Quasinormal modes and the phase structure of strongly coupled matter

P u b l i s h e d f o r SISSA b y S p r i n g e r

Received: April 27, 2016 Accepted: May 26, 2016 Published: June 8, 2016

Quasinormal modes and the phase structure of strongly coupled matter

Romuald A. Janik, Jakub Jankowski and Hesam Soltanpanahi Institute of Physics, Jagiellonian University,

Ł ojasiewicza 11, 30-348 Kraków, Poland

E -m a il: r o m u a l d @ t h . i f . u j . e d u . p l, j a k u b j @ t h . i f . u j . e d u . p l, h e s a m @ t h . i f . u j . e d u . p l

Ab s t r a c t: We investigate th e poles of th e retard ed G reen’s functions of strongly coupled field theories exhibiting a variety of phase stru c tu re s from a crossover u p to different first o rd er phase tra n sitio n s. T hese theories are m odeled by a d u a l g ra v ita tio n a l description.

The poles of th e holographic G reen’s functions appear at th e frequencies of th e quasinorm al m odes of th e dual black hole background. We focus on quantifying linearized level dynam ical response of th e system in th e 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 vicin ity of a phase tran sitio n . For a m odel w ith linear confinement in th e meson spectrum we find degeneracy of hydrodynam ic and non-hydrodynam ic modes close to th e m inim al black hole tem p eratu re, a n d we esta b lish a region of te m p e ra tu re s w ith u n sta b le n o n -h y d ro d y n am ic m odes in a b ranch of black hole solutions.

Ke y w o r d s: A dS-C FT Correspondence, Black Holes, Gauge-gravity correspondence, Holog­

rap hy and quark-gluon plasm as

ArXiv ePr in t: 1603.05950

Open Access, © The Authors.

Article funded by SCOAP3. doi:10.1007/JHEP06(2016)047

J H E P 0 6 ( 2 0 1 6 ) 0 4 7

C o n te n ts

1 I n tr o d u c tio n 1

2 T h e b a c k g r o u n d a n d t h e r m o d y n a m ic s o f t h e s y s te m 3

2.1 M etric A nsatz and equations of m otion 3

2.2 T herm odynam ics 5

3 Q u a s in o r m a l m o d e s 6

3.1 E qu ation s of m otion and b o u n d ary conditions 7

3.2 G eneral rem arks and sum m ary 8

4 T h e c r o sso v e r ca se 11

5 T h e s e c o n d o rd er p h a s e t r a n s it io n ca se 13

6 T h e first o r d e r p h a s e tr a n s it io n ca se 14

7 T h e im p r o v e d h o lo g r a p h ic Q C D 16

8 D is c u s s io n 20

A O n -s h e ll a c tio n a n d F ree E n e r g y 21

B Q N M s e q u a tio n s o f m o tio n a n d n u m e r ic a l d e ta ils 23

1 In tr o d u c tio n

It is alm ost tw enty years since th ere has been discovered a rem arkable new relation between g eom etry a n d physics: w ith in th e A nti-de S itte r/C o n fo rm a l F ield T h eo ry (A d S /C F T ) correspondence [1] we can inv estig ate th e dynam ics of stro ng ly coupled q u a n tu m field theories by m eans of G eneral R elativ ity m ethods. From purely academ ic studies th is field of research evolved to address experim ental system s an exam ple being strongly interacting h a d ro n ic m a tte r [2]. In p a rtic u la r, real tim e response of a th e rm a l eq uilib riu m s ta te has been quantified in th e case of N = 4 super Yang-Mills th eo ry by th e m eans of th e poles of th e re ta rd e d G reen ’s function [3], w hich co rrespond to qu asinorm al m odes (QNM ) in th e dual g rav itatio n a l theory.

W hile th e h y d ro dy n am ic QNM s have been stu d ied in different g ra v ita tio n a l theories dual to non-C FT cases (e.g. ref. [4 , 5]), initial steps tow ards extension were taken in ref. [6 , 7]

where nonhydrodynam ic QN M ’s of an external scalar field were considered in non-conformal field theories, which still ad m it a g ravitatio nal dual description. Subsequent investigations

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J H E P 0 6 ( 2 0 1 6 ) 0 4 7

include different m echanism s of scale gen eratio n [8], different rela x 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 th e letter [13] where we provide m any more details as well as extend th e investigation to a model of an improved holographic QCD type which exhibits novel and interesting phenom ena. We concen trate on investigating linearized real tim e response of stro n g ly coupled non-conform al field theories in th e vicin ity of various types of phase transitio n s and phase structures. Thus th e physical regime of interest in the present pap er is quite d istin ct from th e th e one of interest for ‘early th erm aliz a tio n ’ 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 p ertu rb atio n s and corresponding tw o-point correlation functions. Secondly, we concentrate on th e phenom ena a p p e a rin g in th e vicin ity of a n o n triv ia l ph ase s tru c tu re of various type: a crossover (m otivated by th e lattice QCD equations of sta te [14]), a 2nd o rder phase tra n sitio n and a 1st o rd er ph ase tra n sitio n . T hese cases are m odeled by choosing a p p ro p ria te scalar field self-interaction p o ten tia ls in a holographic gravity -scalar th e o ry used in [15]. A p a rt form th is, we also analyze a p o te n tia l from a different fam ily of m odels, im proved holographic QCD (IH Q C D ), considered in [17, 18]. In th is case th e focus was on g ettin g best possible co ntact w ith properties of QCD, in p a rticu la r asym ptotic freedom and colour confinem ent as well as o btain in g a realistic value of th e bulk viscosity.

D espite th e fact, th a t considered m odels have a r a th e r sim plistic co n stru ctio n , th e resulting near equilibrium response shows a variety of non-trivial phenom ena. Some generic featu res consist of: (i) th e break d ow n of th e ap p licab ility of a h y d rod y nam ic d escrip tio n already at lower m om enta th a n in th e conform al case; (ii) in th e cases w ith a first order phase tra n s itio n we find a generic m in im al te m p e ra tu re , T m , below w hich no u n sta b le solution exists; (iii) w henever th e re exists a th erm o d y n am ical in sta b ility th e re is a corresponding dynam ical instability present in th e hydrodynam ic m ode of th e theory; (iv) th e ultralocality p ro p erty of non-hydrodynam ic m odes, i.e., weak dependence on th e m om entum scale.

T h e n a tu re of th e d u a l g ra v ita tio n a l fo rm u latio n allows for a d e ta ile d q u a n tita tiv e investigation of th e above phenom ena as well as for accessing diverse physical scenarios. In p articu lar, th e first order phase tra n sitio n appears in two different scenarios. T he first one is sim ilar to th e u su al H aw king-Page tra n s itio n [19] in w hich th e tw o ph ases are a black hole geom etry an d a th e rm a l gas geom etry [18]. In th e second one th e tra n s itio n app ears betw een two black hole solutions [15]. T his diversity is trig gered by a different functional d ependence of th e scalar field p o te n tia l in th e deep in fra re d (IR ) region, a n d is reflected in th e co rrespo n ding Q N M sp ectru m . N evertheless th e re is a com m on asp ect in b o th situations. We observe some specific dynam ical response of th e system for a characteristic te m p e ra tu re , Tch > Tm , in th e stab le b ra n c h of EoS. T h e d e ta ils of th is effect d e p e n d on th e case, b u t th e existence of Tch is generic for a first order phase tran sitio n .

P a rtic u la rly in te re stin g effects a p p e a r in IH Q C D m odel, w hich a d m its a first order phase tran sition between a black hole and a therm al gas [18]. F irst, for tem p eratures in th e ran g e T m < T < T ch th e lowest lying e x c ita tio n m odes becom e p u rely im ag in ary for low m om enta, which leads to a u ltralo cality violation. Second, a t T = T m for m om enta higher th a n some threshold value th e hydrodynam ic m ode and th e first non-hydrodynam ic mode

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h a v e t h e s a m e d i s p e r s i o n r e l a t i o n . T h i r d , i n t h e s m a l l b l a c k h o l e b r a n c h t h e r e i s a r a n g e o f t e m p e r a t u r e s w h i c h s h o w s i n s t a b i l i t y i n a n o n - h y d r o m o d e . T h e a p p e a r a n c e o f t h e s e p h e n o m e n a m a k e s t h e I H Q C D m o d e l u n i q u e i n t h e l a n d s c a p e c o n s i d e r e d .

