Quasinormal modes and the phase structure of strongly coupled matter

We investigate the poles of the retarded Green's functions of strongly coupled field theories exhibiting a variety of phase structures from a crossover up to different first order phase transitions. These theories are modeled by a dual gravitational description. The poles of the holographic Green's functions appear at the frequencies of the quasinormal modes of the dual black hole background. We focus on quantifying linearized level dynamical response of the system in the critical region of phase diagram. Generically non-hydrodynamic degrees of freedom are important for the low energy physics in the vicinity of a phase transition. For a model with linear confinement in the meson spectrum we find degeneracy of hydrodynamic and non-hydrodynamic modes close to the minimal black hole temperature, and we establish a region of temperatures with unstable non-hydrodynamic modes in a branch of black hole solutions.

A b s t r a c t : We investigate the poles of th e retarded G reen's functions of strongly coupled field theories exhibiting a variety of phase structures from a crossover up to different first order phase tran sitio n s. These theories are m odeled by a dual grav itatio n al description. The poles of the holographic G reen's functions appear at the frequencies of the quasinorm al modes of the dual black hole background. We focus on quantifying linearized level dynam ical response of the system in the critical region of phase diagram . Generically non-hydrodynam ic degrees of freedom are im p o rta n t for th e low energy physics in th e vicinity of a phase transition. For a model w ith linear confinement in the meson spectrum we find degeneracy of hydrodynam ic and non-hydrodynam ic modes close to the minimal black hole tem perature, a n d we establish a region of tem p e ra tu re s w ith u n stab le non-hydrodynam ic m odes in a branch of black hole solutions. K e y w o r d s: A dS-C FT Correspondence, Black Holes, Gauge-gravity correspondence, Holog raphy and quark-gluon plasm as A r X iv e P r in t : 1603.05950 Open Access, © The Authors. Article funded by SCOAP3. doi:10.1007/JHEP06(2016)047

In tro d u ctio n
It is almost twenty years since there has been discovered a rem arkable new relation between geom etry an d physics: w ithin th e A nti-de S itte r/C o n fo rm a l Field T heory (A d S /C F T ) correspondence [1] we can investigate th e dynam ics of strongly coupled q u a n tu m field theories by m eans of General R elativity m ethods. From purely academ ic studies this field of research evolved to address experim ental system s an exam ple being strongly interacting hadronic m a tte r [2]. In p articu lar, real tim e response of a th e rm a l equilibrium sta te has been quantified in the case of N = 4 super Yang-Mills theory by the m eans of the poles of th e reta rd e d G reen's function [3], which correspond to quasinorm al m odes (QNM ) in th e dual gravitational theory. W hile th e hydrodynam ic QNM s have been studied in different grav itatio n al theories dual to non-C FT cases (e.g. ref. [4,5]), initial steps towards extension were taken in ref. [6,7] where nonhydrodynam ic QNM 's of an external scalar field were considered in non-conformal field theories, which still adm it a gravitational dual description. Subsequent investigations -1 -

