Note on stability of new hyperbolic AdS black holes and phase transitions in R\'{e}nyi entropies

We construct a series of new hyperbolic black hole solutions in Einstein-Scalar system and we apply holographic approach to investigate the spherical R\'{e}nyi entropy in various deformations of dual conformal field theories (CFTs). Especially, we introduce various powers of scalars in the scalar potentials for massive and massless scalar. These scalar potentials correspond to deformation of dual CFTs. Then we solve asymptotically hyperbolic AdS black hole solutions numerically. We map the instabilities of these black hole solutions to phase transitions of field theory in terms of CHM mapping between hyperbolic hairy AdS black hole and spherical R\'{e}nyi entropy in dual field theories. Based on these solutions, we study the temperature dependent condensation of dual operator of massive and massless scalar respectively. These condensations show that there might exist phase transitions in dual deformed CFTs. We also compare free energy between asymptotically hyperbolic AdS black hole solutions and hyperbolic AdS Schwarz (AdS-SW) black hole to test phase transitions. In order to confirm the existence of phase transitions, we turn on linear in-homogenous perturbation to test stability of these hyperbolic hairy AdS black holes. In this paper, we show how potential parameters affect the stability of hyperbolic black holes in several specific examples. For general values of potential parameters, it needs further studies to see how the transition happens. Finally, we comment on these instabilities associated with spherical R\'{e}nyi entropy in dual deformed CFTs.


Introduction
The stability of black holes in anti-de Sitter space has been widely studied in the context of the AdS/CFT correspondence [1] [2] [3] [4].The investigation of thermodynamical stability of black hole provides a novel window on the phase structure of the dual CFTs.In holographic approaches to condensed matter physics the instability of a black hole due to the condensation of scalar hair is dual to a superconducting phase transition [5] [6].The physical relevance is also related to studying phase transitions in AdS/QCD literature [7] [8] [9] [10] [11].These phase transitions correspond to freezing or releasing the degree of freedom in such systems.
Entanglement entropy (EE) can measure the effective degree of freedom in quantum system.
Therefore, one can calculate entanglement entropy or entanglement Rényi entropy (ERE) in gravity side to test phase transition in field theory.
We will firstly focus on the instability of hyperbolic AdS black holes and finally comment on holographic spherical Rényi entropy.For spherical Rényi entropy of the ground state in CFTs, [46] [47] proposed that it is equal to thermal entropy of higher dimensional hyperbolic AdS black hole by so called CHM mapping method [46] [47] [48] [49].The main goal of the present paper is to study how to make use of this dictionary to test the phase structures in deformed field theory side.The relevant deformations in field theory correspond to adding massive scalar or massless scalar in the dual hyperbolic black hole.We review how hyperbolic AdS black holes relate to holographic Rényi entropy.A holographic calculation of Rényi entropy for a spherical entangling surface is derived in [46] [47] [48] [49].Applying this approach, there are many extended studies [19] [50] [51].Following CHM mapping, the density matrix is thermal and we can write the n'th power of ρ as following where n is an integer number.The unitary transformation U and its inverse will be canceled by taking the trace of this expression.Hence the trace of the n'th power of density matrix is Using the definition of the free energy of dual black hole, i.e.F (T ) = −T log Z(T ), the corresponding Rényi entropy becomes in terms of the derivation in [46] [48] [49].Further using S = −∂F/∂T , this expression can be rewritten as where S n is the Rényi entropy while S thermal (T ) denotes the thermal entropy of the CFT on R × H d−1 .The entanglement entropy can be with T 0 given by 1 2πR .Here, R is the curvature scale on the hyperbolic spatial slices H d−1 matching the radius of the original spherical entangle surface, R.This proposal gives us a way to connect the hyperbolic AdS black holes with ERE.
There are several hyperbolic AdS black holes, which have been constructed in [52] [53] [54] [55] [56] [57].In [58], it was shown that static black holes with hyperbolic horizons can become unstable to the formation of uncharged scalar hair on the horizon of the black hole due to the presence of an extremal limit with near-horizon geometry AdS 2 × H 3 [59] [60] [61] [62].
Furthermore, authors of [63] introduced a topological black hole with a minimal coupled scalar field with negative mass-square and showed this new instability appeared.In [18], the authors mapped the instability of this gravity solution to the phase transition happened in dual CFTs by CHM mapping.In [19], they investigated charged hyperbolic black holes, which became unstable to presence of scalar hair at sufficiently low temperature.Such kind of instability is the same as the holographic superconducting instability in boundary hyperbolic space.In summary, scalar fields with masses below the effective Breitenlohner-Freedman bound for the near-horizon AdS 2 will induce instability at sufficient low temperatures.This happens for both charged and uncharged black holes.
In this paper, we will construct a series of general hyperbolic AdS black holes with neutra scalar.More precisely, in this system, we introduce series of specific powers of scalar in scalar potential.In [18], the authors showed that there was an instability in massive scalar hariy hyperbolic AdS black hole.The instability might induce a phase transition and study on entanglement Rényi entropy also confirmed the phase transition.In our setup, we extend the studies in [18] by introducing higher powers of scalar self-interactions to deform dual CFTs.We start with a general gravity setup and see what will happen.Firstly, we work out these gravity solutions in UV region.Secondly, we can find hyperbolic AdS black hole solution numerically in various scalar potentials.In terms of CHM mapping, basing on these hyperbolic AdS hairy black holes, we obtain the spherical ERE in dual deformed CFTs.ERE obtained in our setup shows that there exist phase transitions in dual CFTs.We can extract the condensation of dual operator with respect to temperature in each solution.The condensation of dual operator indicates that the phase transition might happen.To confirm the phase transitions, we compare the free energy between the hyperbolic scalar hairy AdS black hole solutions (HSHAdS) and hyperbolic AdS-SW black hole to reveal the transition.Furthermore, we turn on the in-homogenous linear perturbation to test the stability of HSHAdS and the stability condition highly constrains the potential parameters presented in the massive and massless scalar potential.We will give some explicit examples to show what kinds of scalar potential will give stable HSHAdS.blueFinally, one can make use of the stability to obtain the phase structure of the dual theories.
An overview of the remainder of the paper is as follows: in section 2, we firstly set up the gravity which is our starting point.In section 3, we study the boundary energy momentum tensor of hyperbolic hairy black hole solutions with introducing various of boundary counter terms in massless and massive scalars respectively.Furthermore, we evaluate the free energy of these solutions.In section 4, through the above numerical analysis, we found that there are interesting phase transitions in deformed CFTs.We make use of condensation of dual operators and free energy of each solution to check whether phase transition will really happen in deformed CFTs or not.In section 5, we will demonstrate that the hyperbolic black holes are unstable and entanglement Rényi entropies show a phase transition.Therefore, in section 6, we turn to the physical case of these models which are normalizable on hyperboloid.We will devote section 7 to conclusions and discussions.In Appendix A, we will list the asymptotic AdS boundary behavior which is controlled by Einstein equations for massless and massive scalar respectively.These UV behaviors are useful to obtain the numerical solutions and we also list how to obtain the vacuum expectation value of dual operators.In Appendix B, we show various new hyperbolic scalar hairy AdS black hole solutions numerically as examples to check the validity of our numerical procedure.