T h e o r g a n i z a t i o n o f t h e p a p e r i s a s f o l l o w s . I n t h e n e x t s e c t i o n , 2 w e s h o r t l y d e s c r i b e t h e t h e r m o d y n a m i c s o f c o n s i d e r e d m o d e l s a n d p a r a m e t e r c h o i c e s f o r b u l k s c a l a r i n t e r a c t i o n s . I n s e c t i o n 3 w e d i s c u s s e q u a t i o n s o f m o t i o n f o r t h e l i n e a r p e r t u r b a t i o n s o f t h e b a c k g r o u n d a n d t e c h n i c a l a s p e c t s o f t h e i r s o l u t i o n s . I n t h e f i r s t s u b s e c t i o n w e c l a r i f y t h e r i g h t b o u n d a r y c o n d i t i o n s w h i c h h a v e t o b e c h o s e n f o r t h e Q N M s p e c t r u m . I n t h e s e c o n d s u b s e c t i o n w e g i v e g e n e r a l r e m a r k s a n d l i s t m a i n a s p e c t s o f p h y s i c a l p r o p e r t i e s w e o b t a i n . T h e f o l l o w i n g s e c t i o n s 4 t o 7 c o n t a i n r e s u l t s a n d d e t a i l e d s t u d i e s o f d i f f e r e n t c a s e s . W e c l o s e t h e p a p e r b y a s u m m a r y a n d o u t l o o k i n s e c t i o n 8 . F o r c o m p l e t e n e s s a p p e n d i x e s A a n d B r e s p e c t i v e l y c o n t a i n s o m e t e c h n i c a l d e t a i l s o f t h e F r e e E n e r g y c o m p u t a t i o n , a n d t h e e x p l i c i t f o r m o f t h e Q N M e q u a t i o n s o f m o t i o n .

2 T h e b a ck g ro u n d an d th e r m o d y n a m ic s o f th e s y ste m

I n t h i s s e c t i o n w e f o r m u l a t e t h e b a c k g r o u n d b l a c k h o l e s o l u t i o n s a n d d e t e r m i n e t h e s c a l a r f i e l d p o t e n t i a l b y c o n s i d e r i n g e m e r g e n t e q u a t i o n s o f s t a t e i n t h e d u a l f i e l d t h e o r y .

2.1 M e tr ic A n s a t z a n d e q u a tio n s o f m o tio n

T h i s s e c t i o n d e s c r i b e s t h e b l a c k h o l e b a c k g r o u n d s o l u t i o n s f o r t h e q u a s i n o r m a l m o d e c a l c u l a t i o n s , w h i c h f o l l o w f r o m t h e a c t i o n

L = Lc f t + A 4 A O<

p

w h e r e A i s a n e n e r g y s c a l e , a n d A i s a c o n f o r m a l d i m e n s i o n o f t h e o p e r a t o r O ^ r e l a t e d t o t h e m a s s p a r a m e t e r o f t h e s c a l a r f i e l d a c c o r d i n g t o h o l o g r a p h y , A ( A _ 4 ) = m 2 . W e

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J H E P 0 6 ( 2 0 1 6 ) 0 4 7

S = -~ o [ d5x ^ —g R — 1 (d ó )2 — V(ó) — \ [ d4x V ~ h K , (2.1)

2k5 J M L 2 _l K2 JdM

w here V (ó ) is th u s far a rb itra ry a n d k5 is re la te d to five d im ensional N ew to n c o n sta n t b y = A/8nG5. T h e 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 contribu tio n. These solutions are sim ilar to those studied in ref. [15, 17]. Since our goal is to determ ine th e QNM frequencies, it will be convenient to em ploy E d dington-F inkelstein coordinates, which have been proven useful in th e case of th e scalar field m odes [6]. We will discuss th is in a m ore d etail in th e following section.

W h ereas we are in te rested in asy m p to tically AdS space-tim e geom etry, th e p o te n tia l needs to have th e following small ó expansion

12 1

V (ó) — ^ + 2 m 2Ó2 + O (ó 4) .

H ere, L is th e AdS radiu s, w hich 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 th e boundary conformal field theory

(2.2)

consider 2 < A < 4 w hich co rresp on ds to relevant d efo rm atio ns of th e C F T a n d satisfies th e B reitenlohner-F reedm an bound, m 2 > —4 [20, 21].

T h e A n satz for solutions u n d e r consideration s follows from th e assum ed sym m etries:

tra n s la tio n invariance in th e M inkowski directions as well as SO(3) ro ta tio n sy m m etry in th e sp atial p a rt. T his leads to th e following form of th e line elem ent:

d s 2 = g ttd t2 + gxxdx2 + g „ d r 2 + 2grfd r d t , (2.4) where all th e m etric coefficients ap p earing in (2.4) are functions of th e radial coordinate r alone, as is th e scalar field ¢. This form of th e field A nsatz (determ ined so far only by th e assum ed sym m etries) allows two gauge choices to be m ade. For th e purpo se of com puting th e quasinorm al m odes it is very convenient to use th e Eddington-Finkelstein gauge grr = 0.

It is ty p ically convenient also to im pose th e gauge choice gtr = 1, b u t for o u r p urp o ses it tu rn s o u t to be very effective to use th e rem ainin g gauge freedom to set ę = r. We label th e m etric com ponents as

d s 2 = e2A ( —h d t 2 + d x 2) — 2 eA+ B d t d r , (2.5)

ę = r . (2.6)

In th e above c o o rd in a te sy stem 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 E instein-scalar field equations

R/dv —2 V ^ V v ę —2 V (ę )guv = 0 , (2.7)

= 0 , (2.8)

a ę

takes th e following form

A " — A /B / + 1 = 0 , (2.9) 6

h " + (4 A / — B /) h = 0 , (2.10) 6 A 'h ' + h(24A /2 — 1) + 2e2BV = 0 , (2.11)

h e 2B

4A/ — B / + ---— V / = 0 , (2.12)

h h

w here th e prim e denotes a derivative w ith respect to ¢.

In c o n tra st to m e th o d s p ro p o sed in ref. [15] we solve th is coupled eq u atio n s d irectly using th e sp e c tra l m e th o d [22] in th e N ew ton lin earizatio n algo rith m . We are in te rested 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 :

h ( m ) = 0 . (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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2.2 T herm odynam ics

H a v i n g d e t e r m i n e d t h e g e o m e t r y w e c a n e x t r a c t t h e t h e r m o d y n a m i c q u a n t i t i e s i n a s t a n d a r d w a y . T h e B e k e n s t e i n - H a w k i n g f o r m u l a f o r e n t r o p y , t o g e t h e r w i t h t h e e v e n t h o r i z o n r e g u l a r i t y , l e a d t o t h e f o l l o w i n g e x p r e s s i o n s f o r t h e e n t r o p y d e n s i t y a n d t h e H a w k i n g t e m p e r a t u r e

2n ^{o M X} eA(rH)+B(r H) Iy ' (r )^|

S = ^ e 3A(rH) , T = -----I M M . (2.14)

Ki 4n

c2, ^{= .} ( 2 . 1 5 )

d l o g s

L e t u s e m p h a s i s t h a t t h i s i s t h e s p e e d o f s o u n d i n t h e d u a l f i e l d t h e o r y . T h e c o r r e s p o n d i n g F r e e E n e r g y ( F E ) i s r e l a t e d t o t h e v a l u e o f t h e a c t i o n e v a l u a t e d a t t h e s o l u t i o n [ 2 3 ]

