7)for the free energy, and ˜ G ( ~q, t, H ) = λ y H − p ˜ G ( λ~q, λ y T t, λ y H H ) , (2.8)for the Fourier transform of the equilibrium two-point correlation function of the mag-netization G ( ~r ) = hM ( ~r ) M ( ~ i ∝ ∂ W ∂ H ( ~r ) ∂ H ( ~ . (2.9)The static critical exponents { α, β, γ, δ, ν, η } are defined as c H = − T (cid:18) ∂ W ∂T (cid:19) H ∝ | t | − α , M ∝ | t | β ,χ T = (cid:18) ∂ M ∂ H (cid:19) T ∝ | t | − γ , M ( t = 0) ∝ |H| /δ , (2.10) Some of the holographic phase transitions we discuss are “not standard”, see [24]. G ( ~r ) ∝ e −| ~r | /ξ , t = 0 | ~r | − p +2 − η , t = 0 , with ξ ∝ | t | − ν , (2.11)where c H is the specific heat, χ T is the magnetic susceptibility, and ξ is the correla-tion length. Given (2.4) and the scaling hypothesis (2.7) and (2.8), the two criticalexponents { y T , y H } determine the critical exponents (2.10) which identify the standardstatic universality classes: α = 2 − py T , β = p − y H y T , γ = 2 y H − py T ,δ = y H p − y H , ν = 1 y T , η = p − y H + 2 . (2.12)Note that (2.12) implies the following scaling relations α + 2 β + γ = 2 , γ = β ( δ −
1) = ν (2 − η ) , − α = νp . (2.13)The scaling relations (2.13) can be violated whenever the single-scale hypotheses (2.7)and (2.8) break down. In what follows we assume that this is not the case in ourmodels.A finer classification of the universality classes of systems undergoing second orderphase transition occurs once the dynamical properties of the system (such as hydro-dynamics) are under consideration [27]. Here, the main assumption is that a systemperturbed away from the equilibrium will relax with a characteristic time scale τ q ,which diverges near the phase transition with a dynamical critical exponent z : τ q ∝ ξ z ∝ | t | − νz . (2.14)Correspondingly, the near-equilibrium scaling of the magnetization correlation function(2.8) is modified according to˜ G ( ω, ~q, t, H ) = λ y H − d + z ˜ G ( λ z ω, λ~q, λ y T t, λ y H H ) . (2.15)In this paper we will be interested in the hydrodynamic aspects of the critical phe-nomena, specifically in the scaling of the bulk viscosity at the transition. Althoughsecond order transitions imply spatial scale invariance (2.11), the latter does not pre-clude a non-zero bulk viscosity which necessitates the space-time scale invariance, see(1.4). A large bulk viscosity at the transition signals strong coupling between the5ilatational modes of the system and its internal degrees of freedom. This is a veryinteresting phenomena given that hydrodynamics is often regarded as an effective low-energy description of a system. Without going into details we now mention threeproposals (Models A,B,C) for the scaling of the bulk viscosity at criticality .(Model A): In [9] the authors proposed that the critical behavior of the bulk viscosityis governed by the critical exponent of the specific heat ζ ∝ c H ∝ | t | − α . (2.16)(Model B): In the quasi-particle model under the relaxation time approximation thebulk viscosity was argued to scale as [10] ζ ∝ | t | α +4 β − . (2.17)(Model C): In [11] Onuki argued that the bulk viscosity near the gas-liquid criticalpoint (model H in classification of [27]) scales as ζ ∝ | t | − zν + α . (2.18)Below, we discuss second order phase transitions in holographic models of gauge/gravitycorrespondence. We show that only the scaling (2.18) passes these holographic tests. A second order phase transition in a holographic model of gauge/gravity correspon-dence was first reported in [28]. This phase transition is associated with the sponta-neous breaking of U (1) symmetry, and as a result the hydrodynamics of the modelmust include a long-range Goldstone boson mode. Another type of a holographic sec-ond order phase transition was discussed in [29]. The phase transitions in [28] and [29]occur both at finite temperature and finite chemical potential.We discuss bulk viscosity