Gravity Setup
The action of Einstein-Dilaton system in 5D spacetime in Einstein frame is Here G 5 is the 5D Newton constant, g is the 5D metric determinant and φ, V are the scalar field and the corresponding potential.In this paper, we study the potential of the form From these cases, one can learn how the self interaction in the bulk involve in boundary phase structure.In general scalar potential, our calculations involve in examining the Einstein and scalar field equations together and finding solutions where the scalar has a nontrivial profile reflecting the presence of the relevant deformations in the boundary theory.
The equations of motion are where E µν = R µν − 1 2 Rg µν is Einstein tensor.We would like to choose the following ansatz to solve the Einstein equations of motion, where H 3 is 3 dimensional hyperbolic space and L is AdS radius.In terms of the above ansatz, one can obtain equations, One more constrain equation is (2.5) is not independent on the other three equations in (2.4).Once the gravity solution is obtained from (2.4), one could use (2.5) to check the solution.
Here, we note that (2.4) would impose a natural boundary condition near horizon.If one collects all the terms with a denominator f (z), the results are as following φ).Since the horizon is not a real singularity, the apparent singularity f (z h ) = 0 in Eq.(2.4) should be canceled by requiring Q(z h ) = 0. Later, we will try to solve this boundary value problem using numerical method described in appendix A and developed in Ref. [64].In this paper, we show some details in Appendix A and Appendix B.
In numerical procedure, we set G = L = 1 to simplify our numerical calculation.

Energy Momentum Tensor and Free energy
In this section, we turn to study the stability of hyperbolic AdS black hole solutions.We only focus on two cases.The one is massless scalar case and the other is massive case.The massive and massless neutral scalar correspond to specific QCD operator, e.g.dimension 4 glueball operator O 1 and dimension 2 glueball operator O 21 respectively in this paper.Our studies will be helpful to understand how deconfinement transition from holographic point of view.
The Rényi entropy is very good quantity to mimic these phase transitions in QCD literature.
Furthermore, we extend CHM transformation to calculate Rényi entropy in sub classes of nonconformal theories.These non-conformal theories are obtained by adding simple deformations operators2 which correspond to neutral scalar with self interaction potential.
To obtain well defined energy momentum tensor on the boundary, one should introduce the suitable counter terms.For later use, we will work out a well defined counter term for these gravity solutions and these terms will be also used in studying free energy and spherical Rényi entropy of dual CFTs.

Energy Momentum Tensor
In this subsection, we would like to introduce the counter terms to cancel the UV divergences of the on-shell action and make the energy momentum tensor of dual field theory well defined.
Firstly, we introduce general gauge invariant counter terms with undetermined coefficients in our system.Finally, we can solve these coefficients to cancel the divergences in massless and massive cases respectively in this paper.

Massless Scalar Cases
For massless scalar case, the total action now becomes where γ µν denotes the induced metric, n µ stands for the normal direction to the boundary surface ∂M as well as D λ stands for covariant derivative.Finally, R and R ab are the Ricci scalar and Ricci tensor for the boundary metric respectively.In general cases, one should introduce higher powers of R and various combination of R ab to cancel the total UV divergence.For massive and massless cases in this paper, we just introduce R to cancel all the UV divergence.
That means we can set λ 2 , λ 3 to be vanishing.
In the asymptotical AdS hyperbolic black hole, the boundary surface locates at z = 0 surface, and usually one has to regularized it to a finite z = ǫ surface.So we have the normalized normal vector n µ = δ µ z √ gzz .To regulate the theory, we restrict to the region z ≥ ǫ and the surface term is evaluated at , where the leading term of expansion of g ij (x, ǫ) with respect to ǫ is the flat metric g ij (0) .Then the one point function of stress-energy tensor of the dual CFT is given by [71] [72] [73] [74] The finite part of boundary energy-stress tensor is from the O(ǫ 2 ) terms of the Brown-York tensor T ij on the boundary z = ǫ, with In the massless scalar hair hyperbolic AdS black hole, the coefficients of counter terms can be following where we have fixed these coefficients by removing the UV divergence z → 0 appeared in on-shell action of massless scalar.Directly evaluate (3.7) using (3.6), we get tt component of energy momentum tensor (3.9)

Massive Scalar Cases
For massive scalar, the total action will be different from massless cases.The main reason is that the UV behavior of massive scalar is different from the massless cases.In massive case, we will introduce following counter term to cook up well defined on-shell action.
Here massive scalar potential are chosen to be V are free parameters.In terms of (3.6), the boundary energy momentum tensor would be In the massive scalar hairy hyperbolic AdS black hole, the coefficients of count terms can be following where O 2 corresponds to expectation value of dual operator O 2 of massive scalar φ.We have fixed these coefficients by removing the UV divergence z → 0 appearing in on-shell action of massive scalar.
Directly evaluate (3.7) using (3.6), we get We have introduced counter terms to make well defined boundary stress tensor.With these counter terms, we can obtain on-shell action which will play an important role in judging the phases of theories in the coming section.

