Compactified holographic conformal order

We study holographic conformal order compactified on $S^3$. The corresponding boundary CFT$ _4$ has a thermal phase with a nonzero expectation value of a certain operator. The gravitational dual to the ordered phase is represented by a black hole in asymptotically $AdS_5$ that violates the no-hair theorem. While the compactification does not destroy the ordered phase, it does not cure its perturbative instability: we identify the scalar channel QNM of the hairy black hole with Im$[w]>0$. On the contrary, we argue that the disordered thermal phase of the boundary CFT is perturbatively stable in holographic models of Einstein gravity and scalars.

Conformal order stands for exotic thermal states of conformal theories with spontaneously broken global symmetry group G [1][2][3][4]. For a CFT d+1 in Minkowski space-time R d,1 the existence of the ordered phase implies that there are at least two distinct thermal phases: where F is the free energy density, T is the temperature, C is a positive constant proportional to the central charge of the theory, and O ∆ is the order parameter for the symmetry breaking of conformal dimension ∆. The parameters κ and γ characterizing the thermodynamics of the the symmetry broken phase are necessarily constants. Note that when κ > 1 (κ < 1), the symmetry broken phase dominates (is subdominant) both in the canonical and the microcanonical ensembles. Irrespectively of the value, provided κ > 0, the symmetry broken phase is thermodynamically stable. It is difficult to compute directly in a CFT the values {κ, γ}, thus establishing the present and the (in)stability of the ordered phase. Rather, the authors of [1][2][3][4] established the instability of the disordered (symmetry preserving) thermal phases in discussed CFTs.
The condensation of the identified unstable mode then leads to O ∆ = 0 for the new equilibrium thermal state -the conformal order.
Conformal order states are very interesting in the context of holography [5,6], as they imply the existence of the dual black branes in a Poincare patch of asymptotically AdS d+2 bulk geometry that violate the no-hair theorem. In section 3 we prove a theorem that the disordered conformal thermal states are always stable 1 in dual holographic models of Einstein gravity with multiple scalars. Thus, the mechanism for the conformal order presented in [1] is not viable in these holographic models.
The no-go theorem of section 3 does not imply that the holographic conformal order can not exist. In fact, way before [1], what is now known as a conformal order was constructed in [7] 2 . In the gravitational dual to the holographic conformal order it is straightforward to compute the parameter κ in (1.1), and establish that κ < 1: at best, the holographic conformal order is metastable 3 . In [12] it was established that the holographic conformal order is perturbatively unstable. Specifically, we identified a non-hydrodynamic quasinormal mode (QNM) of the dual hairy black brane with 4 Im[w] > 0. The purpose of this paper is to use the techniques of [13] to investigate the stability of the holographic conformal order compactified on S 3 .
As introduced, the conformal order is associated with the spontaneous breaking of a global symmetry. Thus, one might wonder whether the order can ever survive the compactification. To this end, we point the following: In the infinitely large-N (large central charge) limit spontaneous symmetry breaking can happen on a compact manifold. For a recent example, see a discussion of the spontaneous chiral symmetry breaking of the cascading gauge theory on S 3 [11].
We present an explicit example in this paper where the existence of the holographic conformal order is not associated with the breaking of any global symmetry.
A detailed summary of our study is provided in section 2. We highlight here the new results: • Our main model -M b P W -is a holographic CF T 4 with a single operator O 2 1 We consider AdS 5 /CF T 4 dualities, but the argument can be readily extended to other dimensions. 2 See [8,9] for the recent work. Potentially relevant top-down constructions also include [10,11]. 3 Given the no-go theorem reported here this is not surprising. 4 We define dimensionless frequencies as w ≡ w/(2πT ).
of the conformal dimension ∆ = 2. The order is associated with the thermal expectation value O 2 = 0. The subscript P W indicates that the model is a deformation, b is the deformation parameter, of the top-down N = 2 * gauge theory/gravity correspondence [14][15][16][17]. N = 2 * holographic correspondence is recovered in the limit b → 1. Importantly, our conformal model M b P W does not have any global symmetry. We follow the techniques introduced in [9] and construct conformal order in M b P W , with the holographic boundary CFT in R 3,1 Minkowski space-time, perturbatively as b → ∞. This is the first ever example of the holographic conformal order that is not associated with spontaneous breaking of a global symmetry. We find that κ < 1 (see (1.1)) in the model, making the ordered phase not the preferred one. We identify the unstable QNM mode of the dual hairy black brane in asymptotically AdS 5 geometry, and thus establish that the conformal order in M b P W CFT is unstable.
• We extend M b P W holographic model when the boundary CFT is compactified on S 3 of a radius L, equivalently, with the curvature scale K = 1 L 2 . We show that the conformal order persists in the limit b → ∞ for a wide range of K 1/2 T . The ordered phase is subdominant both the canonical and the microcanonical ensembles. It is always unstable: we identify a QNM mode of the dual hairy black hole in asymptotically AdS 5 geometry with Im[w] > 0.
• In the limit b → ∞, the ordered and the disordered phases become identical: for the parameters introduced in (1.1), we find The absence of the thermodynamically dominant conformal order in M b P W holographic model in this limit is consistent with the perturbative stability of the disordered phase, as dictated by the general no-go theorem of section 3.
• We study compactified conformal order in M b P W model at finite b for select values of K 1/2 T . We establish that the conformal order exists only for b > b crit (K). Since b crit is found to be always larger than 1, the holographic model M b P W with the ordered phase is always a deformation of the N = 2 * /Pilch-Warner top-down holography. In other words, similar to [9], there is no conformal order in N = 2 * theory.
• We establish the compactified conformal order in M b P W model at finite b, when it exists, is thermodynamically subdominant and unstable.  [10,11] in the limit of infinitely high temperature T Λ → ∞, where Λ is the strong coupling scale of the boundary cascading gauge theory. A big if is the existence of these black branes/black holes in the high temperature limit in the supergravity approximation. We expect to report on this question in the near future.