P F = l i m ( S ( e ) _ S c t ( e ) ) , ( 2 . 1 6 )

w h e r e P = 1 / T , S i s t h e E i n s t e i n - H i l b e r t - s c a l a r a c t i o n ( w i t h G i b b o n s - H a w k i n g t e r m ) e v a l u a t e d o n - s h e l l w i t h a c u t - o f f e i n a h o l o g r a p h i c d i r e c t i o n . S c t a r e p r o p e r l y c h o s e n c o u n t e r t e r m s . W e w i l l u s e t h i s f o r m u l a i n t h e c a s e o f p o t e n t i a l s w i t h a f i r s t o r d e r p h a s e t r a n s i t i o n i n o r d e r t o c o m p u t e t h e F r e e E n e r g y d i f f e r e n c e b e t w e e n p h a s e s a s a f u n c t i o n o f t e m p e r a t u r e a n d d e t e r m i n e t h e critical temperature, Tc, f o r t h o s e m o d e l s . I n e v a l u a t i n g t h i s d i f f e r e n c e t h e c o u n t e r t e r m s w i l l c a n c e l t h a t i s w h y w e d o n o t n e e d t o h a v e a d e t a i l e d k n o w l e d g e t h e r e o f .

T h e w a y i n w h i c h c o n f o r m a l s y m m e t r y i s b r o k e n i s d e t e r m i n e d b y t h e c h o i c e o f t h e s c a l a r f i e l d p o t e n t i a l w h i c h i n o u r c a s e i s t a k e n i n a g e n e r i c f o r m [ 1 5 , 1 7 ]

V ( 0 ) = _ 1 2 ( 1 + a 0 2 ) 1 / 4 c o s h ( y 0 ) + b2 0 2 ⁺ b4 0 4 + b 6 0 6 . ( 2 . 1 7 )

T h e c h o s e n p o t e n t i a l s a r e s u m m a r i z e d i n t a b l e 1 . C o r r e s p o n d i n g p l o t s , r e p r e s e n t i n g t e m p e r a t u r e d e p e n d e n c e o f t h e e n t r o p y d e n s i t y , i . e . , t h e e q u a t i o n o f s t a t e ( E o S ) , w i l l b e g i v e n t o g e t h e r w i t h t h e d e t a i l e d d i s c u s s i o n o f e a c h c a s e i n f o l l o w i n g s e c t i o n s . H e r e w e o n l y m a k e a f e w g e n e r a l r e m a r k s . T h e p a r a m e t e r s f o r t h e Vq c d p o t e n t i a l h a v e b e e n c h o s e n t o f i t t h e l a t t i c e Q C D ( l Q C D ) d a t a f r o m r e f . [ 1 4 ] , a n d t h e s y s t e m i s k n o w n t o p o s s e s s a c r o s s o v e r b e h a v i o u r a t z e r o b a r y o n c h a r g e d e n s i t y . P a r a m e t e r s o f p o t e n t i a l s V 1 s t a n d V 2 n d w e r e f i t t e d s o t h a t t h e c o r r e s p o n d i n g e q u a t i o n s o f s t a t e e x h i b i t r e s p e c t i v e l y t h e 1 s t , a n d t h e 2 n d o r d e r p h a s e t r a n s i t i o n s . I n p a r t i c u l a r , f o r t h e 1 s t o r d e r c a s e , i n a c e r t a i n t e m p e r a t u r e r a n g e w e e x p e c t a n i n s t a b i l i t y ( s p i n o d a l ) r e g i o n .

T h i s c o n c r e t e f o r m o f t h e l a s t p o t e n t i a l w a s a l r e a d y u s e d e x p l i c i t l y i n [ 2 4 ] a n d i s b a s e d o n t h e c o n s i d e r a t i o n s i n [ 1 6 ] n e g l e c t i n g l o g a r i t h m i c r u n n i n g i n t h e U V . W e w i l l r e f e r t o i t a s t h e I H Q C D p o t e n t i a l [ 1 7 , 1 8 ] . A s i t w a s m e n t i o n e d i n t h e i n t r o d u c t i o n , a n d w i l l b e e x t e n d e d i n s e c t i o n 7 , i t i s d e s i g n e d t o m i m i c s o m e d y n a m i c a l a s p e c t s o f Q C D . H o w e v e r i t i s i m p o r t a n t t o e m p h a s i z e t h a t t h e v e r s i o n u s e d h e r e i s s i m p l i f i e d a s i t d o e s n o t i n c o r p o r a t e t h e U V l o g a r i t h m i c r u n n i n g .

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In tu rn , th e speed of sound of th e system can be determ ined as

p o ten tial a 7 b2 64 be A

VQCD 0 0.606 1.4 -0.1 0.0034 3.55

V2 nd 0 1 /V 2 1.958 0 0 3.38

Vist 0 ^a/ 7 / 1 2 2.5 0 0 3.41

VIHQCD 1 7 / 3 6.25 0 0 3.58

T ab le 1. Potentials chosen to study different equations of state exhibiting different phase structure and corresponding conformal dimension of the scalar field.

T he m odels d e term in ed by th e p o ten tia ls Vist an d Vihqcd exhibit a first order phase tra n sitio n s. In th e form er case th e tra n s itio n h a p p e n s betw een tw o different black hole solutions, while in th e la tte r th e tran sition happens between a black hole and a horizon-less geom etry. In b o th of tho se cases one can d eterm in e th e tra n s itio n by e v alu atin g th e F E difference according to form ula (2.16) , if one knows th e counter te rm s.1 In this com putation we follow an a lte rn a tiv e m e th o d o f ref. [17] a n d in te g ra te th e th e rm o d y n am ic relatio n, d F = —s d T , w ith p ro p erly chosen b o u n d a ry condition. We c an achieve th is by first choosing some a rb itra ry reference te m p e ra tu re To and w rite

F ( T ) = F (T o) — I s ( T ) d T , (2.18) JTo

w here we assum e to be in one p a rtic u la r class of solutions. To ev alu ate th e in te g ra tio n c o n sta n t, F (T o ), we use th e fact th a t th e Free E nerg y vanishes for th e zero h orizon are a geometry. In general th e small horizon area lim it of th e black hole solutions corresponds to th e vacuum geo m etry w ith “good sin g u larity ” in th e deep IR [25]. B y using th e rela tio n of T a n d s a n d th e h orizon rad iu s (2.14) we can ev alu ate th e Free E n erg y w ith th e d a ta o b ta in e d w ith m eth o d s o u tlin ed in th e previous subsection. T h is am o u n ts to a generic form ula

2n dT

F ( r u ) = ----2 / exp (3A (A h)) -^ — d \ n . (2.19)

k 5 J r H d A u

T he details of th e co m p u tatio ns along w ith th e corresponding plots and predictions for Tc will be given in th e corresponding sections of th e paper.

3 Q u a sin o rm a l m o d es

In th is section we form ulate th e problem of analyzing th e linear p e rtu rb atio n s aro un d th e equilibrium states in considered models. T h e first subsection contains equations of m otion and proper b o u n d ary conditions th a t need to be im posed. T h e second subsection contains a short sum m ary of th e results obtained w ith an em phasis on generic aspects. The detailed case by case discussion is a sub ject of th e rem aining p a rt of th e paper.

Clarification of this point can be found in the appendix A.

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3.1 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

T he linear response of th e system is analyzed by settin g p ertu rb atio n s w ith m om entum in a given d irection and com puting poles of th e resulting G reen functions. In this section we form ulate th e equations and corresponding b o u n d ary conditions to present and discuss th e results in th e following p a rt of th e paper.