The Difference of Free Energy
After introducing the counter term to remove the divergence of the action, we can work out the on-shell action which will be helpful to test the holographic phase structures.Later, we will also make use of condensation of dual operator to get the flavor of phase transitions.
For massless scalar case, the on-shell action can be For massive scalar case, the on-shell action can be where we have to turn off the source p 22 to obtain the expectation value of dual operator in vacuum for later use.
To summarize this section, we have introduced a self-consistent counter term to obtain the well defined free energy by requiring boundary energy momentum to be finite.Once we obtain the free energy of these black hole solutions, we can study phase structures by comparing free energy in the coming section.

Phase Transitions
In this section, by calculating condensation of dual operator and free energy, we will study the stability of the hyperbolic AdS black hole solutions obtained from Appendix A and Appendix B.
We will show temperature dependence behaviors of condensation of operators O 1 and O 2 , which are dual to massless and massive scalar respectively.Firstly, we will make use of free energy to study the stability of these new hyperbolic black hole solutions.In this section, we mainly focus on the constant modes in which we do not turn on the in-homogenous perturbation of these solutions.The constant mode means that the field configurations only depend on holographic direction z.We would emphasize that this analysis is preliminary and later we will turn to go further to check the stability of these solutions in section 5.In section 5, we will go back to the phase structures in these theories studied in this section in terms of linear perturbation.

Condensation
In this subsection, we will figure out all field configurations and extract the condensation of dual operator O 1 , O 2 of massless or massive scalar field to see what will happen with changing related parameters, for example, temperature and coupling constant of scalar self-interaction.
Basically, one can extract the condensation of dual operator by UV expansion of massless and massive scalar shown in Eq.(A.2) Eq.(A.6) in terms of AdS/CFT dictionary.The condensation will imply whether there is phase transition or not.Later, we will use free energy to confirm these phase transitions and determine the transition temperature.

Massless Cases
We would like to introduce several deformations in massless scalar potential, for example, adding φ 3 , φ 4 , φ 6 terms to the potential.We mainly focus on obtaining condensation of the dual ∆ = 4 glueball operator O 1 with respect to temperature.
Firstly, we would like to calculate the condensation in massless scalar with potential like In fig.1(a), we have shown the condensation as a function of temperature.The different colored curves correspond to choosing different model parameter ν 4 .With increasing ν 4 , the condensation at the same temperature will increase gradually.There is a transition temperature defined where the condensation goes to zero.For each colored curve, the condensation is double valued function with respect to temperature from zero temperature to maximal temperature T max .In fig.1(b), we calculate free energy with respect to temperature and it shows that the dashed line part is unstable comparing with solid curve.That means the T max is phase transition temperature T c in terms of free energy.Below the transi-tion temperature T c = T max , the condensation is a monodrome function of temperature.At the transition temperature, the condensation will jump from finite positive value to zero and the massless hairy black hole solution is unstable comparing with hyperbolic AdS-SW black hole.That is to say hyperbolic AdS-SW black hole is favored when T ≥ T c .Up to this stage, we find the instability exists in this case.
Secondly, we would like to calculate the condensation O 1 in massless scalar with potential In fig.2(a), we have shown the condensation as a function of temperature.The different colored curves correspond to choosing different model parameter ν 6 with fixing ν 4 .With increasing ν 6 , the condensation at same temperature will decrease gradually.There is also a transition temperature defined where the condensation goes to zero.
For each colored curve, the condensation is double valued function with respect to temperature from zero temperature to maximal temperature T max .In fig.2(b), we calculate free energy with respect to temperature and it shows that the dashed line part is unstable comparing with solid curve in T < T max .That means the T max is phase transition temperature T c .Below the transition temperature T c , the condensation is a monodrome function of temperature.At the transition temperature, the condensation will jump from finite positive value to zero and the massless hairy black hole solution is unstable comparing with hyperbolic AdS-SW black hole in T > T max .That is also to say hyperbolic AdS-SW black hole is favored when T ≥ T c .
Below the transition temperature, the condensation is a monodrome function of temperature.
At the transition temperature, the condensation will jump from finite positive value to zero.
We can see that ν Finally, we would like to calculate the condensation in massless scalar with potential like In fig.3(a), the condensation as a function of temperature has been presented.The different colored curves correspond to choosing different model parameter ν 3 with fixing ν 4 .With increasing ν 3 , the condensation at same temperature will decrease gradually.For each colored curve, the condensation is double valued function with respect to temperature from zero temperature to maximal temperature T max .In fig.3(b), we also calculate free energy with respect to temperature and it shows that the dashed line part is unstable comparing with solid curve.That means the T max is still phase transition temperature T c in this case.Below the transition temperature T c , the condensation is a monodrome function of temperature.At the transition temperature, the condensation will jump from finite positive value to zero and the massless AdS hairy black hole solution is unstable comparing with hyperbolic AdS-SW black hole.That is to say hyperbolic AdS-SW black hole is favored when T ≥ T c .Below the transition temperature T ≤ T c , the condensation is a monodrome function of temperature.At the transition temperature, the condensation will jump from finite positive value to zero.The deformation from ν 3 φ 3 L 2 does not change the type of phase transition induced by ν 4 φ 4 L 2 qualitatively.In summary, we introduce three types of special deformations like φ 3 , φ 4 , φ 6 separately in massless neutral scalar potential in the bulk.We calculate the condensation of dual operator of the scalar with respect to temperature.From the numerical behaviors of condensations, there exist transition temperatures T c = T max in three deformations.Furthermore, we calculate the free energy of each deformation to confirm the phase transitions.Finally, these phase transitions induced by three kinds of deformations are the same type qualitatively.Therefore, one can naturally expect that there are still same types of phase transitions in those cases with deformation like superposition of these three kinds of deformations.We will turn to be rigid in section 5 to check the stability of these solutions in the low temperature region T ≤ T c .In section 5, one can find that all these massless hyperbolic hairy AdS black hole are not stable.
There exist more stabler solutions, which are in-homogenous solutions.Therefore, from high temperature to low temperature, the AdS hyperbolic black hole will transit to in-homogenous AdS hairy black hole.

