Holographic conformal order with higher derivatives

Conformal order are isotropic and translationary invariant thermal states of a conformal theory with nonzero expectation value of certain operators. While ubiquitous in bottom-up models of holographic CFTs, conformal order states are unstable in theories dual to bulk two-derivative gravity. We explore conformal order in strongly coupled theories with gravitational holographic duals involving higher derivative corrections.


Introduction and summary
Gauge theory/gravity correspondence [1,2] has been a valuable tool in our understanding of strongly coupled matter.It often lead to discoveries of new and unexpected phenomena.One such discovery was a construction of exotic hairy black holes [3], predicting symmetry broken phases of strongly coupled gauge theories that persist to arbitrary high temperatures.These holographic symmetry broken phases exist even for AdS/CFT duals [4][5][6] and present a holographic realization of a conformal order [7][8][9][10].
Specifically, for a CFT 4 with a global symmetry group G in Minkowski space-time R 3,1   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 local order parameter for the symmetry breaking of conformal dimension ∆.The parameters κ and γ characterizing the thermodynamics of the symmetry broken phase are necessarily constants 1 .In all holographic constructions κ was found to be positive, implying that the symmetry broken phases are thermodynamically stable.It was also found that in all models with two-derivative holographic duals κ < 1, implying that the symmetry broken phases are subdominant both in the canonical and microcanonical ensembles.The fact that the symmetry broken phase in the microcanonical ensemble is less entropic than the symmetry preserving phase suggests that is must be unstable [11].While there are no instabilities in the hydrodynamic sector of the strongly coupled conformal order plasma, there is an instability in the non-hydrodynamic sector -one finds a quasinormal mode of the dual hairy black brane with Im [ω] > 0 [12].This instability is not lifted by compactifying the ordered phase on a positive curvature spatial manifold, such as S3 [13].
In this paper we continue pursuit of stable conformal order and consider holographic models with higher-derivative gravitational duals.Weyl 4 higher-derivative corrections to bulk Einstein gravity encode finite 't Hooft coupling corrections of the dual conformal gauge theory [14], and Riemann2 corrections describe finite-N effects in the dual theory [15,16].Corrections of both types modify the relation between the entropy densities of the boundary CFTs and the horizon area densities of the bulk black branes.This allows to engineer models where only the symmetry broken phase triggers the higherderivative corrections, potentially increasing κ.We now review this idea, originally proposed in [17].
Consider a five-dimensional theory of Einstein gravity in AdS, coupled to a scalar of mass 2 m 2 L 2 = ∆(∆ − 4), where the bulk scalar φ is dual to an operator O ∆ of a conformal dimension ∆, of a boundary CFT with a central charge c = π 8G N .Note that the theory has a global G ≡ Z 2 symmetry, φ ↔ −φ.The only thermal states of the boundary CFT described by (1.2) are AdS 5 Schwarzschild black branes, i.e., the thermal expectation value of the operator O ∆ vanishes, leaving the global symmetry G unbroken.The construction of the holographic conformal order proposed in [5] relies on models where the scalar has a nontrivial potential V [φ] instead, with the leading nonlinear correction unbounded from below 3 .As a simplest example we take ∆ = 3 and where b > 0 is a bulk coupling, leading to The claim of [5] is that the thermal conformal order in the model (1.4) always exists in the limit b → +∞, where the thermal ordered phase is holographically realized as an AdS-Schwarzschild black brane, with perturbatively small "scalar hair", where γ is the normalizable coefficient of p 0 near the AdS 5 boundary.The physical origin of the conformal ordered phase is easy to see.Notice that to leading nonlinear order O(b −1 ), Thus, in the limit b → +∞, the effective mass of the bulk scalar φ is shifted due to nonlinear negative quartic term in (1.3).Potentially, when evaluated at the AdS-Schwarzschild horizon 4 it can dip below the Breitenlohner-Freedman bound, triggering the instability and leading to 'hair'.This is precisely what we find: in fig. 1 we present different profiles of p 0 in the model (1.4), realizing distinct conformal order phases.
The bulk radial coordinate r ∈ [0, 1], with r → 0 being the asymptotic AdS 5 boundary, and r → 1 being the regular black brane horizon.Taking into account the bulk scalar backreaction, we compute 5 the thermodynamic coefficient κ in (1.1) for different profiles reported in fig. 1, Note that in both ordered phases κ < 1, so that they are subdominant relative to the symmetry preserving phase with p 0 ≡ 0 and κ = 1.
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 Figure 2: Leading corrections κ [2] to the conformal order thermodynamic parameter , as a function of the bulk coupling α in the higher-derivative holographic model δL 2 .The solid black and blue curves correspond to scalar profiles without a root or with a single zero in the bulk, as in fig. 1.The dashed red and orange curves represent the corresponding values of κ [2] .The vertical green dashed line (the left panel) indicates a critical value of α crit = 1 40 , beyond which the conformal order phases do not exit.
Motivated by [17], we modify the model (1.4), including the higher derivative corrections δL, 5 See section 2 for details. Figure 3: Leading corrections κ [4] to the conformal order thermodynamic parameter , as a function of the bulk coupling α in the higher-derivative holographic model δL 4 .The solid black and blue curves correspond to scalar profiles without a root or with a single zero in the bulk, as in fig. 1.The dashed red and orange curves represent the corresponding values of κ [4] .
for a constant parameter α.In particular, in this paper we consider two classes of models6 : • four-derivative curvature corrections described by: (1.9) • eight-derivative curvature corrections described by: where C is the Weyl tensor.
Note that the coupling constant of the higher-derivative term δL is small in the limit (1.12) Evaluated at the horizon of the AdS-Schwarzschild black brane both δL 2 and δL 4 are positive 7 .Thus, we expect that α < 0 would facilitate the condensation of the bulk scalar; additionally, precisely for α < 0 the Wald entropy [18] density of the higherderivative black brane is larger that its Bekenstein entropy8 , further increasing the value of κ in the ordered phase.
We delegate the technical details of the analysis of model (1.8) to section 2, and report the results only.We can extract the thermodynamic parameter κ in (1.1) independently, either evaluating the entropy density s, or the energy density E, of the corresponding higher-derivative black brane solution, The agreement κ = κ is a nontrivial consistency check on the computations.
• In the higher-derivative holographic model δL 2 the conformal order exists for α ∈ (−∞, 1 40 ).We parameterize κ = κ(α) to order O(b −1 ) as and consider the bulk scalar profiles as in fig. 1.We find that, as predicted above -see the solid black and blue curves in fig. 2 - .
As we see, at least for the models considered, the higher-derivative corrections can not make the holographic conformal order phases to dominate the symmetric phase.
While we have not done the stability analysis as in [12], we do expect that the higherderivative black branes with a scalar hair constructed here have an unstable quasinormal mode.It is possible that the holographic multiverse simply does not allow for a stable thermal conformal order.It would be interesting to rigorously prove this in full generality.