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include different m echanism s of scale generation [8], different relax atio n channels [9,10], baryon rich plasm a [11], and studies of non-relativistic system s [12]. This paper is an extended version of the letter [13] where we provide m any more details as well as extend the investigation to a model of an improved holographic QCD type which exhibits novel and interesting phenom ena. We concentrate on investigating linearized real tim e response of strongly coupled non-conform al field theories in th e vicinity of various types of phase transitions and phase structures. Thus the physical regime of interest in the present paper is quite distinct from the th e one of interest for 'early therm alization' which have been extensively studied w ithin th e A d S /C F T correspondence.
Firstly, we analyze all allowed channels of energy-m om entum tensor perturbations and corresponding two-point correlation functions. Secondly, we concentrate on th e phenom ena a p p earin g in th e vicinity of a n o n trivial phase s tru c tu re of various type: a crossover (m otivated by th e lattice QCD equations of state [14]), a 2nd order phase tran sitio n and a 1st order phase tra n sitio n . These cases are m odeled by choosing a p p ro p riate scalar field self-interaction p o tentials in a holographic gravity-scalar th eo ry used in [15]. A p a rt form this, we also analyze a p o ten tia l from a different fam ily of m odels, im proved holographic QCD (IH Q CD ), considered in [17,18]. In this case th e focus was on getting best possible contact w ith properties of QCD, in p articular asym ptotic freedom and colour confinement as well as obtaining a realistic value of th e bulk viscosity. D espite th e fact, th a t considered m odels have a ra th e r sim plistic construction, th e resulting near equilibrium response shows a variety of non-trivial phenomena. Some generic features consist of: (i) th e breakdow n of th e applicability of a hydrodynam ic description already at lower m om enta th an in the conformal case; (ii) in the cases w ith a first order phase tra n sitio n we find a generic m inim al tem p e ra tu re , T m , below which no u n stab le solution exists; (iii) whenever th ere exists a therm odynam ical instab ility th ere is a corresponding dynam ical instability present in the hydrodynam ic mode of the theory; (iv) the ultralocality property of non-hydrodynam ic modes, i.e., weak dependence on the m om entum scale.
T he n a tu re of th e d u al g rav itatio n al form ulation allows for a d etailed q u a n tita tiv e investigation of the above phenom ena as well as for accessing diverse physical scenarios. In particular, the first order phase transition appears in two different scenarios. T he first one is sim ilar to th e usual H aw king-Page tra n sitio n [19] in w hich th e two phases are a black hole geom etry and a th erm al gas geom etry [18]. In th e second one th e tra n sitio n appears betw een two black hole solutions [15]. This diversity is triggered by a different functional dependence of th e scalar field p o ten tia l in th e deep infrared (IR ) region, an d is reflected in th e corresponding Q N M spectru m . N evertheless th ere is a com m on aspect in b o th situations. We observe some specific dynam ical response of th e system for a characteristic tem p e ra tu re , Tch > Tm , in th e stable b ran ch of EoS. T h e d etails of th is effect d epend on th e case, b u t th e existence of Tch is generic for a first order phase transition. P a rtic u la rly in terestin g effects ap p e ar in IH Q C D m odel, which ad m its a first order phase transition between a black hole and a therm al gas [18]. First, for tem peratures in the range T m < T < T ch th e lowest lying ex citatio n m odes becom e purely im aginary for low m om enta, which leads to a ultralocality violation. Second, a t T = T m for m om enta higher th a n some threshold value the hydrodynam ic m ode and th e first non-hydrodynam ic mode

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is th u s far a rb itra ry an d k 5 is related to five dim ensional N ew ton constant by = A/8nG5. T he last te rm in (2.1) is th e s ta n d a rd G ibbons-H aw king b o u n d a ry contribution. These solutions are sim ilar to those studied in ref. [15,17]. Since our goal is to determ ine the QNM frequencies, it will be convenient to employ E ddington-Finkelstein coordinates, which have been proven useful in the case of the scalar field modes [6]. We will discuss this in a more detail in th e following section. W hereas we are in terested in asym ptotically AdS space-tim e geom etry, th e p o ten tial needs to have th e following small ó expansion Here, L is th e AdS radius, which we set it to one, L = 1, by th e freedom of th e choice of units. Such a gravity dual corresponds to relevant deform ations of the boundary conformal field theory (2.2) consider 2 < A < 4 which corresponds to relevant deform ations of th e C F T a n d satisfies th e B reitenlohner-Freedm an bound, m 2 > -4 [20,21].
T he A nsatz for solutions under considerations follows from th e assum ed sym m etries: tra n sla tio n invariance in th e M inkowski directions as well as SO(3) ro ta tio n sym m etry in th e spatial part. This leads to th e following form of th e line element: d s 2 = g ttd t2 + gxxdx2 + g " d r 2 + 2grfd r d t , (2.4) where all th e m etric coefficients appearing in (2.4) are functions of the radial coordinate r alone, as is th e scalar field ¢. This form of th e field A nsatz (determ ined so far only by the assum ed sym m etries) allows two gauge choices to be m ade. For th e purpose of com puting the quasinorm al modes it is very convenient to use the Eddington-Finkelstein gauge grr = 0. It is typically convenient also to im pose th e gauge choice gtr = 1, b u t for our purposes it tu rn s out to be very effective to use th e rem aining gauge freedom to set ę = r. We label th e m etric com ponents as In th e above coordinate system th e U V b o u n d a ry is a t r = 0, while th e IR region is th e lim it r ^ to. T he system of Einstein-scalar field equations where th e prim e denotes a derivative w ith respect to ¢.
In co n tra st to m eth o d s proposed in ref. [15] we solve th is coupled equations directly using th e sp ectral m eth o d [22] in th e N ew ton linearization algorithm . We are in terested in solutions possessing a horizon, which requires th a t th e blackening function h (r) should have a zero a t some r = r H : (2.13) A sym ptotically we require th a t our geom etry is th a t of th e AdS space-tim e.