Summary
The starting point of our analysis is an example of the gauge theory/string theory holographic correspondence between N = 2 * SU(N) theory (in the planar limit and at large 't Hooft coupling) and type IIB supergravity [14][15][16][17]. The corresponding bulk effective action resulting from the dimensional reduction on the five-sphere of ten-dimensional supergravity takes form with the potential V determined from the superpotential W as The bulk scalars α and χ are (correspondingly) the holographic dual to operators O 2 and O 3 of the N = 4 Yang-Mills [15,18,19]. Finally, five-dimensional Newton's constant G 5 is 5 3) The non-normalizable coefficients of the bulk scalars α and χ are related to the masses of the bosonic and fermionic components of N = 2 hypermultiplet. In this paper we are interested in conformal theories, thus, these parameters are set to zero.
To simplify the discussion, we can consistently set χ ≡ 0, leading to a conformal model Note that M P W does not have any global symmetry. It does not have a conformal order either, as we already alluded to in section 1. Following [9], we generate a class i.e., we recover in this limit the top-down holographic model M P W . We find that there is a thermal ordered phase in M b P W , provided b > b crit > 1. The precise value of b crit below which the ordered phase ceases to exist depends on K 1/2 /T , i.e., the ratio of the S 3 compactification scale and the temperature T .
In the rest of this section we explain the results only; a curious reader can find technical details necessary to reproduce them in section 4.
In fig. 1 we present the results for the conformal order in M b P W model in the limit b → +∞ for different values of K 1/2 /T . Note that the thermal expectation implying that the backreaction of the bulk scalar α on the geometry becomes vanishingly small in the limit b → +∞. Constructed conformal order is perturbatively unstable: in the right panel we identify a QNM of the corresponding hairy black hole with Im[w] > 0. The disordered phase is represented holographically by AdS 5 black hole. There are four distinct regimes of K 1/2 /T in the disordered phase, which we also highlighted with solid/dashed/dotted curves in the PSfrag replacements the solid black curves correspond to with AdS 5 black holes temperature about the Hawking-Page transition [20,21]; the solid blue curves correspond to with AdS 5 black holes temperature below the Hawking-Page transition, but having a positive specific heat; the dashed blue curves correspond to with AdS 5 black holes having a negative specific heat, but being perturbatively stable with respect to localization on S 5 [22,23]; the dotted red curves correspond to  PSfrag replacements The ordered phase of the conformal model M b P W in the limit b → +∞ has a higher free energy density than that of the disordered phase at the corresponding temperature (the left panel). The ordered phase of the conformal model M b P W in the limit b → +∞ has a lower entropy density than that of the disordered phase at the corresponding energy density (the right panel). See (2.11) and (2.12) for the definition of the reduced thermodynamic functions.
with AdS 5 black holes being perturbatively unstable with respect to localization on In fig. 2 we show that the ordered phase of the conformal model M b P W in the limit b → +∞ is subdominant both the in canonical ensemble (the left panel) and the microcanonical ensemble (the right panel). We define the reduced free energy densitŷ F, the reduced energy densityÊ, the reduced entropy densityŝ, and the reduced temperatureT as 6 (2.12) The color/style coding in the left panel is as in fig. 1; in the right panel the coding reflects the values ofÊ of the disordered phase corresponding to (2.7)-(2.10). 6 The normalization is chosen so that limT →∞F T 4 = −1.  (2.13). The left panel demonstrates that the ordered phase has a higher free energy density than that of the disordered phase (at the corresponding temperature).
The right panel identifies the QNM in hairy black holes, holographically dual to the ordered phase, with Im[w] > 0, rendering this thermal ordered phase perturbatively unstable.
In fig. 3 we present the results for the thermal ordered phase in M b P W model for finite values of b and select values of K 1/2 T : The black and the blue curves represent black holes with the positive specific heat; the magenta and the purple curves represent black holes with the negative specific heat.
The left panel indicates that the disordered phase has a lower free energy density, and thus is the preferred one 7 . The right panel demonstrates that the ordered phase is perturbatively unstable: we identify the quasinormal mode in the helicity zero sector of the dual hairy black hole with Im[w] > 0.
Thermal ordered phases in M b P W model exist only for b > b crit . In fig. 4 we present the estimate for b crit for the set of K 1/2 /T in (2.13). Additionally, the red dot indicates (see fig. 8 (2.14) So far we presented the evidence that the holographic conformal order survives the compactification of the dual boundary theory on S 3 , but remains unstable. In the example presented, the order was not associated with the spontaneous breaking of any global symmetry. The reader might reasonable worry whether our conclusions are specific to models without any global symmetry. Rather, we claim that our results are generic: we extend analysis to M b P W,sym model, where the bulk scalar potential is Z 2 -symmetric, frag replacements PSfrag replacements

