The fate of the conformal order

We use holographic correspondence to study transport of the conformal plasma in ${\mathbb R}^{2,1}$ in a phase with a spontaneously broken global $Z_2$ symmetry. The dual black branes in a Poincare patch of asymptotically $AdS_4$ have ``hair'' -- a condensate of the order parameter for the broken symmetry. This hair affects both the hydrodynamic and the nonhydrodynamic quasinormal modes of the black branes. Nonetheless, the shear viscosity of the conformal order is universal, the bulk viscosity vanishes and the speed of the sound waves is $c_s^2=\frac 12$. We compute the low-lying spectrum of the non-hydrodynamic modes. We identify a quasinormal mode associated with the fluctuations of the $Z_2$ order parameter with the positive imaginary part. The presence of this mode in the spectrum renders the holographic conformal order perturbatively unstable. Correspondingly, the dual black branes violate the correlated stability conjecture.


Introduction and summary
Following the general suggestion of [1], we proposed in [2] where F is the free energy density, T is the temperature and c is the central charge.
The constant 0 ≤ κ ≤ 1 and the thermal expectation value of O (in the symmetry breaking phase) depends on the parameter b of the dual gravitational action 3 : The symmetry broken phase exists only b < b crit,0 ≡ − 3 2 . In this paper we continue study of the model (2). First, we point out that there are multiple branches, indexed by i = 0, 1, · · · , of the "hair" -the thermal expectation values of O. In the holographic dual, the index is related to the number of zeros in the radial profile of the holographic bulk scalar χ. For two branches with i < j, i.e., the branches with the higher index are increasingly less thermodynamically favored -all the symmetry breaking phases have a higher free energy density than that of the Z 2 -symmetric phase. We find that the thermodynamics of all the symmetry broken phases resemble that of the symmetric phase in the limit b → −∞, , Notice that the gravitational potential for χ is unbounded from below as b < 0, (naively) implying that the exotic features of the model (2) are due to this 'sickness'. This is not the case: given (4), it is clear that a simple deformation of the scalar potential, i.e., makes it bounded, while not affecting the thermodynamics of the model, at least as b → −∞: We explicitly demonstrate that the symmetry breaking phases at finite b are robust against the deformation (6), for small enough g > 0.
We study the coupled metric-scalar fluctuations in the symmetry broken phases of (2) and compute the spectrum of the quasinormal modes (QNMs) of the black branes dual to the conformal order on the lowest branch 4 . As expected from the conformal theory, irrespectively of the symmetry breaking, we find for the speed of the sound waves c s and the bulk viscosity ζ correspondingly. From the dispersion relation of the hydrodynamic mode in the shear channel [5] we recover the universal result [6][7][8] for the ratio of the shear viscosity η to the entropy density s η s = 1 4π .
Besides the sound wave -a hydrodynamic QNM in the scalar channel [5] there are two branches of the non-hydrodynamic modes coming from the mixing of the helicity zero metric fluctuations in the equilibrium black brane geometry and the gravitational bulk scalar field, whose boundary value determine the order parameter for the Z 2 symmetry breaking. There are two purely dissipative non-hydrodynamic modes 5 : One of these QNMs, w u (q), at least when the spatial momentum q is below some critical value q c = q c (b), has a positive imaginary part, i.e., We further show that approaching zero in the limit b → b crit,0 . The presence of this mode in the spectrum implies that the translationary invariant horizon of the black brane dual to a conformal order is perturbatively unstable to clumping. We expect 6 that the dynamical evolution of the perturbed conformal order will result in a destruction of the ordered phase, with the Z 2 symmetric equilibrium phase being the attractor. We show that the unstable QNM is present on the higher branches of the conformal order as well.
Note that the perturbative instability of the holographic conformal order proposed in [2] implies that the Correlated Stability Conjecture of [9,10] for the dual black branes is violated: while these black branes are thermodynamically stable, they are dynamically unstable 7 . 5 We take the space-time dependence of the QNM fluctuations to be ∝ e −iωt+i k x and introduce w = ω 4πT and q = | k| 4πT . 6 Work in progress. 7 See [11,12] for other examples of the CSC violation.
A challenge remains to find an example of a stable holographic thermal conformal order.

Holographic thermal conformal order
In this section we review the construction of the holographic conformal order proposed in [2] and discuss two generalizations: we show that there are multiple branches of the conformal order; we show that the effective scalar potential in the gravitational dual can be made bounded, without affecting the existence of the ordered phases.