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In tu rn , the speed of sound of th e system can be determ ined as T he m odels determ ined by th e p o tentials Vist and Vihqcd exhibit a first order phase tran sitio n s. In th e form er case th e tra n sitio n happens betw een two different black hole solutions, while in the latter the transition happens between a black hole and a horizon-less geom etry. In b o th of those cases one can determ ine th e tra n sitio n by evaluating th e F E difference according to formula (2.16) , if one knows the counter term s.1 In this com putation we follow an a ltern ativ e m eth o d of ref.
[17] a n d in teg rate th e th erm odynam ic relation, d F = -s d T , w ith p roperly chosen b o u n d a ry condition. We can achieve th is by first choosing some a rb itra ry reference tem p eratu re To and w rite

18)
JTo w here we assum e to be in one p a rtic u la r class of solutions. To evaluate th e in teg ratio n co n stan t, F (T o), we use th e fact th a t th e Free E nergy vanishes for th e zero horizon area geometry. In general the small horizon area limit of the black hole solutions corresponds to th e vacuum geom etry w ith "good singularity" in th e deep IR [25]. B y using th e relation of T an d s an d th e horizon rad iu s (2.14) we can evaluate th e Free E nergy w ith th e d a ta o b tain ed w ith m ethods outlined in th e previous subsection. T his am ounts to a generic form ula 2n dT F ( r u ) = --- The details of th e com putations along w ith th e corresponding plots and predictions for Tc will be given in th e corresponding sections of th e paper.

Q u asin orm al m od es
In this section we form ulate the problem of analyzing th e linear pertu rb atio n s around the equilibrium states in considered models. T h e first subsection contains equations of m otion and proper boundary conditions th a t need to be imposed. T he second subsection contains a short sum m ary of the results obtained w ith an emphasis on generic aspects. The detailed case by case discussion is a subject of th e rem aining p art of th e paper.
Clarification of this point can be found in the appendix A.

E q u a tio n s o f m o tio n a n d b o u n d a r y c o n d itio n s
The linear response of the system is analyzed by setting pertu rb atio n s w ith m om entum in a given direction and com puting poles of th e resulting G reen functions. In this section we form ulate th e equations and corresponding boundary conditions to present and discuss the results in the following p art of th e paper.
We consider pertu rb atio n s of th e background, obtained in th e previous section, in the following form O n th e basis of [3,4,6] we consider infinitesim al diffeom orphism tran sfo rm atio n s, x a ^ x a + £a, of th e form £a = £a (r)e -iwt+ikz, which act on the perturbations in a standard way, In th e above haa(r) = hxx(r) = h yy(r) are transverse m etric com ponents a n d we have factorized th e background from th e m etric p e rtu rb a tio n s in th e following way: htt(r) = h ( r ) e 2A(r)H tt(r ), h tz (r) = e2A(r)H t z (r), haa(r) = e2A(r)H a a (r), hzz(r) = e2A(r)H zz(r). Com paring w ith equation (3.12) of ref. [3] we can see th a t Z 1(r) m ode corresponds to the sound mode, while th e Z 2 (r) m ight be called a non-conform al mode, since it is intim ately rela te d to th e scalar field. T he th ird m ode (which is decoupled) is th e shear one an d is expressed as Z3(r) = H xz(r) + k H x t ( r ) , (3.6) a n d according to th e residual SO(2) sym m etry in xy-plane (after tu rn in g on m om entum along z-direction) is d egenerated w ith th e m ode in which th e index x is replaced by th e index y. T he dynam ics of th e fourth mode, is governed by an eq u atio n of m otion which is sim ilar to th e ex tern al m assless scalar equation, which was studied w ith details in [6]. T he equations of m otion for th e modes Z 1(r) and Z 2(r) have th e generic form

