Thermal order in holographic CFTs and no-hair theorem violation in black branes

We present a large class of holographic models where the boundary ${\mathbb R}^{2,1}$ dimensional conformal field theory has a thermal phase with a spontaneously broken global symmetry. The dual black branes in a Poincare patch of asymptotically $AdS_4$ violate the no-hair theorem.

In [1] the authors asked an interesting question whether an "order" (a thermal phase with a spontaneously broken global symmetry) is always lost at high temperatures 1 ?
There is a number of holographic models known where the answer is "yes", i.e., there is a critical temperature T crit , such that for T > T crit there is a phase with a spontaneously broken global discrete, Z 2 , symmetry [2][3][4] or a continuous, U(1), symmetry [5]. The holographic models mentioned correspond to boundary quantum field theories with a mass scale: a coupling of a relevant operator in the model [2] or a strong coupling scale of the Klebanov-Strassler gauge theory [6] in [5]. Ultimately, it is this mass scale that determines T crit .
But is it possible to have an exotic phenomenon of the global symmetry breaking in the ultraviolet (as first suggested in [2]) in a conformal theory? There is no scale to determine T crit , and the thermal symmetric and the symmetry broken phases must exist for all temperatures. Necessary, the distinct holographic duals to these phasesthe black branes with the translationary invariant horizons -would violate the no-hair theorem. The purpose of this note is to present explicit examples of such holographic models. An impatient reader can simply jump to the discussion of the holographic models in Step3. Rather, we follow the construction route from massive QFTs.
• Step1 2 . Consider the effective four-dimensional gravitational bulk action 3 , dual to a QFT 3 on R 2,1 , where we split the action into a conformal part S CFT 3 ; its deformation by a relevant operator O r ; and a sector S i involving an irrelevant operator O i along with its mixing with O r under the renormalization group flow, specified by a constant g. In all our numerical analysis we set g = −100. The four dimensional gravitational constant κ is related to a central charge c of the UV fixed point as 1 As in [1], we consider equilibrium phases of the theories without a chemical potential for the conserved global U (1) symmetries. 2 This is a review of [2]. 3 We set the radius of an asymptotic AdS 4 geometry to unity.
We assume the scaling dimension of O r to be ∆ r = 2. The scaling dimension of O i is ∆ i = 4 . Generically, we turn on the non-normalizable coefficient of φ, corresponding, the nonzero coupling Λ of the dual operator O r . The effective action (1) has a Z 2 × Z 2 discrete symmetry that acts as a parity transformation on the scalar fields φ and χ.
The discrete symmetry φ ↔ −φ is explicitly broken by a relevant deformation of the CFT 3 ; while the χ ↔ −χ symmetry is broken spontaneously.
The thermal states of the QFT 3 are dual to the black brane solutions in (1) with translationary invariant horizons: where the radial coordinate x ∈ (0, 1). The constant α is an arbitrary scale parameter, and the metric warp factor g xx is determined algebraically from a, φ, ψ. Solving the equations of motion from (1) and (2) with the background ansatz (4), we find, near the AdS 4 boundary x → 0 + , and near the black brane horizon y = 1 − x → 0 + . Apart from the overall scaling factor α, a black brane solution is specified with the three UV coefficients {p 1 , p 2 , χ 4 } and the four IR coefficients {a h 0 , a h 1 , p h 0 , c h 0 }. The UV parameter p 1 determines the coupling constant of the relevant operator O r as while the remaining parameters determine the Hawking temperature T of the black brane, its entropy density s, the energy density E, and the free energy density F as follows: PSfrag replacementŝ The latter expectation value, i.e., χ 4 , serves as an order parameter for the spontaneous breaking of the global Z 2 symmetry.
A selection 4 of thermal phases of the QFT 3 is presents in fig. 1. Notice that the horizontal axes correspond to Λ/(8πT ), so the region close to the origin corresponds to high temperatures, i.e., T ≫ Λ. We show 5 the Z 2 symmetric phases with Ô i = 0 (red curves) and the symmetry broken phases Ô i = 0 (purple curves). The left panel shows the reduced free energy densityF, see (8). Note that for the AdS 4 -Schwarzschild 4 There is a tower of the symmetry broken thermal phases similar to those discussed here [2].
5 In all cases we verified that the first law of thermodynamics 0 = dE/(T ds) − 1 Λ=const is true numerically to ∼ 10 −10 or better.
black braneF denoted by a red dashed horizontal line. The solid purple curves connect to the red curve with the second-order transition, green dots, at denoted by the green dashed vertical lines. As discovered in [2], the symmetry broken phases exist only in the UV, i.e., for T ≥ T crit . They extend to infinitely high temperatures, denoted by the black dot. The black dot is our first example of the CFT 3 with the spontaneously broken global discrete symmetry, in this particular case Z 2 × Z 2 : where the uncorrelated ± signs represent the 4-fold degeneracy of the symmetry broken phases.