We consider p ertu rb atio n s of th e background, obtain ed in th e previous section, in th e following form

9ab(r, t, z) = g(0) (r) + h a b (r)e-iMt+ikz , (3.1)

0 (r, t, z) = r + ^ ( r ) e -iwt+ikz . (3.2)

O n th e basis of [3 , 4 , 6] we consider infinitesim al diffeom orphism tra n sfo rm atio n s, x a ^ x a + £a, of th e form £a = £a ( r) e -iwt+ikz, which act on th e p ertu rb atio n s in a stan d ard way,

gab ^ gab _ Va£b _ V b£a , 0 ^ 0 _ £aV a0 , (3.3) a n d look for lin ear co m b in atio n s of m etric an d scalar p e rtu rb a tio n s w hich are invarian t u n der those tran sfo rm ation s. T here are four such modes, two of which are decoupled and two coupled. W ritte n explicitly, th e coupled m odes read

Z i( r ) = H aa(r) ) + k 2h (r ) _ w2^ + k 2h ( r ) H t t (r) + w (2kH tz(r) + w f f z z ( r) ) , (3.4) and

Z 2(r) = ^ (r) _ . (3.5)

In th e above h aa(r) = h xx(r) = h yy(r) are tran sv erse m etric co m po nen ts a n d we have factorized th e back g ro u n d from th e m etric p e rtu rb a tio n s in th e following way: h t t (r) = h ( r ) e 2A(r)H tt( r ) , h t z(r) = e2A(r)H t z(r), h aa(r) = e2A(r)H a a ( r) , hzz(r) = e2A(r)H zz(r).

C om paring w ith eq u ation (3.12) of ref. [3] we can see th a t Z 1(r) m ode corresponds to th e sound m ode, while th e Z 2 (r) m ight be called a non-conform al m ode, since it is in tim ately re la te d to th e scalar field. T h e th ird m ode (w hich is decoupled) is th e sh ear one a n d is expressed as

Z 3(r) = H xz(r) + k H x t ( r ) , (3.6)

a n d according to th e residu al SO (2) sy m m etry in xy-plane (after tu rn in g on m o m en tu m along z-directio n) is d e g e n era te d w ith th e m ode in w hich th e in d ex x is rep laced by th e index y. T he dynam ics of th e fo u rth mode,

Z4(r) = H x y ( r ) , (3.7)

is governed by an eq u a tio n of m o tio n w hich is sim ilar to th e e x te rn al m assless scalar equation, which was studied w ith details in [6].

T he equations of m otion for th e m odes Z 1(r) and Z 2(r) have th e generic form

M2(r) Z l'(r ) + M 1(r)Z1 (r) + M o (r)Z1(r) + K o (r)Z2(r) = 0 , (3.8) N2(r)Z2' (r) + N 1(r)Z 2 (r) + N o (r)Z2(r) + L1(r)Z1 (r) + L o (r)Z1(r) = 0 , (3.9)

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and have to be solved num erically w ith p ro p er b o u n d ary conditions. T h e explicit form of th e coefficient functions and com m ents about th e num erics are given in th e appendix B . As u su al a t th e horizon we tak e th e incom ing condition, w hich in our coordin ates m eans th e regular solution.

A n analysis of th e equations (3.8) and (3.9) near th e conform al b o u n d ary leads to the asy m p to tic behavior as r ~ 0

4 A

Z i( r ) A 1 + B 1 r4-A , Z 2(r) ~ A 2 r + B 2 r 4-A . (3.10) Transform ation to th e usual Fefferm an-G raham coordinates close to th e boundary, r ^ p4 -A , reveals th a t Z 1(p) has th e asym ptotic of m etric com ponents like th e p ertu rb atio n s considered in [3]. This p e rtu rb a tio n corresponds to th e sound m ode of th e theory. On th e other hand Z 2(p) has th e asym ptotic of th e background scalar field $ and is sim ilar to th e case studied in [4]. T he right boundary conditions for th e QNM spectrum are: A 1 = 0 and A 2 = 0. The shear m ode p e rtu rb a tio n Z 3(r) has th e sam e asym pto tic as Z 1(r) and requires a sta n d a rd D irichlet b o u n d ary condition a t r = 0.

3 .2 G e n e r a l r e m a r k s a n d su m m a r y

In all th e cases th e p roblem em erging from eq uations 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

w(k) = ± |Re w(k)| + i Im w(k). (3.11)

As we will show in th e n e x t section in some cases th e m odes are p u rely im aginary. B u t we w ant to em phasize th a t in all of these cases (except th e h y d ro d y n am ical shear m ode) we still have a p a ir of m odes w ith different values. A n im p o rta n t th in g to n o te here is th a t due to th e coupled n atu re of th e modes Z 1(r) and Z 2(r) th ere is anoth er approxim ate degeneracy in th e spectrum : all m odes, except for th e hydrodynam ical one, come in pairs.

T h e reader is alerted not to confuse th is stru c tu re w ith th e one ap p earing in eq. (3.11) . For all th e potentials we have m ade n a tu ra l consistency checks. For high tem p eratures (i.e., horizon radii closer to th e asym ptotic boundary) in th e sound and th e shear channels we have an agreem ent w ith th e pure gravity results dual to th e C F T case [3]. T he degeneracy related to th e coupled n a tu re of th e m odes is still present a t high tem p e ra tu re s, w here th e system is e x p ected to b e conform al. T h e second m ost d a m p e d n o n h y d ro d y n am ic m ode tu rn s o u t to be th e m ost dam ped nonhydrodynam ic m ode found in ref. [3]

T h e h ydro d y n am ical Q N M ’s are defined by th e co nd ition lim k^ 0 wH (k) = 0, a n d are related to tra n s p o rt coefficients in th e following way

w t t - i ^ n ^ k 2 , w w ± cs k — i r s k 2 , (3.12) respectively in th e shear a n d sound channels. T hose form ulas are ap p ro x im ate in a sense th a t in general higher order tra n s p o rt coefficients should be considered [26]. However, in a ran g e of sm all m o m e n ta , second o rd er expansion is en ough a n d we use it to re a d off th e

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lowest tra n sp o rt coefficients of th e model. T he sound a tten u atio n constant, r s , is related to shear n and bulk Z viscosities by

r = 2 7 ( 4 n + ; ) • (3.13)

T hose form ulas were used to m ake th e second check of th e results: co m p u te th e speed of sound cs and values of th e shear viscosity from th e hydrodynam ic modes and com pare them respectively to th e one o b tain e d from th e background calculations (2.15) a n d predictions known in th e literatu re [27, 28]. B oth of them are always satisfied, for example th e classical result, n / s = 1 /(4 n ) [27], is found in all cases considered in this paper.

In classical gravity, th e spectrum , a p a rt from th e hydro modes, contains of course also an infinite lad d e r of n o n -h y d ro d y n am ical m odes. T hese 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 co n trast to th e hydrodynam ic m odes, we do not have a universal interpretation for th em in gauge theory, however, this cannot stop us from treatin g th e m as physical ex citatio n s of th e p lasm a system . Indeed, even if one is only in te re ste d in an alyzing (high order) h y d rodynam ics, in [26], one finds p o le s /c u ts in th e B orel plane w hich ex a ctly co rresp on d to th e lowest n o n -h y d ro d y n a m ic QNM . T h is shows th a t th ese non-hydrodynam ic excitations have to be included for th e self-consistency of th e theory.

Of course if one is close to equilibrium , th e higher QNM will be more dam ped and may be neglected in practice. However in some cases th e lowest QNM become com parable to th e hydrodynam ic ones and as such provide an applicability lim it for an effective hydrodynam ic d escrip tion . T hese p h en o m en a will b e a t th e focus of th e p resen t p a p e r. In d eed we find th a t th ey become very im p o rta n t in th e vicinity of a phase tran sitio n .

Finally, to dem ystify som ew hat these higher q u asin o rm 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 sta rts w ith an anisotropic m om entum distribu tion for th e gluons. T h en th e initial energy-m om entum tensor is spatially con stant b u t anisotropic.

If we let th e system evolve, th e system will therm alize (w ith th e energy-m om entum tensor becom ing eventually isotropic). However this (homogeneous) isotropization will not excite an y h y d ro d y n am ic m odes as th e sy m m etry of th e p rob lem forbids an y flow. T h u s th e relevant ex citatio n s will be different. A t stro n g coupling th e y corresp o n d ex a ctly to th e higher q uasinorm al modes.