Massive Cases
In this subsection, we would like to deform massive scalar potential by adding φ 3 , φ 4 , φ 6 terms.
We mainly focus on obtaining condensation of the dual ∆ = 2 operator O 2 with respect to temperature.We will see there exist phase transitions in various deformations and how these deformations affect the phase transitions order in details.
Firstly, we will turn to study the condensation in massive scalar with potential like V (φ) = In Fig. 4(a), we have shown the condensation of dual operator as a function of temperature in several cases.Each case corresponds to setting a certain value of self-interaction coupling constants ν 4 .In each case, there is a transition point when the condensation goes to zero.That means the mass hair AdS hyperbolic black hole is more stabler than vanishing condensation solution which is hyperbolic AdS-SW black hole in low temperature region.It implies that there should be a phase transition with increasing temperature in this system.Furthermore, the types of phase transition will be changed with increasing ν 4 , which shows that the ν 4 φ 4 L 2 deformation will play an important role to determine the transition types.In Fig. 4, we increase ν 4 = −0.2,0.0, 1.0 gradually and find that transition temperature is independent on ν 4 .Furthermore, there exists a critical value for ν 4c between ν 4 = −1 and ν 4 = −0.2 .Crossing this critical point, the phase transition order will be changed in ν 4 < ν 4c .As shown in Fig. 4, the transition 4 will be first and second order phase transition in ν 4 < ν 4c and ν 4 ≥ ν 4c respectively.In fig.4(b), the free energy will increase with temperature.All colored curves will converge to one point which corresponds to transition temperature in ν 4 > ν 4c .
The transition temperature is the same as transition temperature given by fig. 4 Now we will turn to study the condensation in massive scalar with potential like V (φ) = We introduce ν 6 φ 6 L 2 deformation and to see what will happen for phase transition.In Fig. 5(a), one can see the condensation with respect to temperature with choosing different values of coupling constant ν 6 .With increasing ν 6 = 0.0, 2.0, the condensation will monotonically decrease from positive finite value to vanishing.In ν 6 < 0.0 region, the 4 Such phase transitions are similar to holographic P-wave Superconductor Phase Transition shown in [17].
condensation is multiple valued function of temperature as shown in Fig. 5(a) and there is a local maximal temperature T max and minimal temperature T min in each curve.For ν 6 = −0.1, the condensation will decrease from T = 0 to T = T min and it will jump to less finite positive value at T min .From T min < T < T max , the condensation will become multivalued function of temperature.For T ≥ T max , the condensation will decrease to zero continuously in Fig. 5(a).
In Fig. 5(b), we have shown various free energy with respect to temperature with gradually changing the ν 6 .We also find that free energy with ν 6 = 0, 2 is monotonically increasing with temperature.They always continuously converge to the transition point T c .The transition point is defined by vanishing of condensation.But in cases with ν 6 = −0.1, the free energy is multiple valued function of temperature.For these cases, there are minimal temperatures T min and local maximal temperature T max .For T > T c , hyperbolic AdS-SW black hole should be stable and there is no massive scalar hair black hole solution.In T max < T < T c and 0 < T < T min , massive scalar hair black hole is more stabler than hyperbolic AdS-SW black hole.In T min < T < T max , the condensation of dual operator is a multiple valued function and the stable solution is marked by solid curve in Fig. 5(a In the third case, we will focus on the condensation with potential V In Fig. 6(a), we can find that the condensation will decrease from positive finite value to vanishing in ν 3 > ν 3c region.In our setup, ν 3c = 0.In ν 3 < 0 region, the condensation will be multiple valued function of temperature.This case is the similar as first massive case.In this region, the transition order will be change.As shown in Fig. 6, the transition5 will be first and second order phase transition in ν 3 < ν 3c and ν 3 ≥ ν 3c respectively.Because the condensation can not continuously decrease to zero at transition temperature and it will suddenly jump from positive finite value to zero.In 6(b), we have shown the free energy as function of temperature.
In ν 3 > 0 region, free energy is monotonically increasing with temperature, while free energy is multiple valued function of temperature in ν 3 < 0. That means the free energy can not converge to the transition temperature continuously, while massive AdS hairy black hole will jump to hyperbolic AdS-SW black hole at transition temperature.Roughly speaking, the phase transitions induced by ν 3 φ 3 L 2 is the similar as ones induced by ν 4 φ 4 L 2 .
( a ) ( b ) Finally, we would like to focus on the condensation in massive scalar with potential V (φ) = The main motivation to study this case is that we expect to find competitive mechanism between ν 4 φ 4 L 2 deformation and ν 6 φ 6 L 2 deformation.In Fig. 7(a), one can vary ν 6 with fixing ν 4 = 1.0 to see that the condensation will monotonically decrease to zero from low temperature to high temperature for ν 6 > ν 6c .In ν 4 = 1.0 case, ν 6c = −0.1 such that T min = T max .For fixing ν 4 , one can tune ν 6 = ν 6c to be a solution in which T max will coincide with T min .One can vary ν 4 to find corresponding ν 6c .While in Fig. 7(b), we confirm that the hyperbolic AdS hairy black hole solution is much stable than hyperbolic AdS-SW in T max < T < T c and 0 < T < T min with ν 6 < ν 6c , ν 4 = 1.Where T c is defined by the point where the condensation is vanishing in Fig. 7(a) and T min , T max are marked in Fig. 7(a).In T > T c , there is no stable hairy black hole solution for ν 6 > ν 6c and hyperbolic AdS-SW solution is stable one.In ν 6 < ν 6c , the condensation will become multivalued function of temperature from T min < T < T max ,.For ν 6 = −0.5 example, the stable configuration in 0 < T < T min is the hyperbolic AdS hairy black hole solution, while in T > T c > T min is hyperbolic AdS-SW black solution.When T min < T < T c < T max as shown in Fig. 7, there is a phase transition between two hairy AdS black holes and the condensation will jump from positive finite value to less positive finite value.Especially at T c , there is phase transition between hyperbolic AdS hairy black hole and hyperbolic AdS-SW black hole due to condensation goes to vanishing.
These numerical studies show that there is competitive mechanism between ν 4 φ 4 L 2 deformation and ν 6 φ 6 L 2 deformation.One can tune ν 4 , ν 6 to see which phase is stable and what type of phase transition happens.One can set ν 4 = 0 and this numerical result will reproduce one in second massive case.
Here we have introduced 4 kinds of deformations in massive scalar potential and study these deformations effects on stability of hyperbolic AdS hairy black holes case by case.In each case, the condensation as a function of temperature implies that there exist phase transitions in deformed theories.The behavior of condensation and free energy with respect to temperature in φ 3 and φ 4 deformed theories will be the same as ones in the massless cases with φ 3 , φ 4 , φ 6 deformations in section4.induced by φ 6 deformation essentially.In massive cases, they imply that the types of phase transitions induced by φ 6 are different from that caused by φ 3 , φ 4 .Essentially, all these phase transitions mainly originate from the effective mass of scalar below the effective BF bound for the near horizon AdS 2 .However, it is not enough to confirm the phase transitions by analyzing the condensation of dual operator and free energy.In section 5, we will see the hyperbolic AdS hairy black hole solutions will be stable in low temperature region T < T c when coupling constants v 3 , v 4 , v 6 live in specific regions.We will see details in section 5. Otherwise, when the coupling constants v 3 , v 4 , v 6 go beyond these specific region, these hyperbolic AdS hairy black hole solutions will not be stable anymore and there exist much more stabler in-homogenous solutions.Further numerical studies are needed to check these stability of in-homogenous solutions.