Stability of the holographic CFT disordered thermal states
In this section we prove that disordered phases of four-dimensional holographic CFTs on R 3 or S 3 , dual to bulk models of asymptotically AdS 5 Einstein gravity with arbitrary scalars, are perturbatively stable. These disordered phases are represented by AdS-Schwarzschild black branes/black holes. It is well-know that AdS-Schwarzschild black branes/black holes are stable with respect to the purely gravitational perturbations [24] -we extend this statement to the stability with respect to the bulk scalar fluctuations.
It is important to stress what potential instabilities are not covered by the analysis below. A top-down conformal holographic model is formulated in ten dimensional type IIB supergravity in asymptotically AdS 5 × V 5 , where V 5 is a compact manifold, V 5 = S 5 in the familiar example of large-N N = 4 SYM. It is known that AdS 5 -Schwarzschild black holes can be unstable with respect to metric fluctuations carrying nonzero momentum on V 5 [22,23,25]. Once we reduce the ten-dimensional holographic correspondence on V 5 we loose access to such fluctuations. However, such instabilities are incorrect to interpret as Schwarzschild black holes growing the scalar hair and violating the no-hair theorem. Rather, these are the instabilities of the initially smeared over V 5 black holes towards localization on the compact transverse space.
Consider a holographic correspondence encoded in an effective five-dimensional gravitational action We will keep the scalar potential V ({φ j }) arbitrary, in particular, we will not assume any global symmetries in the model 9 . To conform with the rest of the discussion in this paper, we set the radius of the asymptotically AdS 5 geometry to 2, then, the potential takes form The disordered phase is the AdS 5 -Schwarzschild black brane/black hole, which we write in infalling Eddington-Finkelstein (EF) coordinates as (compare with (A.1)) where τ is the EF time, and the minus sign in dτ dr term is due to the fact that the AdS boundary is at r → 0, with   9 We explain shortly how the unitarity of the boundary CF T 4 constrains this potential.
(see eq.(A.23) of [13]). Thus, for all practical purposes we can truncate the generic scalar potential in (3.2) to O(φ 2 ). Using the orthogonal rotation in the field space, the equivalent to (3.1) effective action (in regards to computing the spectra of the QNMs) is given by with a new scalar χ j of mass µ 2 j dual to an operator O j of dimension ∆ j of the boundary CFT In (3.7) we assumed the Breitenlohner-Freedman bound [26] on the scalar masses, equivalently the unitarity bound on the operators of the interactive boundary CFT [27].
An advantage of using (3.6) is that the branches of the scalar fluctuations Φ j now completely decouple:  with Eq. (3.11) needs to be solved subject to the boundary conditions In general, the solution of (3.11) results in complex ψ and w.
We present an analytic argument that Im[w] < 0 when ∆ ≥ 5 2 ; for ∆ ∈ (1, 5 2 ) we need to resort to numerics. Establishing Im[w] < 0 implies stability of the disordered phases of holographic CFTs with the effective action (3.1). Note in particular that M b P W and M b P W,sym models discussed in section 2 are the special cases of (3.1).
Assuming {ψ, w} is a solution to (3.11), we define Ultimately, we will take the limit Using the explicit definition of X (3.11) and integrating by parts, .