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and have to be solved num erically w ith proper boundary conditions. T he explicit form of the coefficient functions and comm ents about the numerics are given in the appendix B . As usual a t th e horizon we take th e incom ing condition, which in our coordinates m eans the regular solution.
A n analysis of th e equations (3.8) and (3.9) near th e conform al boundary leads to the asym ptotic behavior as r ~ 0 Transform ation to the usual Fefferm an-Graham coordinates close to the boundary, r ^ p4 -A , reveals th a t Z 1(p) has the asym ptotic of m etric com ponents like the perturbations considered in [3]. This p e rtu rb atio n corresponds to the sound mode of the theory. On the other hand Z 2(p) has the asym ptotic of the background scalar field $ and is similar to th e case studied in [4]. The right boundary conditions for the QNM spectrum are: A 1 = 0 and A2 = 0. The shear m ode p e rtu rb atio n Z 3(r) has th e sam e asym ptotic as Z 1(r) and requires a standard D irichlet boundary condition a t r = 0.

G en er a l rem a rk s an d su m m a r y
In all th e cases th e problem em erging from equations disused in th e previous section is a generalized eigenvalue equation, which for a given k results in a well defined frequency w(k). N ote th a t all m odes, for which Re w(k) = 0, come in pairs, nam ely As we will show in th e next section in some cases th e m odes are purely im aginary. B u t we w ant to em phasize th a t in all of these cases (except th e hydrodynam ical shear m ode) we still have a p air of m odes w ith different values. A n im p o rta n t th in g to note here is th a t due to the coupled nature of the modes Z 1(r) and Z 2(r) there is another approxim ate degeneracy in the spectrum : all modes, except for the hydrodynam ical one, come in pairs. T he reader is alerted not to confuse this stru c tu re w ith th e one appearing in eq. (3.11) . For all the potentials we have m ade natural consistency checks. For high tem peratures (i.e., horizon radii closer to the asym ptotic boundary) in the sound and the shear channels we have an agreem ent w ith the pure gravity results dual to th e C F T case [3]. The degeneracy related to the coupled natu re of the modes is still present a t high tem peratures, w here the system is expected to be conform al. T h e second m ost d am p ed nonhydrodynam ic m ode tu rn s out to be th e m ost dam ped nonhydrodynam ic m ode found in ref. [3] T he hydrodynam ical Q N M 's are defined by th e condition lim k^0 wH (k) = 0, a n d are related to tra n sp o rt coefficients in th e following way w t t -i^n^k 2 , w w ± cs k -i r s k2 , (3.12) respectively in th e shear and sound channels. Those form ulas are approxim ate in a sense th a t in general higher order tra n sp o rt coefficients should be considered [26]. However, in a range of sm all m o m en ta, second order expansion is enough a n d we use it to read off th e Those form ulas were used to m ake th e second check of th e results: com pute th e speed of sound cs and values of the shear viscosity from the hydrodynam ic modes and compare them respectively to th e one obtain ed from th e background calculations (2.15) and predictions known in the literature [27,28]. Both of them are always satisfied, for example the classical result, n / s = 1 /(4 n ) [27], is found in all cases considered in this paper.
In classical gravity, the spectrum , ap a rt from the hydro modes, contains of course also an infinite ladder of non-hydrodynam ical m odes. These are identified w ith th e poles of corresponding retarded G reen's functions [3], and as such correspond to physical excitations of th e holographic field theory. In contrast to th e hydrodynam ic modes, we do not have a universal interpretation for them in gauge theory, however, this cannot stop us from treating th em as physical excitations of th e plasm a system . Indeed, even if one is only in terested in analyzing (high order) hydrodynam ics, in [26], one finds p o le s/c u ts in th e Borel plane which exactly correspond to th e lowest n o n -h y d ro d yn a m ic QNM . T his shows th a t these non-hydrodynam ic excitations have to be included for th e self-consistency of the theory.