Instability for the Normalizable Mode
Previous discussions on the stability of different phases are mainly based on thermodynamical analysis with comparing free energy.Comparing free energy between constant solution 6 and hyperbolic AdS-SW is not enough to make sure these new hyperbolic AdS hairy black hole solutions are stable or not.To be rigorous, in this section we will investigate the instability of these solutions under scalar perturbation δΦ(t, z, ψ, θ, ϕ).The wave function of δΦ(t, z, ψ, θ, ϕ) could be decomposed as .Here, we will consider the 5D case, so d = 5 and λ > 1.More generally, when Σ is a non-trivial quotient of hyperboloid, then the lower bound of λ would be extended to 0. Thus, below we will only consider λ > 0 and ω 2 (λ = 0) for simplicity [63] [18].
Under the ansatz Eq.(5.1), the equation of motion for δφ could be derived as follows where A e , f, φ are associated with background solutions.In our ansatz Eq.(5.1), the time related part behaves as e ωt .The black hole will be unstable if (5.2) has a solution with real and positive ω 2 with the field satisfying specific boundary conditions at infinity and the horizon.
Therefore, if there exist solutions with positive ω 2 in certain background solutions, then the background constant solutions are unstable.This instability is induced by inhomogenous 6 We also call hyperbolic AdS hairy black hole solutions perturbation in boundary special direction.If one can not find such perturbative modes with positive ω 2 , then the background solutions are stable at the level of linear perturbation.This is the key criterion to test the stability of these solutions.In principle, one should construct AdS hairy black holes at the non-linear level which is considerably more difficult.In this paper, it is sufficient to demonstrate that an instability exists by linear perturbation.
The leading expansion of δφ near the horizon z = z h could be derived from Eq.(5.2) as following with δp 1 , δp 2 the two integral constants of the second order derivative equation Eq.( 5.2).Without loss of generality, we assume ω = √ ω 2 > 0, then the δp h1 mode tends to 0 when z approaches z h , while the δp h2 mode is divergent near horizon.Thus, the near horizon boundary condition is easy to be set as δφ(z h ) = 0.
For the UV boundary condition, again, we could calculate the near boundary expansion of δφ from Eq.(5.2).It depends on the dimension of φ.For ∆ = 2 as example, the leading expansion is of the form As in the background solutions, we will require the coefficient of z 2 log(z) to be 0. In general, only for certain groups of (λ, ω 2 ) the solutions of δφ could satisfy both the UV and IR boundary conditions simultaneously.We will try to find such kind of solutions under the background constant solutions solved in previous sections, and to see whether it is stable or not under the linear perturbation.