(3.17)
Note that J (ǫ, y h ) is manifestly positive, and the boundary term bt vanishes in the leading to where we integrated the second term by parts and dropped in the second line the r → 0 boundary term since |ψ| 2 ∝ r 2∆−3 → 0. From (3.19) we find then The analytic proof of section 3.1 fails here since: the potential V (3.12) can be negative for small r when ∆ ∈ ( 3 2 , 5 2 ); the UV boundary term in (3.18) does not vanish for ∆ ≤ 2; the UV boundary term in (3.20) does not vanish for ∆ ≤ 3 2 . Note that our models M b P W and M b P W,sym (and in particular the top-down example M P W ) have disordered phase with ∆ = 2. We will not attempt to find an analytic proof that Im [w] < 0 when ∆ ∈ (1, 5 2 ), and instead present numerical evidence that this is indeed the case.
In fig. 6 we present Re

(4.4)
Eqs. (4.2)-(4.4) are solved with the following asymptotics: in the UV, i.e., as r → 0 in the IR, i.e., as y ≡ 1 r → 0 From (A.15) we find (4.7) The first law of thermodynamics at order O(b −2 ) leads to the constraint , (4.8) where ′ stands for d dK . We verify the first law of thermodynamics of the conformal order at O(b −2 ) in fig. 9.
We now discuss the computation of the QNMs in the perturbative conformal order.
In the limit b → +∞, the differential operator D 2 in (B.2) is Note that the highlighted term in W implies that the QNM spectra of the ordered and the disordered phases are different even in the strict b → +∞ limit.

M b P W model at finite b
Using the equations of motions and the asymptotics for the background and the QNMs in appendices A and B, the numerical solution is routine. We use the computation techniques developed in [29]. To validate numerical solutions we verify the first law (A.17). For example, for K = 0, the first law can be stated as We present the results for δ K=0 in fig. 9.  To reproduce the results reported in fig. 5 we need the equation for α 1 , and the leading order expression for W (computing the spectrum of the QNMs as in (B.1)): 0 =α ′′ 1 + 16Kr 2 (2r 2 − 1) + 4r 4 − 10r 2 − 10r − 3 r(2r + 1)(16Kr 2 + 2r 2 + 2r + 1)(r + 1) α ′ 1 + 4α 1 (r + 1) 2 (2α 2 1 + 1) (16Kr 2 + 2r 2 + 2r + 1)(2r + 1)r 2 , and where the radial coordinate r ranges as where c i = c i (r), and the round 3-sphere metric dΩ 2 (3,K) of radius K −1/2 is From the gravitational effective Lagrangian of M b P W model (see (2.5) for the scalar potential) we obtain the following second order equations: and the first order constraint in the IR, i.e., as y ≡ 1 where we definedĥ ≡ y −4 h . (A.14) Following the holographic renormalization of the related N = 2 * model [30][31][32] we find: B h = 0 QNMs of the hairy black holes in M b P W model We follow the framework of [13] for the computation of the QNMs. We will be interested in the helicity h = 0 quasinormal modes with ℓ = 0. Note that at ℓ = 0 the metric and the bulk scalar fluctuations decouple. We use F (t, r) to denote gauge invariant fluctuations associated with the bulk scalar α of the conformal model M b P W . Using the background parameterization (A.3), we obtain from [13]: where the second-order differential operator D 2 (coming from on the background geometry (A.1)) is (note that k 2 = Kℓ(ℓ + 2) = 0) and W = 1 3hG 2 1536h 5/2 f r 4 (α ′ ) 4 + 96h 1/2 r 2 (h ′ r + 4h Generically, F , as well as w, are complex. We need to impose the normalizable boundary conditions as r → 0, and the incoming wave boundary conditions at the black brane/black hole horizon, i.e., as y ≡ 1 r → 0. We can explicitly factor the boundary conditions, and the harmonic time dependence, redefining F as precisely as needed to specify a solution of a single second order ODE.

C Numerical tests
In fig. 9 we test the first law of thermodynamics (A.17) for the perturbative conformal order (see (4.8), the left panel) and for the conformal order at K = 0 (see (4.11), the right panel) in M b P W model.