Of course if one is close to equilibrium, the higher QNM will be more dam ped and may be neglected in practice. However in some cases the lowest QNM become com parable to the hydrodynam ic ones and as such provide an applicability limit for an effective hydrodynam ic description. These phenom ena will be a t th e focus of th e present p aper. Indeed we find th a t they become very im p o rtan t in th e vicinity of a phase transition.
Finally, to dem ystify som ew hat these higher quasinorm al m odes, one can give a well known simple physical setup when only these modes are relevant. Suppose th a t one considers a spatially uniform plasm a system and starts with an anisotropic m om entum distribution for th e gluons. T hen the initial energy-m om entum tensor is spatially constant b u t anisotropic.
If we let th e system evolve, th e system will therm alize (w ith the energy-m om entum tensor becoming eventually isotropic). However this (homogeneous) isotropization will not excite any hydrodynam ic m odes as th e sym m etry of th e problem forbids any flow. T hus th e relevant excitations will be different. A t stro n g coupling th ey correspond exactly to th e higher quasinorm al modes.
In th e analysis below we m easure th e m om enta an d th e frequencies in th e u n its of tem p e ra tu re by setting T here are a few novel predictions which we m ake from th e QNM frequencies. F irst is to estim ate th e m om entum , or equivalently th e len g th , scale w here th e hydrodynam ic description of the system breaks. For a C FT case this was evaluated to be q = 1.3 where in th e shear channel first non-hydrodynam ic QNM dom inated th e system dynam ics [29]. In th e sam e tim e th is effect did n ot ap p ear in th e C F T sound channel. T he new feature we find is th a t we see this crossing2 not only in th e shear channel but also in the sound channel. This shows th a t th e influence of a non-trivial phase structure of the background affects the applicability of hydrodynam ics in a qualitative way. O ther aspect is th a t the hydrodynam ic description is valid in large enough length scale (the sm aller critical m om entum ) which means th e applicability of hydrodynam ics near the phase transition is more restricted than in th e high tem p e ra tu re case.
In tab le 2 we sum m arize th e critical m om enta in two channels an d hydrodynam ic p a ra m ete rs for different potentials. All q u an tities are evaluated a t corresponding critical tem p eratu res. In th e following subsections we will show th e Q N M 's m ostly for th e sound channel which present characteristic stru c tu re for each potential. Since th e shear channel in all cases has th e sam e form (w ith different critical m om entum ) we restrict ourselves to show only one related plot for th e VqCD potential.
T he second observation is the bubble form ation in the spinodal region in the case of the 1st order phase transition [30]. This happens when c2 < 0 which means th a t hydrodynam ical m ode is purely im aginary. For sm all m om enta, u h = ± i\c s \k -i T sk 2, th e m ode w ith the plus sign is in th e u n stab le region, i.e., Im u h > 0. For larger m om enta th e o th er term starts to dom inate, so th a t there is kmax = \cs \ / r s for which the hydro mode becomes stable again. T he scale of th e bubble is th e m om entum for which positive im aginary p a rt of the hydro mode attains the m aximal value. Im aginary part of the unstable hydro mode is called th e grow th rate [30]. T hird observation is th a t the hydrodynam ical mode of the sound channel in 1 st order case near the critical tem perature T c, and in the IHQCD case also the first non-hydrodynam ical modes, become purely im aginary for a range of m omenta. Interpretation of this fact is th a t th e corresponding wavelengths cannot propagate at a linearized level and correspondingly there is a diffusion-like m echanism for those modes. It is im portant to note th a t generically th e ultra-locality [6] of the non-hydrodynam ic m ode is still present in th e critical region of th e phase diagram . T h e only exception observed is th e IH Q C D p o ten tial, where th e m odes exhibit a non trivial behaviour. M ost of th e interesting dynam ics and effects observed are due to th e different behaviour of the hydrodynam ical m odes an d how th e y cross th e m ost d a m p e d non-hydrodynam ic m odes. T h is includes th e in stab ility an d th e bubble fo rm ation in th e case of th e 1 st order phase transition.
2 In this paper by crossing between the modes we mean crossing in the imaginary part of the hydrodynamic and the most damped non-hydrodynamic modes.

