aa r X i v : . [ h e p - t h ] J a n UWO-TH-10/8
Correlated stability conjecture revisited
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
Correlated stability conjecture (CSC) proposed by Gubser and Mitra [1, 2] linked thethermodynamic and classical (in)stabilities of black branes. The classical instabilities,whenever occurring, were conjectured to arise as Gregory-Laflamme (GL) instabilitiesof translationally invariant horizons. In [3] it was shown that the thermodynamic insta-bilities, specifically the negative specific heat, indeed result in the instabilities in the hy-drodynamic spectrum of holographically dual plasma excitations. A counter-exampleof CSC was presented in the context of black branes with scalar hair undergoing asecond-order phase transition [4]. In this paper we discuss a related counter-exampleof CSC conjecture, where a thermodynamically stable translationally invariant horizonhas a genuine tachyonic instability. We study the spectrum of quasinormal excitationsof a black brane undergoing a continuous phase transition, and explicitly identify theinstability. We compute the critical exponents of the critical momenta and the fre-quency of the unstable fluctuations and identify the dynamical critical exponent of themodel.October 2010 ontents
Black holes with translationally invariant horizons (also referred to as black branes)are ubiquitous in string theory [5]. They are particularly important in the context ofholographic gauge theory/string theory correspondence [6], where they describe thethermal equilibrium states of the dual gauge theory plasma. In case of d = p + 1space-time dimensional non-conformal gauge theory plasma, the dual black branesof the ( d + 1)-dimensional supergravity are rather complicated, and have multiplescalar hair, see for example [7, 8]. Being dual to a finite temperature gauge theoryplasma, these black branes have nonvanishing Hawking temperature and necessarilyare non-supersymmetric. Thus, their classical stability is not assured. The linearstability analysis of black branes is complicated by mixing the scalar hair and the metricfluctuations. A much simpler exercise is to compute the thermodynamic potentials ofthe black branes, i.e., the free energy density, the energy density, etc., and analyze athermodynamic stability of the resulting potentials. It was a very welcome conjectureby Gubser and Mitra [1, 2] — Correlated Stability Conjecture — that identified thethermodynamic stabilities or instabilities of translationally invariant horizons with theclassical stabilities or instabilities correspondingly. Moreover, the classical instabilities,whenever occurring, were conjectured to appear as Gregory-Laflamme instabilities [9,10]. Recall that GL instabilities of the translationally invariant horizons are linearizedfluctuations δ Φ ∝ e − iωt + i~k · ~x with a dispersion relation ( w , q ) such thatIm ( w ) (cid:12)(cid:12)(cid:12)(cid:12) q < q c > , (1.1)where w ≡ ω πT , q ≡ | ~k | πT , (1.2)and T is the temperature. Note that we always assume that Im ( q ) = 0, i.e., fluctuationamplitudes do not grow exponentially along the translational directions of a horizon.2 standard claim in classical thermodynamics is that a system is thermodynam-ically stable if the Hessian H E s,Q A of the energy density E = E ( s, Q A ) with respect tothe entropy density s and charges Q A ≡ { Q , · · · Q n } , i.e., H E s,Q A ≡ ∂ E ∂s ∂ E ∂s∂Q B ∂ E ∂Q A ∂s ∂ E ∂Q A ∂Q B ! , (1.3)does not have negative eigenvalues. In the simplest case n = 0, i.e., no conservedcharges, the thermodynamic stability implies that0 < ∂ E ∂s = Tc v , (1.4)that is the specific heat c v is positive. In the context of gauge theory/string theorycorrespondence black holes with translationary invariant horizons in asymptoticallyanti-de-Sitter space-time are dual (equivalent) to equilibrium thermal states of certainstrongly coupled systems. Thus, the above thermodynamic stability criteria should bedirectly applicable to black branes as well. CSC asserts that it is only when the Hessian(1.3) for a given black brane geometry is positive, the spectrum of on-shell excitationsin this background geometry is free from tachyons.CSC is trivially true in a holographic context, whenever the specific heat of thecorresponding black brane is negative [3]. Indeed, in the absence of conserved charges,the speed of hydrodynamic (sound channel) modes in the dual plasma is c s = sc v , (1.5)where s , E are the entropy and the energy densities, and c v c v = (cid:18) ∂ E ∂T (cid:19) v , (1.6)is the specific heat. Thus whenever c v <