in second order phase transitions at finite density in [2].Here, we focus on phase transitions at vanishing chemical potential for the conservedcharges. An example of such transition was reported in [24] (we review the main Here we indicate only the scaling of the potentially singular part of the bulk viscosity. A finitecontribution is always implicit. .03515 0.03520 0.03525 - PSfrag replacements c s ρ m b T PSfrag replacements c s ρ m b T Figure 1: (Colour online) The speed of sound c s (left plot) and the reduced temperature m b T (right plot) of the strongly coupled N = 2 ∗ plasma with m f = 0 and m b = 0 asa function of the dual gravitation parameter ρ . We identify the blue curve with the”ordered” phase and the red curve with the ”disordered” phase, see (2.1).features of this transition in section 3.3). We show that both the N = 2 ∗ gauge theoryand the cascading gauge theory plasma undergo the second order phase transitionswhich are in the same static universality class. The latter universality class includesthe model of [29], but is different from the universality class of the model [24]. Thestatic universality classes of the holographic models we discuss below are not of themean-field type since the critical exponent η (see (2.11)) is nonzero. N = 2 ∗ plasma The holographic renormalization and the thermodynamics of the N = 2 ∗ gauge theoryplasma in the planar limit and at (infinitely) large ’t Hooft coupling was discussed pre-viously [30, 31, 12] . This gauge theory is obtained by giving a mass to one of the three N = 2 hypermultiplets of N = 4 SU ( N ) supersymmetric Yang-Mills theory. At finitetemperature the supersymmetry is broken, and one can study the theory with differ-ent masses for bosonic and fermionic components of the massive hypermultiplet. In[12] the different mass deformations were called the “bosonic” m b , and the “fermionic” m f . For the range of parameters ( T, m b , m f ) studied in [12, 13] the N = 2 ∗ plasma isalways in deconfined phase . It was further shown that whenever m f < m b the theory Thermodynamics of weakly coupled N = 2 SU (2) Yang-Mills theory was recently discussed in[32]. An interesting phase transition in this model was conjectured in [30], see also [32]. .03515 0.03520 0.03525 - - - - - PSfrag replacements Ω / (cid:0) π N T (cid:1) ρ m b T - - - - - PSfrag replacements Ω / (cid:0) π N T (cid:1) ρ m b T Figure 2: (Colour online) Free energy densities Ω o of the “ordered” phase (blue curves)and Ω d of the “disordered” phase (red curves) as a function of ρ (left plot) and m b T (right plot) of the N = 2 ∗ plasma with m f = 0.undergoes a phase transition with the vanishing speed of sound. Although the criticaltemperature T c of the transition depends on the ratio m f m b , the general thermodynamicand the hydrodynamic features of the model are the same [25]. Thus, without theloss of generality we consider the mass deformation m f = 0 and m b = 0. Here, thetransition occurs at [12] m b T c ≈ . . (3.1)We now identify this phase transition as a second order phase transition with the staticcritical exponents ( α, β, γ, δ, ν, η ) = (cid:18) , , , , , (cid:19) . (3.2)It is convenient to present the thermodynamic data as a function of the dual grav-itational parameter ρ . For T ≫ m b we have [30] ρ = √ π (cid:16) m b T (cid:17) + O (cid:18) m b T (cid:19) , (3.3)for all other temperatures see section 3.3 of [12]. The left plot on Fig. 1 representsthe square of the speed of sound c s as a function of ρ while the right plot on Fig. 1represents the reduced temperature m b T as a function of ρ . The critical — the lowest —temperature corresponds to the maximum on the plot. The plots on Fig. 2 represent the(reduced) free energy of the system Ω as a function of ρ (left plot) and the reducedtemperature (right plot). For a given temperature T > T c there are two phases of N = 2 ∗ plasma denoted by a blue/red dot. From Fig. 2 it is clear that the transition8etween the two phases is a continuous one. The “red” phase is perturbatively unstableas it has c s (cid:12)(cid:12)(cid:12)(cid:12) red < W = Ω o − Ω d = Ω blue − Ω red < , (3.4)Note that in this system, contrary to the standard ferromagnet (2.6), the dimensionlesstemperature, defined by (2.5), t > ρ c the critical value of ρ , i.e. , m b T (cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ c = m b T c . (3.5)Introducing ∆ ρ = ρ − ρ c , (3.6)from Fig. 1 it is clear that t ∝ (∆ ρ ) , c s (cid:12)(cid:12)(cid:12)(cid:12) blue ∝ (cid:0) − c s (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) red ∝ | ∆ ρ | ∝ t / . (3.7)Thus, c H = − T (cid:18) ∂ W ∂T (cid:19) = sc s (cid:12)(cid:12)(cid:12)(cid:12) bluered ∝ c − s (cid:12)(cid:12)(cid:12)(cid:12) bluered ∝ t − / , (3.8)where we used the fact that the entropy density s is continuous across the phasetransition. Comparing (3.8) and (2.10) we determine the critical exponent α [13] α = 12 . (3.9)To determine the critical exponent β we need to identify the control parameter corre-sponding to the external magnetic field H of the effective ferromagnet of section 2. Wepropose to identify H = m b . (3.10)Given (3.1), it follows from (3.7) that ∂ H ∝ − ∂ t ∝ − ρ ∂ ∆ ρ . (3.11)The left plot on Fig. 2 implies that W = o (∆ ρ ) , (3.12)9 - - - PSfrag replacements π
81 ΩΛ T Λ k s - - - - PSfrag replacements π
81 ΩΛ T Λ k s Figure 3: (Colour online) The reduced temperature T Λ (left plot) and the free energydensities Ω o of the “ordered” phase (blue curve, right plot) and Ω d of the “disordered”phase (red curve, right plot) , see (2.1), of the strongly coupled cascading plasma as afunction of the dual gravitational parameter k s .at the very least. Actually, the best fit to Ω shows that W ∝ −| ∆ ρ | . (3.13)Thus, using (3.11) and (3.13), M = − (cid:18) ∂ W ∂ H (cid:19) ∝ ρ ∂ ∆ ρ W ∝ −| ∆ ρ | ∝ − t / , (3.14)Comparing (3.14) and (2.10) we conclude that β = 12 . (3.15)The rest of the critical exponents in (3.2) follow from the scaling relations (2.13).The second order phase transition in N = 2 ∗ plasma we just exhibit is remarkablysimilar to the second order phase transition in N = 4 supersymmetric Yang-Millsplasma with a single U (1) ⊂ SU (4) R R-symmetry chemical potential discussed in [29].In [29] this chemical potential was identified with the external magnetic field H of theeffective ferromagnet of section 2. The holographic renormalization and the thermodynamics of the cascading gauge the-ory plasma in the planar limit and at (infinitely) large ’t Hooft coupling was discussed This is not surprising since Ω on the left plot of Fig. 2 can not be an even function of ∆ ρ . N = 1 supersymmetric SU ( K + P ) × SU ( K ) gauge theory where the effectivenumber of colours K is not constant along the renormalization group flow, but changeswith energy according to [20, 34, 35] K = K ( µ ) ∼ P ln µ Λ , (3.16)at least when the energy scale µ is much larger than the strong coupling scale Λ of thegauge theory. At zero temperature, and when K ( µ ) is a multiple of P , cascading gaugetheory confines in the infrared with the spontaneous breaking of chiral symmetry. Thehigh temperature phase of the theory is expected to be that of the deconfined chirallysymmetric plasma [20]. In was shown in [23] that at T = T confinement = 0 . T c = 0 . × T confinement < T < T confinement , (3.18)the deconfined phase of the cascading plasma, although non-perturbatively unstabledue to the nucleation of bubbles of the confined phase, remains perturbatively (andthermodynamically) stable [14]. In the vicinity of T c the thermodynamics of the cas-cading gauge theory plasma is identical to that of the N = 2 ∗ plasma. Fig. 3 presentsthe reduced temperature T Λ (left plot) and the free energy density of the cascadingplasma as a function