T h e crossover case
T he results for th e QNM w ith a QCD-like equations of s ta te are sum m arized below. P a ram e te rs of th is p o te n tia l have been chosen to fit th e tem p e ra tu re dependence of th e speed of sound obtained in lattice QCD com putations w ith dynam ical quarks at zero baryon chemical potential [14].
In our com putations from th e hydrodynam ic m ode we estim ate th e value of th e bulk viscosity, w hich is in agreem ent w ith ref. [24] (cf. tab le 2) . It is im p o rta n t to note, th a t despite th e fact th a t th e EoS of QCD are correctly reproduced in th e m odel tra n s p o rt coefficients are lower th an th e lattice predictions [31,32]. For exam ple only the qualitative tem p e ra tu re dependence of bulk viscosity is correct, nam ely th a t it rapidly raises near th e Tc [24].3 In th is analysis we tak e an o th er step, a n d stu d y th e tem p e ra tu re an d m om entum behaviour not only of the hydrodynam ic mode but also of the first and second of the infinite tow er of higher m odes. In p a rtic u la r th is allows us to e stim ate th e applicability of th e hydrodynam ic approxim ation in th e critical region of tem peratures where we find crossing of th e modes in sound channel.
Firstly, before we move to the new results, using the example of the Vqcd potential, let us discuss the high tem perature quasinorm al modes. The results com puted for T = 3Tc are shown in figure 2. T he speed of sound, shear a n d bulk viscosities read of from th e lowest QNM are very close to results expected for a conform al system , i.e., n /s -1 /(4 n ), -0.321, Z /s -0.003. M odes com puted for th is te m p e ra tu re in th e sound an d th e shear channels are in agreement w ith the conformal results of ref. [3]. As we m entioned in previous section, since Z i(r) and Z 2(r) modes are coupled the nonhydrodynam ic QNM 's are in pair in all range of tem peratures, and th e second m ost dam ped nonhydrodynam ic m ode tu rn s out to be th e m ost dam ped one found in ref. [3]. Now let us tu rn our a tte n tio n to th e opposite case of lower tem peratures. T he results co m p u ted for th e pseudo-critical tem p e ra tu re , T = Tc, are shown in figure 3. T he m ost 3 We define the pseudo-critical temperature as the lowest value for the speed of sound (2.15). Corresponding lQCD definition refers to peaks of chiral and Polyakov loop susceptibilities [33,34].

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num erical instability, while th e sta n d a rd th erm odynam ic relation, d F = -s d T , suffers a problem of correct choice of th e reference configuration. A possible candidate for reference geom etry is the one of vanishing horizon area in the unstable black hole branch. However it has infinite tem perature and it is not clear for us w hether it can be used as a proxy for the therm al gas geometry. Also the unstable branch black holes exhibit a variety of pathologies (which will be described later) with increasing T . Due to the above m entioned difficulties we refrained from estim ating th e value of Tc in this p articular model. Nevertheless we expect th a t th ere exists a critical te m p e ra tu re Tc > T m w here th e tra n sitio n takes place [15,17]. This tran sitio n changes the geom etry substantially.
It is im portant to note th a t in this case there exists a m inim al tem p eratu re T m below which a black hole solution does not exist. As in th e case of V1st th e onset of in stab ility appears a t T > Tm (for configurations w ith c2( T ) < 0).
T he different stru c tu re of th e EoS is reflected in th e behaviour of QNM frequencies, which indicate th e existence of second ch aracteristic te m p e ra tu re Tch ^ 1.102Tm. T he novel effect observed in this system is th a t for tem peratures near th e minimal tem perature th e u ltralo cality p ro p erty of th e first non-hydrodynam ic m ode is violated. T he m ode tu rn s out to be purely im aginary for very low m om enta and for tem peratures of the range T m < T < Tch, and it does not have a structure described in eq. (6.1) . There are two purely im aginary modes which have the following form In figure 12 we show th e tem p e ra tu re dependence of those m odes a t k = 0 in th e range where th ere are purely im aginary. As th e system is h eated fu rth er th e real p a rt develops, and the mode becomes the least dam ped non-hydrodynam ic mode of the high-T limit, with th e usual s tru c tu re (3.11) . It directly comes from th e presence of th e background scalar field, which breaks th e conform al invariance.
In figure 13 we show Q N M 's in th e sound channel co m puted for V ihqcd a t T = T m . T he m ode s tru c tu re is different th a n th e one generically present in previous cases. F irst th in g which is ap p a re n t is th a t hydrodynam ic m odes are purely im aginary for a range of sm all m om enta. In addition, th ere is a sm all gap betw een th e hydrodynam ic and non-hydrodynam ic degrees of freedom a t a rb itra ry low m om entum , which in tu rn implies th a t th e crossing h ap p en s a t very low value of qc ~ 0.14 (see th e insert in figure 13) . As a m a tte r of fact, in th is case near th e T m one m ust always take into account th e non hydrodynam ic degrees of freedom in th e description of th e system dynam ics. A nother absolutely fascinating effect observed exactly a t Tm is th a t th e non-hydrodynam ic modes, which are purely im aginary for low m om enta, join w ith the hydrodynam ic modes a t some finite m om entum q j , and follow them w ith increasing q. This effect is illustrated in figure 13, where the non-hydro1 m ode which has two branches joins w ith the two branches of the hydro modes respectively at qJ ~ 0.14 and qJ ~ 1.5. In the same tim e the real part develops with both signs, as expected from general considerations (see eq. (3.11)). This effect implies the ultralocality violation observed generically in other models, and joining does not happen for tem peratures higher th an the minimal one. The final observation from figure 13 is th a t the