In th e analysis below we m easure th e m o m e n ta a n d th e frequencies in th e u n its of te m p e ra tu re by settin g

q = 2 ^ , - = 2 p f (3-14)

T h ere are a few novel p red ictio n s w hich we m ake from th e QNM frequencies. F irs t is to e stim a te th e m o m e n tu m , or equivalently th e le n g th , scale w here th e h yd ro d y n am ic description of th e system breaks. For a C F T case this was evaluated to be q = 1.3 where in th e shear channel first n o n -h yd ro d yn am ic QNM d o m in a te d th e system dynam ics [29]. In th e sam e tim e th is effect d id n o t a p p e a r in th e C F T sound channel. T h e new featu re we

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p o ten tial sound channel qc shear channel qc c2cs ( / s

VQCD 0.8 1.1 0.124 0.041

V2nd 0.55 0.9 0.0 0.061

Vlst 0.8 1.15 0.0 0.060

VIHQCD 0.14 1.25 0.0 0.512

T able 2. The momenta for which the crossing phenomena in different channels and the corresponding values of the speed of sound and bulk viscosity read of from the hydrodynamic mode. Values given at corresponding critical temperatures (Tm for Vist and Vh q c d).

find is th a t we see this crossing2 not only in th e shear channel b ut also in th e sound channel.

This shows th a t th e influence of a non-trivial phase stru c tu re of th e background affects th e applicability of hydrodynam ics in a qualitativ e way. O th er aspect is th a t th e hydrodynam ic d escrip tio n is valid in large enough len g th scale (th e sm aller critical m o m en tu m ) w hich m eans th e applicability of hydrodynam ics near th e phase tran sitio n is m ore restricted th an in th e high te m p e ra tu re case.

In ta b le 2 we sum m arize th e c ritic a l m o m e n ta in tw o channels a n d h ydro d y n am ic p a ra m e te rs for different p o ten tia ls. All q u a n titie s are e v alu ated a t correspon d ing critical tem p e ra tu re s. In th e following subsections we will show th e Q N M ’s m ostly for th e sound channel w hich p resent ch aracteristic stru c tu re for each p o ten tia l. Since th e shear channel in all cases has th e sam e form (w ith different critical m om entum ) we restric t ourselves to show only one related plot for th e VqCD potential.

T h e second observation is th e bubble form ation in th e spinodal region in th e case of th e 1st order phase transitio n [30]. This happens when c2 < 0 which m eans th a t hydrodynam ical m ode is p u rely im aginary. For sm all m om enta, u h = ± i \ c s \k — i T sk 2, th e m ode w ith th e plus sign is in th e u n sta b le region, i.e., Im u h > 0. For larg er m o m e n ta th e o th e r te rm sta rts to dom inate, so th a t th ere is kmax = \cs \ / r s for which th e hydro mode becomes stable again. T he scale of th e bubble is th e m om en tu m for w hich positive im aginary p a rt of th e hydro m ode a tta in s th e m axim al value. Im aginary p a rt of th e unstable hydro m ode is called th e grow th ra te [30].

T h ird observation is th a t th e hydrodynam ical mode of th e sound channel in 1 st order case near th e critical tem p e ra tu re T c, and in th e IHQCD case also th e first non-hydrodynam ical modes, become purely im aginary for a range of m om enta. In terp retatio n of th is fact is th a t th e corresponding w avelengths can n ot p ropag ate at a linearized level and correspondingly th ere is a diffusion-like m echanism for those modes.

It is im p o rta n t to note th a t generically th e ultra-locality [6] of th e non-hydrodynam ic m ode is still p resen t in th e critical region of th e ph ase d iag ram . T h e only exception observed is th e IH Q C D p o te n tia l, w here th e m odes exhibit a non triv ia l behaviour. M ost of th e in terestin g dynam ics an d effects observed are due to th e different behavio ur of th e h y d ro d y n am ica l m odes a n d how th e y cross th e m ost d a m p e d no n -h y d ro d y n am ic m odes.

T h is includes th e in sta b ility a n d th e bu b b le fo rm a tio n in th e case of th e 1 st o rd er phase tran sitio n .

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.

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F ig u re 1. Left panel: entropy density for Vqcd potential with k5 = 1. Right panel: speed of sound squared as a function of temperature. Dots are the lQCD equation of state [14].

4 T h e cro sso v er ca se

T h e resu lts for th e Q N M w ith a Q C D -like eq u atio n s of s ta te are su m m arized below.

P a ra m e te rs of th is p o te n tia l have b een chosen to fit th e te m p e ra tu re d ependence of th e speed of sound obtained in lattice QCD com putations w ith dynam ical quarks at zero baryon chem ical p o ten tial [14].

In our c o m p u tatio n s from th e hydrod yn am ic m ode we e stim a te th e value of th e bulk viscosity, w hich is in agreem ent w ith ref. [24] (cf. ta b le 2 ) . It is im p o rta n t to note, th a t d esp ite th e fact th a t th e EoS of QCD are co rrectly rep ro d u c e d in th e m odel tra n s p o rt coefficients are lower th a n th e lattice predictions [31, 32]. For exam ple only th e q ualitative te m p e ra tu re dependence of b u lk viscosity is correct, n am ely th a t it rap id ly raises n e a r th e Tc [24].3

In th is analysis we tak e a n o th e r step, a n d stu d y th e te m p e ra tu re a n d m o m en tu m behaviour not only of th e hydrodynam ic m ode b u t also of th e first and second of th e infinite tow er of h igher m odes. In p a rtic u la r th is allows us to e stim a te th e ap p licab ility of th e hydrodynam ic approxim ation in th e critical region of tem p e ra tu re s where we find crossing of th e m odes in sound channel.

Firstly, before we move to th e new results, using th e exam ple of th e Vqcd potential, let us discuss th e high tem p eratu re quasinorm al modes. T he results com puted for T = 3Tc are show n in figure 2 . T he speed of sound, shear a n d b ulk viscosities re a d of from th e lowest Q N M are very close to resu lts ex p ected for a conform al system , i.e., n / s — 1 /(4 n ), — 0.321, Z /s — 0.003. M odes co m p u te d for th is te m p e ra tu re in th e sou nd a n d th e shear channels are in agreem ent w ith th e conformal results of ref. [3]. As we m entioned in previous section, since Z i( r ) and Z 2(r) m odes are coupled th e nonhydrodynam ic Q N M ’s are in pair in all range of te m p e ra tu re s, an d th e second m ost d a m p e d nonhyd ro dy nam ic m ode tu rn s o u t 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 p e ra tu re s. T he results co m p u te d for th e p seud o -critical te m p e ra tu re , T = Tc, are show n in figure 3 . T h e 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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Figure 3 . The real (left) and imaginary (right) parts of the quasinormal modes in the sound channel for the potential VQcd at T = Tc.

i m p o r t a n t d i f f e r e n c e w i t h r e s p e c t t o h i g h -T c a s e i s a c h a n g e i n l a r g e m o m e n t u m d e p e n d e n c e o f t h e i m a g i n a r y p a r t o f t h e h y d r o d y n a m i c m o d e . I n s t e a d o f a p p r o a c h i n g s o m e c o n s t a n t v a l u e t h e i m a g i n a r y p a r t o f t h e m o d e f l o w s t o m i n u s i n f i n i t y a s m o m e n t u m i n c r e a s e s . T h i s i m p l i e s a n o v e l e f f e c t i n t h e s o u n d c h a n n e l : c r o s s i n g b e t w e e n t h e h y d r o d y n a m i c a n d n o n - h y d r o d y n a m i c m o d e a p p e a r s . A t t h e p s e u d o c r i t i c a l t e m p e r a t u r e t h i s h a p p e n s f o r c r i t i c a l m o m e n t u m q c ~ 0 . 9 . W h i l e i n t h e c o n f o r m a l c a s e t h i s w a s p r e s e n t o n l y i n t h e s h e a r c h a n n e l f o r q c ~ 1 . 3 [2 9 ] , a s s h o w n i n f i g u r e 4 f o r t h e c r o s s o v e r p o t e n t i a l q c ~ 1 . 1 5 i n t h e s a m e c h a n n e l . I n c o n t r a s t , n o n h y d r o d y n a m i c m o d e s a r e n o t m u c h a f f e c t e d o b e y i n g u l t r a l o c a l i t y p r o p e r t y [ 6 ] .