Massless Scalar Cases
Firstly, we focus on the stability in the massless cases.In terms of previous arguments in last section, we only study the sign of ω 2 (λ = 0) and we can test stability of these solutions solved in previous several sections.
In Fig.
. In all these cases, one can see that ω 2 (λ = 0) always positive from low to high temperature region.These solutions shown in Fig. 1 Fig. 2 Fig. 3 should be unstable configurations, although these configurations are much more stable than hyperbolic AdS-SW black hole with comparing free energy.One can see that there should exist in-homogenous black hole solutions which break the hyperbolic symmetry.
That is also means that hyperbolic AdS-SW black hole will transit to in-homogenous black hole solutions.In-homogenous black hole solutions are hard to be constructed which will be interesting to be studied in the near future.
L 2 at the same parameter values as in Fig. 2.Where we have scanned all relevant region of ν 4 , ν 6 in our setup and we just show most important characteristic qualitative behavior by choosing specific value of ν 4 , ν 6 as examples.
Figure 10: ω 2 (λ = 0) as a function of temperature in massless scalar case with potential L 2 at the same parameter values as in Fig. 3.Where we have scanned all relevant region of ν 3 , ν 4 in our setup and we just show most important characteristic qualitative behavior by choosing specific value of ν 3 , ν 4 as examples.

Massive Scalar Cases
In this subsection, we turn to focus on the stability of new hyperbolic black hole solutions with massive scalar potentials.Here we have studied four cases which are shown in Fig. 11 L 2 respectively.For massive scalar cases, there are something new presented.Up to linear perturbative analysis, some homogenous solutions are still stable.
For example, as shown in Fig. 11, we can tune v 4 gradually and then we can find a critical value of v 4c = 0. Once v 4 > 0, ω 2 (λ = 0) are always negative definite function of temperature and which also means v 4 > 0 these solutions found in Fig. 4 might be stable at level of linear perturbation analysis.That means hyperbolic AdS-SW black hole will transit to inhomogenous solution from high temperature to low temperature when v 3 = 0, v 6 = 0, v 4 > 0.
One can also tune the v 6 gradually to find the critical value of v 6 = 0.When v 6 becomes positive, one can not find positive definite ω 2 (λ = 0) which implies that solutions with positive v 6 might be also stable and phase transition might happen in Fig. 5.That means hyperbolic AdS-SW black hole will transit to homogenous solution from high temperature to 7 Here we confirm the transition in terms of linear perturbative analysis.low temperature when v 3 = 0, v 4 = 0, v 6 ≥ 0. The high temperature solution will transit to in-homogenous solutions for v 3 = 0, v 4 = 0, v 6 < 0. In Fig. 13, one can also tune the v 3 gradually to find critical value v 3 = 0. From high temperature to low temperature, the hyperbolic AdS-SW black hole will transit to in-homogenous solution will transit to in-homogenous solutions for v 3 < 0, v 4 = 0, v 6 = 0.For v 3 ≥ 0, v 4 = 0, v 6 = 0, it will transit to homogenous solution constructed in section 4.1.2.
Finally, we consider more complicated situation with potential For simplifying our study, we fix v 3 = 0, v 4 = 1 and gradually tune v 6 to obtain the critical value v 6 = −0.1 such that there is no positive ω 2 (λ = 0) existing in the black hole solution.
We expect that the solutions found in Fig. 7 with positive v 6 are stable.In terms of criterion mentioned above, hyperbolic AdS-SW black hole might transit to homogenous solution from high temperature to low temperature when it will transit to in-homogenous solution as confirmed in Fig. 14.Further, one can also see interesting phenomenon that change of v 4 will affect the critical value of v 6 .
Figure 11: ω 2 (λ = 0) as a function of temperature in massive scalar case with potential L 2 at the same parameter values as in Fig. 4. When v 4 = 0, 1, we did not find positive ω 2 at λ = 0.Where we have scanned all relevant region of ν 4 in our setup and we just show most important characteristic qualitative behavior by choosing specific value of ν 4 as examples.

Comments on Holographic Spherical Rényi Entropy
In this section, we would like to connect the instability of hyperbolic AdS black hole with holographic Rényi Entropy, as we reviewed in the introduction.
Figure 12: ω 2 (λ = 0) as a function of temperature in massive scalar case with potential L 2 at the same parameter values as in Fig. 5.When v 6 = 0, 2, we did not find positive ω 2 at λ = 0.Where we have scanned all relevant region of ν 6 in our setup and we just show most important characteristic qualitative behavior by choosing specific value of ν 6 as examples.
Figure 13: ω 2 (λ = 0) as a function of temperature in massive scalar case with potential V (φ) = − 12 L 2 − 16φ 2 3L 2 + ν 3 φ 3 L 2 at the same parameter values as in Fig. 6.When v 3 = 0, 1, we did not find positive ω 2 at λ = 0.Where we have scanned all relevant region of ν 3 in our setup and we just show most important characteristic qualitative behavior by choosing specific value of ν 3 as examples.