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down of th e hydrodynam ic description near th e QCD critical region. One specific feature is th e u ltra lo c a lity p ro p erty obeyed by nonhydrodynam ic m odes. K eeping in m ind th e qualitative n a tu re of those considerations, it would be a very interesting task to com pute sim ilar spectrum in lattice QCD.
We stu d y two system s which exhibit different ty p es of th e 1st order phase tra n sitio n as determ in ed by p o ten tials V1st an d V ihqcd. We explicitly determ in ed th e in stab ility in th e spinoidal b ran ch of b o th of them , a n d for V1st we e stim a ted th e length scale for bubble form ation. O n to p of th a t, b o th m odels posses generic m inim al tem p e ra tu re , T m , below which certain solution cease to exist. From the tem perature dependence of the QNM spectrum , in the stable region of the corresponding EoS, one can see the existence of another characteristic tem p eratu re, related to th e appearance of th e diffusive-like m odes, which is slightly higher th a n T m .
N um ber of novel phenom ena is found in th e case of IHQCD potential. At T = T m the hydrodynam ic a n d th e first non-hydrodynam ic m odes becom e purely im aginary for low m om enta. This implies th e violation of ultralocality property generically observed in other cases [6,13]. One more surprising observation is th a t instead of th e crossing of th e modes, found generically in the studied models, there is a " jo in in g " phenomenon. From some value of the m om entum both the hydrodynam ic and first-non hydrodynam ic mode obey the same dispersion relation. In th e sam e tim e, higher non-hydrodynam ic m odes adm it u ltralo cal m om entum dependence.
W hat makes the VIHQCD potential exceptional among the studied cases is a rich structure of th e sm all black hole branch solutions. T he spectrum of quasinorm al m odes shows two type of instabilities. O ne is th e usual spinodal instability, sim ilar to one found in th e V1 st case. This appears exactly when the systems shows therm odynam ic instability in equations of state. Second is an in stab ility triggered by th e non-hydrodynam ic m ode. In th is case th e relatio n betw een EoS a n d instabilities is th a t for configurations which have c2 > 1/3 the first non-hydrodynam ic m ode becomes unstable. Two regions are separated and do not overlap. Up to our knowledge this is the first exam ple where such a dynam ical mechanism has been presented.

A ck n o w led g m en ts
R J a n d HS were su p p o rte d by N C N grant 2 0 1 2 /0 6 /A /S T 2 /0 0 3 9 6 , J J by th e N C N p o st d o cto ral internship grant D E C -2 0 1 3 /0 8 /S /S T 2 /0 0 5 4 7 . We th a n k Ju ergen Engels for providing us w ith th e lattice d a ta for the speed of sound squared in th e pure gluon sector. We would like to th an k D. Blaschke and P. W itaszczyk for interesting discussions.

A O n -sh ell a ctio n and Free E n ergy
In this appendix we give some details about asym ptotic behaviour of our black hole solutions an d we show how, in principle, one can use th is expansion to com pute th e Free Energy.
In the k = 0 case all equations, except for the Z 2(r) which rem ains coupled to Z i(r), reduce to scalar field equations. This case has been studied in ref. [6].
A t k = 0 th is equations reduces to a eq u ation of m otion of m inim ally coupled m assless scalar field which was studied in [6]. In order to solve equations (B.1) an d (B.2) , w hich are linear o rd in ary differential equations, we use spectral discretization w ith Chebyshev polynom ials [22]. T he resulting m a trix eq u ation is of polynom ial c h aracter in th e m ode frequency w an d we determ ine QNM s by evaluating th e d eterm inant of th e m atrix and setting it to zero. Corresponding