Branches of the conformal order
The starting point is the effective action (2). The thermal ordered states are dual to black brane solutions where all the metric warp factors c i as well as the bulk scalar χ are functions of the radial coordinate r, where r 0 is a location of a regular Schwarzschild horizon, and r → +∞ is the asymptotic AdS 4 boundary. Introducing a new radial coordinate and denoting we obtain the following system of ODEs (in a radial coordinate x, ′ = d dx ): where the scalar potential V is given by (5), and ∂V ≡ δV δχ . Notice that r 0 is completely scaled out from all the equations of motion. Eqs.(17)- (19) have to be solved subject to the following asymptotics: in the UV, i.e., as x → 0 + , in the IR, i.e., as y ≡ 1 − x → 0 + , where V (c h 0 ) implies that the scalar potential (5) is evaluated on the horizon value of the bulk scalar χ.
In total, given b, the asymptotic expansions are specified by 4 parameters which is the correct number of parameters necessary to provide a solution to a system of a single second order and two first order equations, 1×2+2×1 = 4. It is straightforward to extract the thermodynamics of the resulting black brane: for the free energy density F , the energy density E, the temperature T , and the thermal expectation value of the conformal order Ô . We explicitly indicated the r 0 dependence -all the parameters in (22), as well as κ and V (c h 0 ) depend only on b. From (23), the speed of the sound waves in the holographic conformal order plasma is Additionally, since that temperature T depends on r 0 , and the horizon value of the scalar is r 0 -independent, implying that the bulk viscosity of the corresponding plasma must vanish 8 [15] ζ T 2 = 0 .
In section 3.1 we reproduce (24) and (26) from the dispersion relation of the hydrodynamic QNMs.
Note that the "disordered phase" corresponds to identical vanishing of the bulk scalar field, in this case To understand the ordered phases it is the easiest to consider b → −∞ limit. From (17)- (19) it is straightforward to see that and correspondingly (from (20) a 1,3 ≡ −a 3,3 ) i.e., in this limit the "hairy" black branes approach the standard AdS 4 -Schwarzschild solution 9 . To leading order, i.e., n = 1, we have: .
(32) 8 Eling-Oz formula implies that the holographic plasma bulk viscosity is proportional to the square of the derivative of the bulk scalar field(s) evaluated at the horizon with respect to the temperature, keeping all the mass parameters fixed [13,14]. 9 As we will see in section 3.2, this is not the case for the spectrum of the non-hydrodynamic QNMs: some of the QNMs of the hairy black branes remain distinct from the AdS 4 -Schwarzschild black brane QNMs in the limit b → −∞. There is a discrete set of solutions of (30) subject to the boundary conditions inherited from (20) and (21), characterized by the number of zeroes in the function f [1] . In the left panel of fig. 1 we present with i = 0, 1, 2 (green, orange, brown curves correspondingly) zeros. The discrete set of solutions for (30) leads to discrete sets of a 3, [1] and a 1, [1] . We just constructed the lowest 10 (i = 0) and the higher (i = 1, 2) overtones of the holographic conformal order: # of zeros f 1, [4]  where ± for the parameters specifying f [1] reflects the spontaneously broken global Z 2 symmetry. To leading order as b → −∞ (see (23)) which is represented by the dashed green/orange/brown (for i = 0, 1, 2 correspondingly) curves in the right panel of fig. 1. Once the overtones of the conformal order are Note that lim b→b crit,i and for each i < j, Thus, all the ordered phases are subdominant (have the higher free energy density) compare to the Z 2 -symmetric phase, see (27). Additionally, the free energy density of the conformal order overtones increases (at fixed b) with its index.
Equilibrium thermal phases with or without the global Z 2 symmetry are thermodynamically stable. There is a latent heat ∆E associated with the transitions between the symmetry broken phases i < j, and the transitions to the Z 2 symmetric phase, typically indicative of the first-order phase transition. Rather, as we show in section 3.2, each of the ordered phases suffers the perturbative instability for any value of b it 11 In all cases as b → b crit,i the order parameter Ô i diverges, see also [2]. exists. A natural guess is that the end point of the dynamical evolution will bring us from the ordered phase to a disordered, Z 2 -symmetric, phases. However, to confirm this, a numerical simulation is necessary 12 .