I n v i e w o f p o s s i b l e r e l a t i o n s t o Q C D w e c o u l d e x p e c t o n l y q u a l i t a t i v e p r e d i c t i o n s f r o m o u r c o m p u t a t i o n s . H o w e v e r , l a t t i c e Q C D c o m p u t a t i o n s c o u l d , i n p r i n c i p l e , v e r i f y t h e u l t r a - l o c a l i t y p r o p e r t y a n d t h e g e n e r i c c r o s s i n g o f t h e m o d e s . T h e m a i n o b s t r u c t i o n i n t h i s c a s e w o u l d b e t h e n e c e s s i t y o f r e a l t i m e f o r m u l a t i o n o f t h e p r o b l e m , w h i c h i s n o t y e t a v a i l a b l e o n t h e l a t t i c e .

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F ig u re 2. Sound channel quasinormal modes for the potential VQcd at T = 3Tc. Real part (left panel) and imaginary part (right panel).

F ig u re 4. The real (left) and imaginary (right) parts of the quasinormal modes in the shear channel for the potential VQcd at T = Tc.

Figure 5 . Left panel: equation of state for V2nd. Right panel: equation of state for V2nd near the Tc (blue points). Magenta line is the fit (5.1) with ^a ~ 0.65. In both plots we set ^k5 = 1.

5 T h e sec o n d ord er p h a se tr a n s itio n ca se

I n t h i s s e c t i o n w e p r e s e n t r e s u l t s f o r t h e c a s e o f a s y s t e m w i t h 2 n d p h a s e t r a n s i t i o n E o S , w h i c h c a n b e a c h i e v e d b y a s u i t a b l e c h o i c e o f p a r a m e t e r s . W e d o n o t f i t t o a n y p a r t i c u l a r s y s t e m c o n s i d e r e d i n t h e l i t e r a t u r e - w e o n l y r e q u i r e a p a r t i c u l a r s h a p e o f t h e e n t r o p y a s a f u n c t i o n o f t e m p e r a t u r e ( c f . l e f t p a n e l o f f i g u r e 5 ) w h i c h l e a d s t o v a n i s h i n g s p e e d o f s o u n d a t t h e c r i t i c a l t e m p e r a t u r e T = Tc [ 1 5 ] . N e a r t h e Tc e n t r o p y o f t h e s y s t e m t a k e s t h e f o r m

s ( T) ~ s o + s i t 1-a , ( 5 . 1 )

w h e r e t = ( T — T c ) / T c , a n d a ~ 0 . 6 5 i s t h e s p e c i f i c h e a t c r i t i c a l e x p o n e n t ( c f . r i g h t p a n e l o f f i g u r e 5 ) . T h i s v a l u e i s v e r y c l o s e t o a = 2 / 3 f r o m r e f . [ 1 5 ] .

T h e r e s u l t s f o r Q N M a t c r i t i c a l t e m p e r a t u r e a r e d i s p l a y e d i n f i g u r e 6 . S i n c e t h e r e i s n o n e w p h e n o m e n a i n t h e s h e a r c h a n n e l , o n l y t h e s o u n d m o d e i s s h o w n . G e n e r i c t e m p e r a t u r e d e p e n d e n c e o f Q N M f r e q u e n c i e s i s v e r y s i m i l a r t o t h e c r o s s o v e r c a s e . T h e m a i n d i f f e r e n c e c o m p a r e d t o t h e c r o s s o v e r p o t e n t i a l ( f i g u r e 3 ) i s t h a t a t T c t h e h y d r o d y n a m i c d e s c r i p t i o n o f t h e s y s t e m b r e a k s d o w n a l r e a d y a t s m a l l e r m o m e n t a s c a l e s .

W e w o u l d l i k e t o m e n t i o n t h a t i n h i g h t e m p e r a t u r e r e g i m e w e r e c o v e r e d t h e C F T r e s u l t s i n b o t h c h a n n e l s w i t h t h e p a i r s t r u c t u r e e x p l a i n e d i n t h e p r e v i o u s s u b s e c t i o n i n t h e s o u n d c h a n n e l d u e t o c o u p l i n g o f t h e m o d e s .

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Figure 6 . Quasinormal modes for the potential V2nd at Tc. Real part (left panel) and imaginary part (right panel).

6 T h e first ord er p h a se tr a n s itio n ca se

I n t h i s s e c t i o n w e d i s c u s s t h e m o s t f a s c i n a t i n g c a s e o f a s y s t e m w h i c h e x h i b i t s a 1 s t o r d e r p h a s e t r a n s i t i o n . T h e r e a r e t w o p o s s i b l e s c e n a r i o s f o r s u c h a t r a n s i t i o n : o n e i s s i m i l a r t o H a w k i n g - P a g e c a s e w h e r e t h e r e i s a t r a n s i t i o n f r o m a b l a c k h o l e t o t h e v a c u u m g e o m e t r y w i t h o u t a h o r i z o n [ 1 9 ] . T h e s e c o n d o n e , m e n t i o n e d i n r e f . [ 1 7 ] , i s a t r a n s i t i o n f r o m o n e b l a c k h o l e s o l u t i o n t o a n o t h e r . I n t h i s s e c t i o n w e c o n s i d e r t h e l a t t e r c a s e w h i l e t h e f o r m e r a p p e a r s i n t h e s t u d i e s o f I H Q C D m o d e l s ( c f . s e c t i o n 7 ) . T h e o n s e t o f t h e a p p e a r a n c e o f a n o n p r o p a g a t i n g s o u n d m o d e i n t h e d e e p l y o v e r c o o l e d p h a s e h a s b e e n o b s e r v e d e a r l i e r i n a r e l a t e d m o d e l [ 5 ] .

I n t h e V i s t p o t e n t i a l c a s e t h e r e e x i s t t h r e e c h a r a c t e r i s t i c t e m p e r a t u r e s . T h e f i r s t o n e i s t h e m i n i m a l t e m p e r a t u r e Tm , b e l o w w h i c h n o u n s t a b l e s o l u t i o n e x i s t s . T h e o n s e t o f i n s t a b i l i t y i s s e e n a t t e m p e r a t u r e s T > Tm ( i n t h e b r a n c h w h e r e c2(T) < 0 ) , a n d g e n e r i c a l l y w e e x p e c t t h e 1 s t o r d e r p h a s e t r a n s i t i o n t o a p p e a r a t a c r i t i c a l t e m p e r a t u r e T c > T^{m ,}

w h i c h i s d e t e r m i n e d b y t h e t e m p e r a t u r e d e p e n d e n c e o f t h e F r e e E n e r g y . T o e v a l u a t e t h i s o n e c a n e i t h e r u s e d i r e c t o n - s h e l l a c t i o n s o r o n e c a n u s e t h e m e t h o d o u t l i n e d i n s e c t i o n 2 . T h e l a t t e r u s e s t h e s t a n d a r d t h e r m o d y n a m i c r e l a t i o n d F = — s d T, w h e r e t h e i n t e g r a t i o n c o n s t a n t c a n b e f i x e d b y t h e c h o i c e o f t h e r e f e r e n c e g e o m e t r y w i t h v a n i s h i n g h o r i z o n a r e a , w h i c h i n t h i s c a s e c o r r e s p o n d s t o T = 0 s o l u t i o n . T e m p e r a t u r e d e p e n d e n c e o f t h e F E f o r t h i s c a s e i s s h o w n i n t h e r i g h t p a n e l o f f i g u r e 7 a n d w e d e t e r m i n e d T c ~ 1 . 0 5 T m . T h e o t h e r c h a r a c t e r i s t i c t e m p e r a t u r e i s e s t i m a t e d t o b e T c h ^ 1 . 0 0 0 1 T m , w h i c h i s b a s e d o n t h e o b s e r v a t i o n , t h a t f o r a r a n g e o f m o m e n t a t h e h y d r o d y n a m i c m o d e s b e c o m e p u r e l y i m a g i n a r y a n d d o n o t p r o p a g a t e i n t h e p l a s m a ( c f . f i g u r e 8 ) . T h i s e f f e c t a p p e a r s f o r t e m p e r a t u r e s

Tm < T < T c h , i n t h e stable r e g i o n o f t h e E o S ( g r e e n l i n e i n l e f t p a n e l o f f i g u r e 7 ) . L e t u s n o t e t h a t i n t h i s m o d e l Tm < T c h < Tc.