of the dual gravitational parameter k s . We use the same colourcoding as in Fig. 1 and Fig. 2: the “blue” dots indicate stable (ordered) phase and the”red” dots indicate unstable (disordered) phase. As in the case of N = 2 ∗ plasma, inthe ordered (disordered) phase the square of the speed of sound is positive (negative)[14]. Once we identify the external magnetic field H of the effective ferromagnet ofsection 2 as H = Λ , (3.19)we can literally repeat the arguments of section 3.1 and arrive at the same set of staticcritical exponents (3.2). Following Gubser’s suggestion [36], we constructed an exotic model of the second or-der phase transition in d = 3 at finite temperature and zero chemical potentials in11 .90 0.95 1.00 1.05 - - - - - - PSfrag replacements Ω( πT ) T c T hO i i > PSfrag replacements Ω( πT ) T c T hO i i > Figure 4: (Colour online) The free energy densities Ω of the “ordered” phase (redcurve, left plot) and Ω d of the “disordered” phase (purple curve, left plot) as a functionof the reduced temperature TT c in gauge theory plasma dual to holographic RG flow in[24]. The right plot represents the square of hO i i (which we use an an order parameterfor the transition) as a function of reduced temperature. The dashed green line is alinear fit to hO i i .[24]. Specifically, we considered relativistic conformal field theory in 2+1 dimensions,deformed by a relevant operator O r : H CF T → ˜ H = H CF T + λ r O r . (3.20)Such a deformation softly breaks the scale invariance and induces the renormalizationgroup flow. We further assumed that the deformed theory ˜ H has an irrelevant op-erator O i that mixes along the RG flow with O r . In the explicit holographic modelrealizing this scenario [24], the irrelevant operator O i developed a vacuum expectationvalue in the high temperature phase, i.e. , for T > T c , spontaneously breaking a dis-crete Z symmetry of the model. The unusual part of this phase transition was thatthe symmetry broken phase occurs at high temperatures (rather then the low temper-atures) and that the broken phase (although perturbatively and thermodynamicallystable) has higher free energy density than the unbroken phase with hO i i = 0. Thevarious high temperature phases which spontaneously break Z symmetry differ in theholographic wavefunction of the condensate and their free energies [24]. Without theloss of generality we focus here on the model with dim[ O r ] = 2, and the broken phaseof the lowest free energy. Fig. 4 represents the energy densities of the ordered (redline, left plot) and the disordered (purple line, left plot) phases as a function of thereduced temperature TT c . The expectation value of O i is the order parameter of the12 .995 1.000 1.005 1.0100.9900.9951.0001.0051.010 PSfrag replacements 2 c s T c T Figure 5: (Colour online) The speed of sound c s in gauge theory plasma dual to holo-graphic RG flow in [24] as a function of the reduced temperature T c T .phase transition — the spontaneous magnetization M of the effective ferromagnet insection 2. An excellent linear fit (the dashed green line) to hO i i (see the right plot onFig. 4) implies that in this model M ≡ hO i i ∝ | t | / , (3.21)resulting in the critical exponent β = 12 . (3.22)From Fig. 5 it is clear that the speed of sound is finite at the transition , thus c H ∝ c − s ∝ | t | , (3.23)resulting in the critical exponent α = 0 . (3.24)The remaining critical exponents can be obtained from the scaling relations (2.13):( α, β, γ, δ, ν, η ) = (cid:18) , , , , , (cid:19) . (3.25)Note that the universality classes (3.2) and (3.25) are different. The technique for computing the bulk viscosity in non-conformal holographic modelsfrom the dispersion relation of the sound waves was developed in [8] . It was used to Notice that in the disordered phase the speed of sound violates the bound proposed in [37, 38]. A much simpler