0, the sound modes in plasma, and corre-spondingly the dual quasinormal modes in the gravitational background, are unstable .The situation is more complicated in holographic examples in the presence of con-served charges. Consider a strongly coupled N = 4 SU ( N ) supersymmetric Yang-Millstheory in the planar limit at finite temperature T and a chemical potential µ for a sin-gle global U (1) ⊂ SO (6) R-symmetry. The dual gravitational geometry is that of the Assuming that the temperature is positive. Such instability occurs in N = 2 ∗ holographic plasma [7, 11]. AdS black brane [12–14]. For any temperature largerthan the critical one, T c = µ √ /π , there are two phases of the plasma. Althoughone of the phases has a negative specific heat, the sound modes in plasma are alwaysstable [15]: c v = 4 π N T (1 + κ )(3 − κ )(2 + κ ) (2 − κ ) , w = ± √ q − i κ + 22 κ + 2 q + O ( q ) , (1.7)where 2 πTµ = √ κ + 2 √ κ , (1.8)so that given T > T c there are two possible values of κ distinguishing two differentphases: one with κ < κ > c s = (cid:18) ( E + P ) ∂ ( P, ρ ) ∂ ( T, µ ) + ρ ∂ ( E , P ) ∂ ( T, µ ) (cid:19)(cid:18) ( E + P ) ∂ ( E , ρ ) ∂ ( T, µ ) (cid:19) − , (1.9)where P is the pressure. Remarkably, there is no contradiction with CSC conjectureas well! Indeed, it was shown in [15] that a two-point retarded correlation function ofthe charge density fluctuations has a pole (and thus the SYM plasma has a physicalexcitation with this dispersion relation) at w = A i ( κ − q + O ( q ) , A = 0 . , (1.10)so that a phase with κ > q . We emphasize that even though the gravitational instabilityis in the quasinormal sound channel mode, it is not a hydrodynamic sound, which ischaracterized by the dispersion relation w ∝ q , see (1.7).So far, the examples we discussed provide support for CSC conjecture. It is believedhowever, that CSC conjecture is false [4] . The idea advocated in [4] is to considergravitational backgrounds with translationally invariant horizon that are dual to gaugetheory plasmas undergoing a continuous phase transition. In the vicinity of the phasetransition the condensate does not noticeably modify the thermodynamics, and thusshould not affect the thermodynamic stability of the system. On the other hand, the See also [16]. i.e., the GL instability(1.1).In this paper we discuss an explicit example of the physical scenario suggested in [4]from the perspective of the quasinormal spectrum. We identify the GL tachyon andcompute the critical exponents for the critical momentum and the frequency of the cor-responding fluctuations. Our model is the ’exotic hairy black hole’ introduced in [17].The dispersion relation for the sound waves in this model was studied in [18]. Muchlike in case of the R-charged N = 4 SYM plasma, there are no instabilities in hydro-dynamic modes. We review the thermodynamics and the hydrodynamics of the modelin section 2. In section 3 we identify non-hydrodynamic quasinormal excitations in thesound channel of the gravitational background with the dispersion features identicalto those of the GL instabilities (1.1). The GL tachyons exist in the symmetry brokenphase only, and disappear from the spectrum at the second-order phase transition. Wecompute critical exponents associated with vanishing of q c and w c ≡ w ( q = 0) at thetransition. Finally, we comment on the dynamical universality class of the model. The model considered here is not a string theory derived example of gauge/gravity cor-respondence — rather, it should be viewed as a phenomenological model of holography.Of course, whether or not the model can be embedded into the full string theory isirrelevant in so far as one is interested in CSC. The holography here is simply a usefultool to think physically about mathematical problem of (in)stabilities of translationallyinvariant horizons.Following [19], consider a 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 . (2.1)Such a deformation softly breaks the scale invariance and induces the renormalizationgroup flow. We further assume that the deformed theory ˜ H has an irrelevant operator O i that mixes along the RG flow with O r . The explicit holographic model realizing5 .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 1: (Colour online) The free energy densities Ω of the symmetric phase (redcurve, left plot) and Ω d of the symmetry broken phase (purple curve, left plot) asa function of the reduced temperature TT c in the gauge theory plasma dual to theholographic RG flow in [17]. The right plot represents the square of hO i i (which weuse as an order parameter for the transition) as a function of the reduced temperature.The dashed green line is a linear fit to hO i i .this scenario was discussed in [17]: S = S CF T + S r + S i = 12 κ Z dx √− γ [ L CF T + L r + L i ] , (2.2) L CF T = R + 6 , L r = −