In any thermal phase, symmetric or symmetry broken, the equation of state of a In the right panel of fig. 1 we plot E/F in the QFT 3 as a function of Λ/8πT : both the red curve and the purple curves pass through (−2) in the UV (with a numerical accuracy of ∼ 10 −11 ).
The symmetry broken phases (solid purple curves) can be smoothly extended "past the infinite temperature" -the dashed purples curves. We have been able to reliably construct the dashed purple phases only for |Λ|/(8πT ) 0.2043.
In fig. 2 we present the order parameter Ô i for the spontaneous breaking of the global Z 2 symmetry (the left panel), and the thermal expectation value of Ô r (the right panel). The color coding is the same as in fig. 1. Notice that as one lowers the temperature along the dashed purple curves, the absolute value of the order parameter sharply increases as |Λ|/(8πT ) → 0.2043. This is the main reason why we could not extend these phases to low temperatures 6 .
In fig. 3 we present the speed of the sound waves in various thermal phases of the QFT 3 plasma. The color coding is the same as in fig. 1. All the equilibrium phases, symmetric and with the spontaneously broken Z 2 symmetry, are thermodynamically and dynamically stable [7]. The speed of the sound waves approach a conformal value in the limit T /Λ → ∞. 6 We expect that the corresponding black branes have a singular horizon in this limit. An example of a CFT 3 with a spontaneously broken Z 2 × Z 2 -the black dot phase in fig. 1 -has a larger free energy density than that of the symmetric phase, dual to the AdS 4 -Schwarzschild black brane: Thus, this phase is metastable 7 . In what follows, we ask the question whether we can introduce an additional "knob" in a holographic model (1) to potential reverse the inequality (13), and have a symmetry broken phase to dominate in a canonical ensemble 8 . This leads up to Step2.
• Step2. Consider a smooth constant b parameter deformation of the model (1): with the remaining parts of the effective action left unchanged. Note that the QFT 3 b has the same global symmetries as the QFT 3 . Additionally, the second-order phase transitions associated with the spontaneous breaking of Z 2 symmetry (χ ↔ −χ) happens at T crit given by (11), independent of the deformation parameter b.
7 It would be extremely interesting to understand the dynamics of the first order phase transition, in particular, to compute the wall tension of the symmetric phase bubble in the symmetry broken thermal phase. 8 No examples of such models known. It is straightforward to repeat the analysis of the thermal phase diagram of the model. In the thermodynamic relations (8) there is only one change 9 : In fig. 4 we show the effect of the deformation parameter b on the reduced free energy densityF b of the symmetry broken phase in the QFT 3 b model, compare to the corresponding free energy density in the QFT 3 model, denoted byF 0 (this is the solid magenta curve in the left panel of fig. 1). The dashed curve corresponds to b = 1 and the dotted curve corresponds to b = −1. They originate from the same green dot, representing the critical temperature T crit , as in (11). Following the symmetry broken phases in the QFT 3 b model to infinitely high temperature, we identify conformal phases, CFT 3 b , with the spontaneously broken Z 2 × Z 2 global symmetry -these are the two black dots in fig. 4. Since we want to reduced the free energy density as much as possible, we now study CFT 3 b models for b < 0 10 . 9 Of course, all the UV and IR parameters {p 2 , χ 4 , a h 0 , a h 1 , p h 0 , c h 0 } will develop an implicit b dependence.
10 Notice that the gravitational scalar potential in (14) is unbounded for b < 0. This does not immediately signal any pathologies -many scalar potentials in top-down supersymmetric holographic models are unbounded. See [4] for further discussion.
The critical value of the deformation parameter is denoted by a vertical dashed blue line.
The origin of b crit is easy to understand as we follow the symmetry breaking order parameters Ô r and Ô i in the CFT 3 b model, see fig. 6. We find that while Ô i remains finite in the limit b → b crit , Thus, precisely at b = b crit the black brane dual to the CFT 3 b symmetry broken phase "losses" the φ-hair, while maintaining the non-vanishing ψ-hair. We call this "terminal" where again the ± signs indicate the 2-fold degeneracy due to the spontaneously broken Can we do better with relaxing b = b crit constraint directly in the CFT 3 ψ model? This leads us to Step3. • Step3. Consider a gravitational dual to CFT 3 ψ model, Clearly, the S CFT 3 ψ model is a consistent truncation of the S QFT 3 b model with φ ≡ 0.
The CFT Notice thatF for all value of b, see fig 10. We find that as b → −∞, where C ≈ 1.81.