Holographic conformal order with the bounded gravitational potential
The gravitational potential for a scalar field dual to a conformal order parameter is unbounded from below, see (5) and note that the existence of the conformal order requires b < − 3 2 . We used simple scaling arguments in section 1 to suggest that the conformal order exists once the scalar potential is made bounded as in (6), at least for sufficiently small g and in the limit b → −∞.
What happens at finite b? First, note that all the analysis in section (2) allow for a simple generalization as V is replaced with V g of (6) -see eqs. (17)- (19) and (23).
In the left panel of fig. 2 we show the results for the parameter κ characterizing the thermodynamics of the lowest branch of the conformal order (see (1)) evaluated at b = −10 as a function of g > 0. We find that the holographic conformal order persists 12 Another possibility is the evolution to a naked singularity in a dual gravitational description, see [16,17].
As g → g crit , the order parameter | Ô | diverges, see the right panel of fig. 2.
It is clear that the conformal order constructed in section 2 is robust against the deformation of the bulk scalar potential as in (6) for 0 ≤ g < g crit (b) since the basic equations determining it, i.e., eqs. (17)- (19) and (23), are analytic in the limit g → 0 + .

QNMs of the conformal order
In this section we study the spectrum of the linearized fluctuations about hairy black branes, dual to the conformal order -the spectrum of the quasinormal modes [5]: In section 3.1 we consider the hydrodynamic modes, i.e., QNMs such that w(q) → as q → 0. We confirm the conformal speed of the sound waves (24), the vanishing of the bulk viscosity (26), and the universality of the ratio of the shear viscosity to the entropy density [6][7][8].
In section 3.2 we consider the spectrum of the non-hydrodynamic modes. We discuss different branches of the spectra and exhibit the QNM with Im [w] > 0, for q ≤ q c , see eq. (11). This mode make holographic conformal order of [2] perturbatively unstable.
Its existence is yet another counterexample of the correlated stability conjecture [9,10].
For the most part we focus on the QNM spectra of the lowest branch of the conformal order with a dual gravitational action (2). We discuss however the instability of the higher overtones of the conformal order.
Consider fluctuations of the background geometry (13) For convenience, we partially fix the gauge by requiring We orient the coordinate system in such a way that the x 2 axis is directed along the spatial momentum, and assume that all the fluctuations depend only on (t, x 2 , r), i.e., we have a Z 2 parity symmetry along the x 1 -axis. At a linearized level, the following sets of fluctuations decouple from each other Let are defined in (16). Following [5,18], we introduce fluctuations invariant under the residual diffeomorphisms preserving (40): • Z 2 -even, the sound channel, Using the radial coordinate (15), we find the following equations of motion: for the sound channel, with where we introduced with V being the scalar potential (5); and for the shear channel, with

Hydrodynamic modes and the transport
We begin with the q → 0 limit of the dispersion relation for the shear and the sound channels QNM modes. From the former, we extract the shear viscosity, and from the latter, the speed of the sound waves and the bulk viscosity of the holographic conformal order. We discuss the dispersion relation w = w(q) for the sound waves in section 3.1.3 for different values of b.

The shear viscosity
The shear mode dispersion relation in the limit q → 0 takes form We now evaluate (58) in the holographic conformal order. To this end, we set Using (55), we find where Equations (60) have to be solved subject to the following boundary conditions: in the UV, i.e., as x → 0 + , in the IR, i.e., as y ≡ 1 − x → 0 + , Given (59), these asymptotes reflects the incoming boundary conditions as the horizon, and the Dirichlet boundary conditions at the asymptotic boundary [5].
Ultimately, we need to extract β = β(b). First, we need to solve numerically the first equation in (60), producing the data sets {z h 0,0 (b), z h 0,1 (b)}. Remarkably, we do not need to solve for Z s,2 : direct series expansion of the second equation in (60) in the IR leads to with the parameters of the asymptotic expansion of Z s,2 (see (63)) entering only in O(y). Thus, we find In the disordered phase, see eq. (27), the equation for Z s,0 is very simple: leading to the universal result for the ratio of the shear viscosity to the entropy density [6]. In the symmetry broken phase, from the perspective of the shear channel QNM discussed here, we do not have a general proof why β must be unity as well. In fig. 3 we present β − 1 of (65), from the numerical solution of the equation for Z s,0 .
We would like to stress that we already know from [6] that 13 β = 1; the analysis presented here should be viewed as highly nontrivial check on the QNM equations and our numerical construction of the holographic order parameter.