12 ( ∇ φ ) + φ , L i = −
12 ( ∇ χ ) − χ − gφ χ , (2.3)where we split the action into (a holographic dual to) a CFT part S CF T ; its deformationby a relevant operator O r ; and a sector S i involving an irrelevant operator O i alongwith its mixing with O r under the RG dynamics. The four dimensional gravitationalconstant κ is related to the central charge c of the UV fixed point as c = 192 κ . (2.4)In our case the scaling dimension of O r is 2 and the scaling dimension of O i is 4. Inorder to have asymptotically AdS solutions, we assume that only the normalizablemode of O i is nonzero near the boundary. Finally, we assume that g < Z × Z discrete symmetry that acts as a parity trans-formation on the scalar fields φ and χ . The discrete symmetry φ → − φ is softly brokenby a relevant deformation of the AdS CFT; while the χ → − χ symmetry is broken6 .995 1.000 1.005 1.0100.9900.9951.0001.0051.010 PSfrag replacements 2 c s T c Tζη
PSfrag replacements2 c s T c Tζη
Figure 2: (Colour online) The speed of sound c s and the ratio of bulk-to-shear viscositiesin the gauge theory plasma dual to the holographic RG flow in [17] as a function ofthe reduced temperature T c T .spontaneously. Spontaneous breaking of the latter symmetry is caused by the develop-ment of the condensate for the irrelevant operator O i . The unusual part of this phasetransition is that the symmetry broken phase occurs at high temperatures (rather thanat low temperatures) and that the broken phase has a higher free energy density thanthe unbroken phase with hO i i = 0. The free energy densities of the Z -symmetricphase ( hO i i = 0) and the symmetry broken phase ( hO i i 6 = 0) are presented in Figure1. The hydrodynamic sound channel quasinormal modes of both phases were studiedin details in [18]. There is a jump discontinuity at the transition in the speed of sound;with the speed of sound in the symmetry broken phase being ∼
1% higher than inthe symmetric phase. Both the shear and the bulk viscosities in the two phases arepositive. Figure 2 collects the results for the speed of sound and the ratio of bulk-to-shear viscosities in the exotic plasma. In the symmetry broken phase the bulk viscositydiverges in the vicinity of the phase transition [18] ζη (cid:12)(cid:12)(cid:12)(cid:12) broken ∝ |hO i i| − ∝ ( T − T c ) − . (2.5)Note that since c s >
0, the specific heat (see (1.5)) of both the symmetric and thesymmetry broken phases is positive — the two phases are thermodynamically stable . The shear viscosity is continuous and finite at the transition: η/s = 1 / π [20–22]. We also verified that the susceptibility ∂ E ∂ Λ (cid:12)(cid:12)(cid:12)(cid:12) s = constant is positive — Λ is the scale introduced bythe relevant operator O r , see [17] for more details. .2 0.4 0.6 0.8 1.02468 PSfrag replacements q T c T PSfrag replacements q T c T Figure 3: (Colour online) Dispersion relation of quasinormal modes in the symme-try broken (classically unstable) phase of the exotic black branes. The red line (leftplot) represents the modes at the threshold of GL instability: ( w , q ) = (0 , q c ). Thegreen/blue dots (left plot) are the stable/unstable quasinormal modes with w = − . i and w = 0 . i correspondingly. The right plot presents the modes at the threshold ofinstability in the vicinity of the critical point. The dashed green line (right plot) is thebest quadratic in (cid:0) − T c T (cid:1) fit to data.The dispersion relation for the sound mode takes the form w = ± c s q − i (cid:18) ζη (cid:19) q + O ( q ) , (2.6)implying that the sound modes are stable as well.Since the free energy of the symmetry broken phase is bigger, this phase is notthermodynamically preferable and one expects a classical instability driving the systemto a true (symmetric) ground state. Such instability is expected to be of GL type, see(1.1), and thus is not expected to show up in the hydrodynamic limit. In the nextsection we explicitly identify the Gregory-Laflamme instability, and thus prove thatCSC is false. Physical excitations in a gauge theory plasma are dual to quasinormal modes of acorresponding black brane solution [23]. For a translationally invariant horizon, awave-function of a generic background field fluctuation δ Φ takes form δ Φ = F ( r ) e − iωt + i~k · ~x , (3.1)8 .2 0.4 0.6 0.8 1.0 - - - PSfrag replacements i w c T c T - - - - - - - PSfrag replacements i w cT c T Figure 4: (Colour online) Frequency of the homogeneous and isotropic unstable quasi-normal modes ( w , q ) = ( w c ,