Spherical Rényi Entropy as Thermal Entropy
Following [18], we can compute the Rényi entropy from these thermal entropies, via (1.4) where S Eh thermal (T ) is the entropy of the hairy black hole and S E thermal (T ) is the entropy of the Einstein black hole.
L 2 at the same parameter values as in Fig. 7.When v 6 = −0.1,0, 0.5, we did not find positive ω 2 at λ = 0.Where we have scanned all relevant region of ν 4 , ν 6 in our setup and we just show important characteristic qualitative behavior by choosing specific value of ν 4 , ν 6 as examples.
In terms of the above formulas, the Rényi entropy as a function of n.Because the derivative of the thermal entropy with respect to the temperature is discontinuous, the second derivative with respect to n of the Rényi entropy is discontinuous.Such kind of discontinuous is closely related to instability of hyperbolic AdS black hole.Such instability has been carefully studied in section 4 and section 5. Therefore, the discontinuous of Rényi entropy implies a phase transition in dual field theory by holography.In order to determine the precise value of n c at which this transition occurs, one should study numerically the scalar wave equation within the black hole background as shown in [50] [19].The critical temperature is defined by 2ncπR .As we shown in section 4 and section 5, the order of phase transition will be different from the second order phase transition8 presented by [18] due to the higher powers of neutral scalar with self-interactions.
The first term in left hand side of Eq.(6.1) can work which highly depends on the wether the CHM mapping can work or not in our the hairy black holes.The CHM mapping is a kind of coordinate transformation which simplifies for a spherical or planar entangling surface in CFTs.It works even when conformal symmetry is partially broken as long as you take into account the conformal factor eq.( 2.3) correctly.It is also clear in the holographic picture where the CHM mapping is a coordinate transformation from the Poincare to the AdS hyperbolic coordinates.It works not only for CFT, but also for any theories with UV fixed points.Indeed, due to the condensation of the scalar, the stable phase will be hyperbolic AdS hairy black hole.
In our cases, the scalar depends only on the holographic direction z and it just modify the conformal factor e Ae(z) in (2.3).The conformal factor only modify the unitary transformation in eq.(1.1).Therefore, CHM mapping is still meaningful in our cases.
However, if the scalar depends on the boundary coordinates called inhomogenous cases in this paper, CHM mapping can not be applicable in condensation phase with large value of condensation.But in these cases, nearby the transition point T ∼ T c 9 from hyperbolic AdS-SW to hyperbolic AdS hairy black hole, the hyperbolic AdS hairy black hole (condensation phase) can be regarded as hyperbolic AdS-SW with neutral scalar perturbatively.And then the condensation of scalar will gradually be turn on and condensation phase will be stable.
Near the transition point, the CHM mapping can be still workable due to the small value of condensation.In this sense, CHM mapping gives us an important insight of phase transition in the dual field theory.When hyperbolic AdS hairy black hole become dominant with decreasing temperature, CHM mapping will break down due to large value of condensation and the spherical entanglement surface in dual field can not be identified with thermal entropy of hyperbolic AdS black hole directly.
To be more precisely, once we take n ≫ n c , CHM mapping does not hold any more due to To close this section, we offer a short summary to make our claim to be clear.For the cases of Einstein Dilaton system shown in eq.(2.1) eq.( 2.3), if the scalars in hyperbolic AdS

Conclusions and Discussions
In this paper, we have constructed several new hyperbolic asymptotic AdS gravity solutions in Einstein Dilaton system numerically.Motivated by studying ERE with spherical entangling surface in deformed CFTs by CHM mapping, we work out the hyperbolic hairy AdS with series powers of neutral scalar in potential.In this paper, we focus on potential with φ 3 , φ 4 , φ 6 .
Especially, we focus on two kinds of special scalar potentials.The one is massless scalar with higher powers of scalar self interaction and the other is that we choose square of the scalar mass to be −2.In terms of AdS/CFT, the first kind of scalar could correspond to dimension 4 gluon sector in gauge field theory side and the other scalar is dual to dimension 2 gluon operator in field theory side.In general, to calculate the ERE with complicated entanglement surface is very hard.For spherical entanglement surface, one can make use of proposals [46] [48] [49] to relate the ERE to the thermal entropy in hyperbolic AdS black hole.We have shown the configuration of these new hyperbolic AdS solutions and also extract the condensation of operators which are dual to massless and massive scalars respectively.Through studying condensation with respect to temperature, we find that there exist phase transitions.We list the well defined boundary energy momentum tensor by introducing proper boundary counter terms in each solution.With these counter terms, the finite free energy can be achieved.
We compare free energy between the new hyperbolic AdS solutions and hyperbolic AdS-SW solution to check the stability of these solutions.To be more rigid, we turn on inhomogenous perturbation on these new hyperbolic AdS black holes to check the stability.We tune the potential parameters to figure out the stable region of potential parameters for these solutions, for example, the coefficients of the cubic, quartic and sextic scalar interactions v 3 , v 4 , v 6 .For massless scalar cases, we can not find stable homogenous solutions with turning on φ 3 , φ 4 , φ 6 in scalar potentials respectively.Therefore, we can not safely say phase transition shown in Fig. 1 Fig. 2 Fig. 3 really happens.There must exist stable inhomogenous solutions.That means hyperbolic AdS-SW black hole will transit to inhomogenous solutions in massless cases from high temperature to low temperature.For massive scalar cases with positive potential parameters v 3 , v 4 , v 6 respectively, φ 3 , φ 4 will induce similar phase transition qualitatively shown in Fig. 4 Fig. 6, while φ 6 term in scalar potential will induce different kinds of phase transition in Fig. 5.If one turns on superposition of φ 3 , φ 4 and φ 6 in scalar potential, there exists competitive mechanism between phase transitions induced by φ 3 , φ 4 and φ 6 in Fig. 7. To be rigid, when we choose negative potential parameters v 3 , v 4 , v 6 respectively, our studies show that all these hyperbolic hairy AdS black hole solutions are not stable ones anymore.With negative potential parameters v 3 , v 4 , v 6 separately, there may exist stable inhomogenous solutions which are much more stabler than hyperbolic AdS-SW black hole solutions.In these cases, hyperbolic AdS-SW black hole solutions will transit to inhomogenous solutions from high temperature to low temperature.Once v 4 , v 6 are turn on simultaneously, the critical values of v 4 , v 6 will be changed accordingly respectively.From this phenomenon, one can expect that there exist competition mechanism to determine the critical value of potential parameters v 3 , v 4 , v 6 .Once we know the phase structures of various black hole solutions, we make use of [46] [48] [49] to comment on the phase structure in the dual field theory in terms of spherical entanglement entropy.In this sense, the stability of these black hole solutions is closely related to the spherical ERE in holographic dual CFTs and it gives some insight of phase transitions in dual field theory.
In this paper, we focus on massless and massive scalar cases with higher powers of self interaction in potentials.In general, such kinds of deformations will lead to various types of phase transitions which are highly sensitive to the operators chosen and types of deformations.
ERE can be also regarded as an order parameter to give some insight on phase transitions in dual field theories.Finally, we analysis how CHM can work in our cases.For our gravity setup, we offer a circumscribed criterion to judge whether CHM can work or not.For generic setup, we offer an idea to use CHM mapping to study Rényi entropy in perturbative sense in section 6. 1