N o w w e t a k e a l o o k a t Q N M s t r u c t u r e a t t h e m i n i m a l t e m p e r a t u r e Tm , i n w h i c h t h e g r e e n l i n e a n d r e d - d a s h e d l i n e m e e t i n f i g u r e 7 a n d t h e s p e e d o f s o u n d v a n i s h e s . T h e r e i s n o n e w s t r u c t u r e i n t h e s h e a r c h a n n e l a n d w e o n l y p l o t t h e s o u n d c h a n n e l Q N M ’s i n f i g u r e 9 . O n e m a y s e e a n e w p a t t e r n a t t h i s p o i n t c o m p a r e d t o t h e c r o s s o v e r a n d t h e 2 n d o r d e r p h a s e t r a n s i t i o n c a s e s , i . e . , t h e h y d r o d y n a m i c m o d e s a r e p u r e l y i m a g i n a r y ( d i f f u s i v e - l i k e ) f o r q < 1 .

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Figure 7 . Left panel: entropy density for ^V1st potential. Green line is the stable region, while red dashed line displays an instability. Right panel: free Energy difference between two black hole solutions as a function of tem perature. Estim ated critical tem perature is ^Tc ~ 1^.05^Tm. In both plots we set k5 = 1.

Figure 8 . Quasinormal modes for the potential ^Vist at ^T = 1^.00004^Tm. Real part (left panel) and imaginary part (right panel).

Figure 9 . Quasinormal modes for the potential ^V1st at ^T = ^Tm. Real part (left panel) and imaginary part (right panel).

T h e m o s t e n g r o s s i n g p h y s i c s i s d i s c o v e r e d i n t h e spinodal r e g i o n ( r e d - d a s h e d l i n e f i g u r e 7 ) w h e r e t h e e q u a t i o n o f s t a t e s u g g e s t s t h e r m o d y n a m i c a l i n s t a b i l i t y , i . e . , c 2 < 0 ( c f . f i g u r e 7 ) . I t w a s a l r e a d y a n t i c i p a t e d i n l i t e r a t u r e [ 3 5 , 3 6 ] t h a t i n t h i s r a n g e o f t e m p e r a t u r e s a c o r r e s p o n d i n g d y n a m i c a l i n s t a b i l i t y s h o u l d a p p e a r i n t h e l o w e s t Q N M m o d e .

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Figure 10. Sound channel quasinormal modes for the potential Vist at T ~ 1.06Tm. An instability of the spinodal region is shown. The speed of sound at th at tem perature is c?s ~ -0.1.

W e s t u d y t h e i n s t a b i l i t y p h e n o m e n o n i n d e t a i l b y o b s e r v i n g t h e b u b b l e f o r m a t i o n i n t h e s p i n o d a l r e g i o n . I t i s g e n e r i c a l l y e x p e c t e d i n t h e c a s e o f t h e 1 s t o r d e r p h a s e t r a n s i t i o n [3 0 ] a n d a s i m i l a r e f f e c t w a s o b s e r v e d i n t h e g r a v i t y c o n t e x t b y G r e g o r y a n d L a f l a m m e [3 7 ] . T h e f o r m a t i o n h a p p e n s w h e n < 0 , w h i c h m e a n s t h a t h y d r o d y n a m i c m o d e i s p u r e l y i m a g i n a r y

= ± i | c s | k — i r s k 2 . F o r s m a l l e n o u g h k t h e m o d e w i t h t h e p l u s s i g n i s i n t h e u n s t a b l e r e g i o n , i . e . , I m w g > 0 . F o r l a r g e r m o m e n t a t h e o t h e r t e r m s t a r t s t o d o m i n a t e , s o t h a t t h e r e i s k m a x = | c s | / r s f o r w h i c h t h e h y d r o d y n a m i c m o d e b e c o m e s a g a i n s t a b l e . T h e s c a l e o f t h e b u b b l e i s t h e m o m e n t u m f o r w h i c h p o s i t i v e i m a g i n a r y p a r t o f t h e h y d r o d y n a m i c m o d e a t t a i n s t h e m a x i m a l v a l u e . I m a g i n a r y p a r t o f t h e u n s t a b l e h y d r o d y n a m i c m o d e i s c a l l e d t h e g r o w t h r a t e [3 0 ] . I t i s i n t r i g u i n g t o n o t e , t h a t f o r t h e V 1 s t p o t e n t i a l t h e h y d r o m o d e i s p u r e l y i m a g i n a r y u p t o m o m e n t a q w 5 , i . e . , i n a l l i n v e s t i g a t e d r a n g e . A n i n t e r e s t i n g o b s e r v a t i o n i s t h a t a l l h i g h e r m o d e s r e m a i n s t a b l e i n t h i s c a s e . P l o t i l l u s t r a t i n g t h e s e w o r d s i s p r e s e n t e d i n f i g u r e 1 0 .

N o t o n l y i n t h e c a s e w h e n t h e r e i s a n i n s t a b i l i t y r e g i o n i n t h e E o S , b u t a l s o f o r t h e t e m p e r a t u r e c l o s e t o t h e Tm i n t h e s t a b l e r e g i o n a n d a t Tm t h e h y d r o d y n a m i c m o d e s b e c o m e p u r e l y i m a g i n a r y . T h e s e c a s e s a r e s h o w n i n f i g u r e s 1 0 , 8 , 9 r e s p e c t i v e l y . W h e n a h y d r o d y n a m i c m o d e , ( k ) , i s p u r e l y i m a g i n a r y , o n e c a n e x p r e s s i t a s

( k ) = ± i O ( k ) — i E ( k ) , ( 6 . 1 )

w i t h O ( — k ) = — O ( k ) a n d E ( — k ) = E ( k ) . T h e n t h e r e a r e t w o s e p a r a t e d b r a n c h e s o f t h e h y d r o d y n a m i c a l m o d e s , a s s e e n i n f i g u r e s 8 , 9 a n d 1 0 . W h e n t h i s h a p p e n s h y d r o d y n a m i c a l m o d e i s n o t a p r o p a g a t i n g o n e , b u t h a s s o m e s o r t o f a “ d i f f u s i v e - l i k e ” b e h a v i o u r .