0) in the symmetry broken (classically unstable) phase ofthe exotic black branes. The dashed green line (right plot) is the best quadratic in (cid:0) − T c T (cid:1) fit to data in the vicinity of the critical point.where F is a radial wave function. Fluctuations of different fields will typically mix witheach other. It is convenient to decompose the fluctuations in irreducible representationswith respect to rotations about the ~k -axis. In the case of the exotic black branes, theGL instabilities are expected to arise as fluctuations in the symmetry-breaking (scalar)condensate. Thus, they appear in the scalar quasinormal mode channel. It is exactlythe same set of fluctuations which in the hydrodynamic limit ( w → q → wq → constant) describe the propagation of the sound waves in plasma. The techniquefor computing the scalar channel quasinormal modes in non-conformal holographicmodels was developed in [24]. It is straightforward to generalize the method of [24] tothe computation of the scalar channel quasinormal modes of the gauge theory plasmadual to the holographic RG flow [17]. The details of the latter analysis will appearelsewhere [25] and here we report only the results, focusing on the GL mode .The red solid line (left plot) in Figure 3 presents the dispersion relation for thequasinormal modes in the symmetry broken (classically unstable) phase at the thresh-old of instability: ( w , q ) (cid:12)(cid:12)(cid:12)(cid:12) threshold = (0 , q c ) , (3.2)as a function of the reduced temperature T c T . The modes with q > q c attenuate, i.e., they have Im ( w ) <
0, while those with q < q c represent a genuine GL instability — Sound waves were discussed in [18] and reviewed in the previous section. w ) >
0. The green dots on the left plot have w = − . i and the bluedots have w = 0 . i . The right plot on Figure 3 represents the dispersion relation of thequasinormal modes in the symmetry broken phase at the threshold of instability in thevicinity of the critical point. The green dashed line is the best quadratic in t ≡ (1 − T c T )fit to the data: q c (cid:12)(cid:12)(cid:12)(cid:12) fit = 3 . (4) × − + 24 . (3) t − . (1) t + O ( t ) . (3.3)The data suggests that q c ∝ ( T − T c ) , (3.4)in the vicinity of the critical point.The left plot in Figure 4 represents the unstable homogeneous and isotropic quasi-normal modes, i.e., GL instabilities with q = 0. The right plot shows the frequencydependence of these modes in the vicinity of the critical point. The green dashed lineis the best quadratic in t fit to the data: i w c (cid:12)(cid:12)(cid:12)(cid:12) fit = − . (8) × − − . (2) t + 102 . (9) t + O ( t ) . (3.5)The data suggests that w c ∝ i ( T − T c ) , (3.6)in the vicinity of the critical point.Notice that since w c ∝ i q c in the vicinity of the critical point, it is natural to identifythe dynamical critical exponent of the model with z = 2. The symmetries of our modelidentify it as ’model A’ according to classification of Hohenberg and Halperin (HH) [26],assuming a natural extension of the classification to classically unstable phases. Mean-field models (with vanishing anomalous static critical exponent) in the HH universalityclass A are predicted to have the dynamical critical exponent z model − A = 2, as the onewe obtained .In this section we explicitly identified the Gregory-Laflamme instabilities in thesymmetry broken phase of the exotic black branes. Since this phase is thermody-namically stable, our model presents a counter-example to the correlated stabilityconjecture. The second-order phase transition in exotic black branes is of the mean-field type [25]. cknowledgments We would like to thank Martin Kruczenski for valuable discussions. Research atPerimeter Institute is supported by the Government of Canada through IndustryCanada and by the Province of Ontario through the Ministry of Research & Inno-vation. AB gratefully acknowledges further support by an NSERC Discovery grantand support through the Early Researcher Award program by the Province of Ontario.
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