The speed of the sound waves and the bulk viscosity
The sound mode dispersion relation in the limit q → 0 takes form We now evaluate (67) in the holographic conformal order. To this end, we set It is straightforward to derive the corresponding equations of motion from (45) and (46) -they are too long to be presented here. We explain the boundary conditions only: in the UV, i.e., as x → 0 + , in the IR, i.e., as y ≡ 1 − x → 0 + , (70) 13 The universality of the shear viscosity in the holographic plasma was never proven from the perspective of the shear channel QNMs dispersion relation.
where we used the universal result for the ratio of the shear viscosity to the entropy density [6]. These results are presented in fig. 4. As expected, the transport in the ordered phase is conformal.

Dispersion of the sound waves in holographic conformal order
We now present results for the dispersion relation of the sound waves at finite q in holographic conformal order. We set and obtain from (45) and (46) equations 14 for {Z H,r , Z H,i , Z F,r , Z F,i }. These equations have to be solved subject to the boundary conditions: 14 The equations are too long to be presented here. The dashed black lines indicate the hydrodynamic q → 0 and the q → ∞ limits.
in the UV, i.e., as x → 0 + , in the IR, i.e., as y ≡ 1 − x → 0 + , Given (72), these asymptotes reflects the incoming boundary conditions as the horizon, and the Dirichlet boundary conditions at the asymptotic boundary [5].
In figs. 5-6 we present results for w r ≡ Re The dashed black line indicates the hydrodynamic q → 0 limit.
There is a noticeable deviation in the sound QNM mode dispersion between the ordered and the disordered phases for b −5; this is a reflection of the fact that in this regime some non-hydrodynamic modes becoming light, see fig. 9.

Non-hydrodynamic QNMs and the instability
In this section we discuss the spectrum of low-lying non-hydrodynamic QNMs. We consider only the non-hydrodynamic modes in the sound channel. We identify the QNM with Im[w u ] > 0, see (11), rendering the holographic conformal order discussed in this paper unstable. This mode is present in the spectrum for all values of b, whenever the conformal order is present. We explain why this mode is present in the ordered phases in the limit 15 b → −∞, and is absent in the disordered phase.
To begin, we set q = 0. There are two distinct branches of the sound channel QNMs: we call them BR metric and BR hair .
BR metric branch of the sound channel QNMs. Notice that, see (49), which decouples Z H and Z F fluctuations. On this branch, the spectrum is completely determined by the equation for Z H fluctuations, even though Z F fluctuations are 15 Recall that in this limit there is no distinction between the ordered and the disordered phases thermodynamics.  (72), but for lim q→0 w(q) = 0, we find The boundary conditions are as in the first lines of eq. (73) and (74). Note that the limit b → −∞ is trivial here (see (54) and (28)): precisely reproducing the equations for the q = 0 non-hydrodynamic QNMs of the AdS 4 Schwarzschild black brane -there is no distinction between the ordered and the disordered QNMs in this limit. We present the spectrum of the quasinormal modes on this branch for b = −4 as blue dots in fig. 7.
BR hair branch of the sound channel QNMs. Setting identically Z H ≡ 0, in addition to q = 0, we satisfy the QNM equation (45). Using the decomposition as in (72) we The boundary conditions are as x → 0 + , and as y ≡ 1 − x → 0 + . Now, there is a clear distinction between the ordered and the disordered phases, even in the limit b → −∞. Indeed, the QNM equations (79) and (80) contain ∂ 2Ṽ (highlighted in red), and where we used (28). In the ordered phase f [1] (x) is a nontrivial function, see fig. 1, while in the disordered phase it vanishes identically. We present the spectrum of the quasinormal modes on this branch for b = −4 as black dots in fig. 7.
There is a non-propagating sub-branch of BR hair branch, we call it BR Im hair . QNM equation of motion on this sub-branch is a consistent truncation of eqs. (79) and (80) with Solving (85) with the boundary conditions for Z F,r as in (81) and (82) In the rest of this section we focus on the BR Im hair branch. First, we identify the BR Im hair branch at q = 0. Using the decomposition (72), we find that the truncation is a consistent one, even for q = 0: with Eqs. (87) have to be solved subject to the following boundary conditions: in the UV, i.e., as x → 0 + , in the IR, i.e., as y ≡ 1 − x → 0 + , In fig. 8 represented by vertical dashed line, such that for i.e., this QNM becomes stable. In fig. 9 we present results for w(q = 0) as a function of b for the QNMs on the The spectral results (99) are obtained solving eq. (85) in the limit b → −∞. Taking and using (28) we find the limiting QNM equation where we highlighted again the crucial difference between the ordered and the disordered phases: nontrivial, given by eq. (30) , in the ordered phase .
The asymptotic representation (101) explains why the ordered phase is always unstable, while the disordered phase is not. Indeed, introducing a new radial coordinate PSfrag replacements Figure 10: Existence of the unstable QNM as b → −∞ is equivalent to the existence of the bound state in an effective one-dimensional potential (106) with E < 0. We present this potential, U 0 (x), for the ordered phase of the overtone 0 (the left panel).
The horizontal dashed lines indicate the effective energies of the QNMs (99). In the right panel we present the effective potential in the disordered phase, U dis . Since U dis (x) ≥ 0 for x ∈ (0, 1], there can not be any unstable QNM. and rescaling we obtain a Schrödinger-like equation for Ψ F : where the effective 1d potential U is and For an unstable QNM, i.e., for w i,0 > 0, the boundary conditions for Z F,0 , i.e., Z F,0 ∝ x 4 as x → 0 + and Z F,0 ∝ (1 − x) 0 as y = 1 − x → 0 + , imply that given the definition (105), The potential U(x) of (106) is divergent as x → 0 + , automatically enforcing the first of the boundary conditions in (108); it is vanishing as x → 1 − , so to enforce the second i.e., with f [1] (x) ≡ 0 (the right panel). In the former case the potential dips below zero, allowing for the bound states (represented by the dashed horizontal lines for the QNM frequencies (99) ), while in the latter case the potential is always non-negative for x ∈ (0, 1], thus excluding the instability. We demonstrated above that there is a perturbative instability on an index 0 ordered phase branch/overtone for any value of b ∈ (−∞, − 3 2 ). In fact, there is an instability on the excited branches/overtones of the conformal order. In fig. 11 we present the spectrum of the QNMs on the BR Im hair branch for the index i = 0, 1, 2 background phase overtones in the limit b → −∞. These results are obtained solving (101) with the bulk scalar profile function f [1] corresponding to the overtone index i = 0, 1, 2, see the left