Technical details
In this section we collect the technical details, necessary to reproduce the results reported in section 1.We heavily rely on a recent construction of the higher-derivative AdS-Schwarzschild black brane solutions in [19].

Black brane geometry dual to thermal states of the boundary theory
The background geometry dual to a thermal equilibrium state of a boundary gauge theory takes form where c i = c i (r), and additionally φ = φ(r).The radial coordinate is r ∈ [0, r h ], with r h = 1 being the location of the regular black brane horizon, lim Notice that at this stage we do not fix the residual diffeomorphism associated with the reparametrization of the radial coordinate.
One can efficiently compute the background equations of motion from the effective one dimensional action, obtained from the evaluation of (1.8) on the ansatz (2.1).Here ( ′ ≡ d dr ), and in model (1.10) by From ( 2.3) we obtain the following equations of motion: where E j are functionals 9 of {c i , φ} such that for a constant λ.We verified that the constraint (2.9) is consistent with the remaining equations.
On-shell, i.e., evaluated when (2.7)-(2.10)hold, the effective action (2.3) is a total derivative.Specifically, we find .12) 9 For readability we will not present their explicit expressions here.
with the higher derivative terms δB given by In what follows we will need the entropy density s, the energy density E, and the temperature T of the boundary thermal state.The temperature is determined by requiring the vanishing of the conical deficit angle of the analytical continuation of the geometry (2.1), 2πT = lim where to obtain the last equality we used (2.2).The thermal entropy density of the boundary gauge theory is identified with the entropy density of the dual black brane [20].Since our holographic model contains higher-derivative terms, the Bekenstein entropy s B , must be replaced with the Wald entropy s W [18], i.e., s = s W .The simplest way to compute the Wald entropy density is instead to use the boundary thermodynamics: According to the holographic correspondence [1,2], the on-shell gravitational action S 1 , properly renormalized [21], has to be identified with the boundary gauge theory free energy density F as follows, where we used (2.12).S GH is a generalized Gibbons-Hawking term [22], necessary to have a well-defined variational principle, and S ct is the counter-term action.
Eq.(2.18) can be rearranged to explicitly implement the basic thermodynamic relation −F = sT − E between the free energy density F , the energy density E and the entropy density s [23]: From (2.19) we identify where in the second line {• • • } denote higher-derivative (generalized) GH terms that vanish in the limit r → 0; we also included the only relevant (non-vanishing in the limit) counterterm in [ ].

Figure 1 :
Figure 1: Condensation of the bulk scalar φ(r) = p 0 (r) √ b in AdS-Schwarzschild background leads to perturbative in the limit b → +∞ conformal order in the holographic model (1.4).The AdS black brane radial coordinate r ∈ [0, 1] runs from the boundary to the horizon.Different p 0 profiles, the black and the blue curves, represent distinct conformal order phases.
see the right panel of fig. 2. For α > 0 the conformal order phases become very subdominant compare to the symmetry preserving phase, and cease to exist as α → 1 40 , represented by the dashed vertical green line in the left panel of fig. 2. The dashed red and orange curves (the left panel)