7 T h e im p ro v ed h o lo g ra p h ic Q C D

T h i s p o t e n t i a l i s i n a c l a s s d e s i g n e d t o g r a s p t h e d y n a m i c a l f e a t u r e s o f Q C D : t h e a s y m p t o t i c f r e e d o m a n d c o l o u r c o n f i n e m e n t [ 1 7 , 1 8 ] . I n t h o s e a s p e c t s i t i s a m o r e d e t a i l e d m o d e l t h a n t h e o n e u s e d i n s e c t i o n 4 . A s y m p t o t i c f r e e d o m i s i m p l e m e n t e d b y l o g a r i t h m i c c o r r e c t i o n s t o t h e p o t e n t i a l i n t h e U V r e g i o n , w h i l e c o n f i n e m e n t i s d e t e c t e d b y a l i n e a r d e p e n d e n c e o f t h e g l u e b a l l s m a s s e s o n t h e c o n s e c u t i v e n u m b e r , i . e . , m ^ ~ n f o r l a r g e n . T h i s i s s o m e t i m e s

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Figure 11. Upper panel: entropy density as a function of tem perature for U i h q c d potential with

« 5 = 1. Lower panel: speed of sound squared for the U i h q c d potential (green line), and pure gluon SU(3) lattice data (orange dots) [39]. Red and blue dashed lines on the right hand side plot correspond to small black hole solutions, which always turn out to be unstable (see text).

r e f e r r e d t o a s a l i n e a r c o n f i n e m e n t [ 3 8 ] . T h e p o t e n t i a l w e c h o o s e h a s c o n f i n i n g I R a s y m p t o t i c , b u t d o e s n o t i n c l u d e t h e l o g a r i t h m i c c o r r e c t i o n s i n t h e U V .

T h e I H Q C D p o t e n t i a l d e t e r m i n e s u n i q u e e q u a t i o n o f s t a t e , w i t h a r i c h s t r u c t u r e d i s p l a y e d i n f i g u r e 1 1 . T h e t w o b r a n c h e s o f b l a c k h o l e s o l u t i o n s a r e d i v i d e d a s u s u a l i n t o l a r g e ( s t a b l e ) , a n d s m a l l ( u n s t a b l e ) c o n f i g u r a t i o n s . S t a b l e c o n f i g u r a t i o n s s h o w b e h a v i o u r w i t h t h e u s u a l f e a t u r e s c h a r a c t e r i s t i c f o r a s y s t e m w i t h a f i r s t , o r d e r p h a s e t r a n s i t i o n , a n d t h e c o r r e s p o n d i n g s p e e d o f s o u n d i s q u a l i t a t i v e l y s i m i l a r t o t h e p u r e g l u e s y s t e m [ 3 9 ] . O n t h e c o n t r a r y , u n s t a b l e b r a n c h c o n s i s t s o f t w o d i s t i n c t s u b b r a n c h e s . O n e o f t h e m i s i n a d i s c o n n e c t e d r a n g e o f t e m p e r a t u r e s , Tm < T < T \ = 1 . 0 1 4 T m a n d = ^{5 . 6 7}Tm < T ,

a n d d i s p l a y s s p i n o d a l i n s t a b i l i t y s i g n a l e d b y t h e i m a g i n a r y s p e e d o f s o u n d . T h i s i n t u r n i m p l i e s b u b b l e f o r m a t i o n a s d e s c r i b e d i n s e c t i o n 6 . S e c o n d s u b - b r a n c h , T\ < T < T2, s h o w s a n o m a l o u s l y l a r g e s p e e d o f s o u n d , b u t d o e s n o t s h o w a n y i n s t a b i l i t y o n t h e l e v e l o f e q u a t i o n s o f s t a t e . H o w e v e r , a s w i l l b e s h o w n b e l o w , i n t h i s r a n g e o f t e m p e r a t u r e s t h e r e e x i s t s a n u n s t a b l e n o n - h y d r o m o d e i n t h e Q N M s p e c t r u m .

T h i s s y s t e m i s e x p e c t e d t o h a v e a p h a s e t r a n s i t i o n o f a 1 s t o r d e r b e t w e e n a b l a c k h o l e g e o m e t r y w i t h a n e v e n t h o r i z o n , a n d t h e v a c u u m c o n f i n i n g g e o m e t r y i n t h e s p i r i t o f H a w k i n g - P a g e p h a s e t r a n s i t i o n [ 1 9 ] . I n p r i n c i p l e , t o e s t i m a t e Tc w e c a n f i n d t h e t e m p e r a t u r e d e p e n d e n c e o f t h e F E a l o n g t h e l i n e s m e n t i o n e d i n s e c t i o n 2 . I n t h i s c a s e n o n o f t h e m e t h o d s b r i n g s u p a d e c e n t r e s u l t . T h e d i r e c t e v a l u a t i o n o f t h e o n - s h e l l a c t i o n i s c o r r u p t e d b y a

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num erical in stab ility , while th e s ta n d a rd th e rm o d y n am ic relation , d F = —s d T, suffers a problem of correct choice of th e reference configuration. A possible can did ate for reference geom etry is th e one of vanishing horizon area in th e unstable black hole branch. However it has infinite tem p e ra tu re and it is not clear for us w hether it can be used as a proxy for th e th erm al gas geometry. Also th e unstable branch black holes exhibit a variety of pathologies (which will be described later) w ith increasing T . Due to th e above m entioned difficulties we refrained from estim atin g th e value of Tc in this p a rticu la r model. N evertheless we expect th a t th e re exists a critical te m p e ra tu re Tc > T m w here th e tra n s itio n tak es place [15, 17].

This tra n sitio n changes th e geom etry substantially.

It is im p o rta n t to note th a t in th is case th ere exists a m inim al te m p e ra tu re T m below w hich a black hole solu tio n does n o t exist. As in th e case of V1st th e onset of in sta b ility app ears a t T > T m (for configurations w ith c2( T) < 0).

T h e different s tru c tu re o f th e EoS is reflected in th e beh av io u r of Q N M frequencies, w hich in d ic a te th e existence of second c h a ra c te ristic te m p e ra tu re Tch ^ 1.102Tm. T he novel effect observed in this system is th a t for tem p eratu res near th e m inim al tem p eratu re th e u ltra lo c a lity p ro p e rty of th e first no n -h y d ro d y n am ic m ode is violated. T h e m ode tu rn s out to be purely im aginary for very low m om enta and for tem p e ra tu re s of th e range T m < T < Tch, and it does not have a stru ctu re described in eq. (6.1) . There are two purely im aginary m odes which have th e following form

u ±(k) = i x ( k ) ± i£ (k ) • (7.1)

In figure 12 we show th e te m p e ra tu re dependence of th o se m odes a t k = 0 in th e range w here th e re are p u rely im aginary. As th e system is h e a te d fu rth e r th e real p a rt develops, and th e mode becomes th e least dam ped non-hydrodynam ic mode of th e high- T limit, w ith th e u su al s tru c tu re (3.11) . It d irectly com es from th e presence of th e back g ro u n d scalar field, which breaks th e conform al invariance.

In figure 13 we show Q N M ’s in th e sou nd channel c o m p u te d for V ihqcd a t T = T m . T h e m ode s tru c tu re is different th a n th e one generically p resen t in previous cases. F irs t th in g w hich is a p p a re n t is th a t h y d ro d y n am ic m odes are p u rely im ag in ary for a range of sm all m om en ta. In ad d itio n , th e re is a sm all gap betw een th e h y d ro d y n am ic an d n on-hydrodynam ic degrees of freedom a t a rb itra ry low m om entum , w hich in tu r n implies th a t th e crossing h a p p e n s a t very low value of qc ~ 0.14 (see th e in se rt in figure 13) . As a m a tte r of fact, in th is case n e a r th e T m one m u st alw ays tak e in to account th e n o n­

h y d ro d y n am ic degrees of freedom in th e d escrip tio n of th e system dynam ics. A n o th e r absolutely fascinating effect observed exactly a t T m is th a t th e non-hydrodynam ic modes, which are purely im aginary for low m om enta, join w ith th e hydrodynam ic m odes a t some finite m om entum q j , and follow them w ith increasing q. This effect is illustrated in figure 13, where th e non-hydro1 m ode which has two branches joins w ith th e two branches of th e hydro modes respectively at qJ ~ 0.14 and qJ ~ 1.5. In th e same tim e th e real p a rt develops with bo th signs, as expected from general considerations (see eq. (3.11) ). This effect implies th e ultralocality violation observed generically in other models, and joining does not happen for tem p eratu res higher th a n th e m inim al one. T he final observation from figure 13 is th a t the

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