Numerical tests
Results reported in this paper involve numerical computation of the QNMs. While some of the conclusions are robust and can be understood in the semi-analytic fashion, e.g., the existence of the unstable QNM in each conformal order phase in the limit b → −∞ from the effective 1d Schrödinger problem, the bulk of the computations involves the heavy numerics. It is thus important to address the question of the numerical tests.
We summarize here the implicit and the explicit test.
• Thermal equilibrium phases of the holographic model (2) are conformal. This fact along predicts the hydrodynamic transport, i.e., the speed of the sound waves c s and the bulk viscosity both in the symmetric and the symmetry broken phases.
In fig. 4 we presented deviations of these quantities, obtained from the QNM computations, from the expected values dictated by the conformal symmetry (8).
• The universality theorem of [6] implies that the shear viscosity of the holographic conformal order is s/(4π). This result was obtained evaluating the retarded two-point correlation function of the boundary stress-energy tensor in a generic holographic model. In fig. 3 we presented the deviation of the shear viscosity from the universal result obtained from the dispersion relation of the QNMs in the shear channel.
• In [2] the holographic conformal order was constructed using the background metric ansatz where we denoted a radial coordinate asx to distinguish it from the radial coordinate x used here, see (15). The two radial coordinates are related as Of course, the results should not depend whether we use x orx as a radial coordinate. We emphasize though that the two computational frameworks are very different; this is particularly profound in the computation of the spectrum of the QNMs. Indeed, while the incoming-wave boundary condition for a typical gauge-invariant fluctuation Z is Z(x) ∝ (1 − x) −iw , the same boundary condition takes form Since generically the dispersion relations w(q) are complex, the Re − Im part splits as in (72) Of course, ultimately, this should not affect the computed spectrum w(q). In figs. 12 and 13 we compare the spectra of the sound waves computed using the x radial coordinate (solid curves) and using thex radial coordinate (dots) in the holographic conformal order at b = −5 and b = −3 correspondingly. There is an excellent agreement.
• In fact, almost all computations presented in this paper were duplicated in x and x radial coordinates. For example, the fractional difference of the unstable QNM, the red dot in fig. 7, evaluated in two schemes is ∝ 10 −9 .