computational framework was recently proposed in [39, 40]. N = 2 ∗ plasma in [8, 13, 25], and the bulk viscosity ofthe cascading gauge theory plasma in [26, 14]. It is straightforward to generalize themethod of [8] to the computation of the bulk viscosity of the gauge theory plasma dualto the holographic RG flow [24]. The details of the latter analysis will appear elsewhere[41], and here we report only the results.In the rest of this section for each of the holographic models with a second or-der phase transition discussed above we analyze the scaling of the bulk viscosity atcriticality in the framework of Models A,B,C — (2.16)-(2.18). N = 2 ∗ plasma The main result of [13, 25] was that the ratio of the bulk-to-shear viscosities in N = 2 ∗ plasma remains finite across the second order phase transition : ζη ∝ | t | . (4.1)Since in holographic phase transitions under the consideration both the entropy densityand the shear viscosity are finite, we conclude from (4.1) that the bulk viscosity staysfinite. Thus:Model A is inconsistent with holographic analysis as it predicts divergent bulk vis-cosity, ζ ∝ | t | − / ;Model B does not contradict our holographic analysis as it predicts that ζ singular ∝| t | / ;Model C agrees with holographic analysis provided the dynamical exponent z is z = 1 . (4.2) It was argued in [14] that the bulk viscosity of the cascading gauge theory plasmaremains finite across the second order phase transition much like in the case of N = 2 ∗ plasma. Thus, cascading gauge theory plasma appears to share not only the staticuniversality class with N = 2 ∗ model, but also the dynamical one — in particular, thecritical exponent z is given by (4.2). 14 PSfrag replacements ζη ln (cid:16) ζη (cid:17) (cid:0) − c s (cid:1) ln |hO i i| Figure 6: (Colour online) The ratio of bulk-to-shear viscosities in gauge theory plasmadual to the RG flow in [24]. The red curve corresponds to the “ordered” phase of thetheory, and the purple one to the “disordered” phase. The dashed blue line indicatesthe bulk viscosity bound proposed in [13]. The dashed green line corresponds to thehigh-temperature approximation to the viscosity ratio given by (4.4).
Fig. 6 represents the ratio of the bulk-to-shear viscosities of the gauge theory plasmadual to the RG flow in [24]. The red curve corresponds to the ”ordered” (symmetric)phase that the purple curve corresponds to the “disordered” (broken) phase. Thedashed blue line represents the bulk viscosity bound ζη ≥ (cid:18) − c s (cid:19) , (4.3)proposed in [13]. The dashed green line represents the bulk-to-shear viscosity ratio fora symmetric phase at high temperatures: ζη (cid:12)(cid:12)(cid:12)(cid:12) ordered = 2 π √ (cid:18) − c s (cid:19) + O (cid:18) − c s (cid:19) ! . (4.4)Notice that the bulk viscosity bound is satisfied both in the symmetric and in thebroken phases. In fact, in the disorder phase (purple curve) the bound is satisfiedtrivially as here c s > , see also Fig. 5.Fig. 6 shows a rapid rise of the bulk-to-shear viscosity ratio in the disordered (purplecurve) phase as one approaches the transition. In fact, a detailed analysis presented in The second-order transport coefficient is nonetheless divergent [42]. - - - - - PSfrag replacements ζη ln (cid:16) ζη (cid:17)(cid:0) − c s (cid:1) ln |hO i i| Figure 7: (Colour online) The ratio of bulk-to-shear viscosities in gauge theory plasmadual to the RG flow in [24] in the vicinity of the second order phase transition as afunction of the order parameter. The dashed green line represents the linear fit to thelog-log data plot with the slope of ( − . (cid:16) ζη (cid:17) versus ln |hO i i| . A dashed green line given by y = − . x − . ζη (cid:12)(cid:12)(cid:12)(cid:12) disordered ∝ |hO i i| − ∝ | t | − , (4.6)where we used (3.21). From (4.6) we conclude:once again, Model A is inconsistent with our holographic analysis as it predicts afinite bulk viscosity at the transition, ζ ∝ | t | ;Model B is inconsistent as well, as it predicts ζ singular ∝ | t | ;Model C agrees with holographic analysis provided the dynamical exponent z is z = 1 . (4.7)Rather intriguingly, even though the static universality classes of the N = 2 ∗ (and thecascading) gauge theory plasma and the exotic plasma discussed in this section aredifferent, when the bulk viscosity is interpreted in the framework of Model C, bothclasses have the same dynamical critical exponent z .16 Conclusion