Figure 4 :
Fig. 4(b), the dominate phase should be hyperbolic AdS-SW black hole above the transition temperature.The free energy can continuously converge to the transition point in Fig. 4(b)with ν 4 > ν 4c .But free energy will jump to the transition point with ν 4 < ν 4c .That is also means the order of phase transition should change suddenly and the transition temperature will be T max , for example, curves shown in ν 4 = −1.4,−1.2, −1.0.This phenomenon is also consistent with a condensation jump from finite value to vanishing in Fig.4(a).

1 . 1 . 8 Fig. 9 Fig. 10 , 5 Fig. 7 .
Comparing with Fig.it has different behaviors with respect to temperature in φ 6 deformed theories shown in Fig.Such exotic behavior is

. 1 )
with Y the eigenfunction of Laplacian in certain manifold Σ and λ the corresponding eigenvalues.When Σ is just the hyperboloid H d−2 , λ has the lower bound λ > (d−3) 24

Fig. 9 Fig. 10 ,
we show the ω 2 (λ = 0) as a function of temperature numerically with turning on the linear perturbation of hyperbolic black hole solution with V

Figure 8 : 2 + ν 4 φ 4 L 2
Figure 8: ω 2 (λ = 0) as a function of temperature in massless scalar case with potential V (φ) = − 12 L 2 + ν 4 φ 4 L 2 at the same parameter values as in Fig.1.Where we have scanned all relevant region of ν 4 in our setup and we just show most important characteristic qualitative behavior by choosing specific value of ν 4 as examples.

Figure 9 :
Figure 9: ω 2 (λ = 0) as a function of temperature in massless scalar case with potential V(φ) = − 12 L 2 + ν 4 φ 4 L 2 + ν 6 φ 6L 2 at the same parameter values as in Fig.2.Where we have scanned all relevant region of ν 4 , ν 6 in our setup and we just show most important characteristic qualitative behavior by choosing specific value of ν 4 , ν 6 as examples.

Fig. 13 Fig. 14 .
Fig.13 Fig.14.Here we summarize final results in the following.In Fig.11 Fig.12 Fig.13 Fig.14, we show the ω 2 (λ = 0) as a function of temperature numerically with turning on the linear perturbation of hyperbolic black hole solution with
the presence of in-homogenous operator condensation.When |n − n c | ∼ 0 or condensation is very small, as argued in last paragraph, CHM mapping can be still hold approximately.In this sense, CHM map spherical entanglement entropy to thermal entropy of hyperbolic AdS black hole perturbatively.When |n − n c | increases gradually, the CHM mapping will be deformed the mapping by two main aspects.The first is that the dual quantum states in field theory will be excited states because of introducing the operator dual to neutral scalar.That means we have to study the Rényi entropy of low excited states by CHM mapping approximately.This is closely related to the first law of entanglement entropy 10 once the subsystem is very small.The other one is the shape of entangle surface will be deformed 11 from sphere to other general geometrical shape gradually with increasing condensation or n → ∞, due to dominant phase occupied by inhomogenous scalar.In the future, quantitative studies of the two main aspects are needed to show whether and how CHM mapping works in excited states and inhomogenous configurations.However, in our cases, once we take |n − n c | ≫ 0 large enough, what we have calculated by (1.2) should correspond to spherical Rényi entropy of states excited by operators which dual to the various homogenous scalars respectively.

9
|n − nc| ∼ 010 The first law of entanglement entropy has been well studied in the holographic literature, for example,[75] [76] [77][78].11The entanglement entropy with deformation of entangle surface has been also studied extensively by[79]  [80]  [81][82].hairyblack hole only depend on holographic coordinate z (called homogenous solutions) and CHM transformation only change the scale transformation by global conformal factor e As(z) in (2.3) which preserve the asymptotic AdS boundary conditions, we can make use of CHM mapping to compute Rényi entropies.Otherwise, for example, in-homogenous scalar will break CHM transformation and one can not absorb conformal transformation for scalar into global conformal factor anymore. valid of CHM transformation highly depends on whether we can find CHM transformation up to global conformal factor which correspond to U unitary transformation in eq.(8.1).

Figure 15 : 2 + ν 4 φ 4 L 2
Figure 15: Solutions when V (φ) = − 12 L 2 + ν 4 φ 4 L 2 with ν 4 = −8, f 4 = 0.2445.The blue lines give the results when p 4 = 0.3 while the purple lines are for p 4 = 0.8.In Panel.(a) and Panel.(b), the solutions of φ and A e are given.In Panel.(c), the solutions of f is shown in red solid line, while the corresponding Q(z) is shown in blue dashed line(Here, in order to put the two in the same figures, we plot Q(z)/50, which is zero at the same z as Q(z)).

Figure 16 : 2 + ν 4 φ 4 L 2 with ν 4 = − 8 .
Figure 16: Characteristic solutions when V (φ) = − 12 L 2 + ν 4 φ 4 L 2 with ν 4 = −8.To get these solutions, we have taken f 4 = 0.2445, p 2 = 0.36734....In Panel.(a) and Panel.(b), the solutions of φ and A e are given.In Panel.(c), the solutions of f is shown in red solid line, while the corresponding Q(z) is shown in blue dashed line(Here, in order to put the two in the same figures, we plot Q(z)/50, which is zero at the same z as Q(z)).