In this paper we studied bulk viscosity at criticality of the second order phase tran-sition in non-conformal models of gauge theory/string theory correspondence at finitetemperature, but at zero chemical potential for the conserved charges. We identifiedthe second order phase transitions in N = 2 ∗ and the cascading gauge theory plasmaand showed that the two models are in the same static universality class. Somewhatsurprisingly, this universality class is not of the mean-field theory type. Incidentally,the same universality class contains a second order phase transition in N = 4 su-persymmetric Yang-Mills plasma with a single U (1) ⊂ SU (4) R R-symmetry chemicalpotential discussed in [29]. The static universality class of the N = 2 ∗ plasma is differ-ent from the static universality class of the gauge theory dual to the 3+1 dimensionalasymptotically AdS RG flow discussed in [24]. Having different static universalityclasses in our holographic laboratory allowed us to test different proposals for the scal-ing of bulk viscosity in the vicinity of the second order phase transition. We foundthat neither the model of [9] or [10] conforms to the tests. If dynamical scaling of bulkviscosity proposed by Onuki [11] is correct, then all our holographic models must havethe same dynamical critical exponent z = 1.In the future, it is important to explicitly verify the scaling relations (2.13) in ourholographic models. It would be extremely interesting to subject Onuki’s theory [11] toa direct holographic test by independently computing the dynamical critical exponent z from the scaling relation (2.14). Finally, it is interesting to understand the dynamicaluniversality class of the holographic models. We already know that in such theoriesthe ratio of shear viscosity to the entropy density is universal. Is it possible that theuniversality of the holographic transport at critically is more robust? Acknowledgments
We would like to thank Pavel Kovtun for valuable discussions and correspondence.Research at Perimeter Institute is supported by the Government of Canada throughIndustry Canada and by the Province of Ontario through the Ministry of Research& Innovation. AB gratefully acknowledges further support by an NSERC Discoverygrant and support through the Early Researcher Award program by the Province ofOntario. 17 eferences [1] J. M. Maldacena, “The large N limit of superconformal field theories and super-gravity,” Adv. Theor. Math. Phys. , 231 (1998) [Int. J. Theor. Phys. , 1113(1999)] [arXiv:hep-th/9711200].[2] A. Buchel and C. Pagnutti, to appear.[3] P. K. Kovtun and A. O. Starinets, Phys. Rev. D , 086009 (2005) [arXiv:hep-th/0506184].[4] A. Kogan and H. Meyer, J. Low Temp. Phys. , 899 (1998)[5] A. Buchel and J. T. Liu, Phys. Rev. Lett. , 090602 (2004) [arXiv:hep-th/0311175].[6] P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. Lett. , 111601 (2005)[arXiv:hep-th/0405231].[7] A. Buchel, Phys. Lett. B , 392 (2005) [arXiv:hep-th/0408095].[8] P. Benincasa, A. Buchel and A. O. Starinets, Nucl. Phys. B , 160 (2006)[arXiv:hep-th/0507026].[9] F. Karsch, D. Kharzeev and K. Tuchin, Phys. Lett. B , 217 (2008)[arXiv:0711.0914 [hep-ph]].[10] C. Sasaki and K. Redlich, Phys. Rev. C , 055207 (2009) [arXiv:0806.4745 [hep-ph]].[11] A. Onuki, Phys. Rev. E
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