# Anti-de-Sitter-Maxwell-Yang-Mills Black Holes Thermodynamics from Nonlocal Observables Point of View

Authors: H. El Moumni
Journal: Advances in High Energy Physics
 (2019)

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## Abstract

In this paper we analyze the thermodynamic properties of the Anti-de-Sitter black hole in the Einstein-Maxwell-Yang-Mills-AdS gravity (EMYM) via many approaches and in different thermodynamical ensembles (canonical/grand canonical). First, we give a concise overview of this phase structure in the entropy-thermal diagram for fixed charges and then we investigate this thermodynamical structure in fixed potentials ensemble. The next relevant step is recalling the nonlocal observables such as holographic entanglement entropy and two-point correlation function to show that both observables exhibit a Van der Waals-like behavior in our numerical accuracy and just near the critical line as the case of the thermal entropy for fixed charges by checking Maxwell’s equal area law and the critical exponent. In the light of the grand canonical ensemble, we also find a newly phase structure for such a black hole where the critical behavior disappears in the thermal picture as well as in the holographic one.


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## Body

## 1. Introduction

Over the last years, a great emphasis has been put on the application of the Anti-de-Sitter/conformal field theory correspondence [1, 2] which plays a pivotal role in recent developments of many physical themes [3–5]; in this particular context the thermodynamics of Anti-de-Sitter black holes become more attractive for investigation [6].In general, black hole thermodynamics has emerged as a fascinating laboratory for testing the predictions of candidate theories of quantum gravity. It has been figured that black holes are associated thermodynamically with a entropy and a temperature [7] and a pressure [8]. This association has led to a rich structure of phase picture and a remarkable critical behavior similar to van der Waals liquid/gas phase transition [9–19]. Another confirmation of this similarity appears when we employ nonlocal observables such as entanglement entropy, Wilson loop, two-point correlation function, and the complexity growth rate [20–33]. Meanwhile, these tools are used extensively in quantum information and to characterize phases and thermodynamical behavior [21, 23, 34–41].The black hole charge finds a deep interpretation in the context of the AdS/CFT correspondence linked to condensed matter physics; the charged black hole introduces a charge density/chemical potential and temperature in the quantum field theory defined on the boundary [42]. In this background, the charged black hole can be viewed as an uncondensed unstable phase which develops a scalar hair at low temperature and breaks U(1) symmetry near the black hole horizon reminiscing the second-order phase transition between conductor and superconductor phases [43]; this situation is called the "s-wave" holographic superconductor. It has also been shown that "p-wave" holographic superconductor corresponds to vector hair models [44, 45]. The simplest example of p-wave holographic superconductors may be provided by an Einstein-Yang-Mills theory with SU(2) gauge group and no scalar fields, where the electromagnetic gauge symmetry is identified with an U(1) subgroup of SU(2). The other components of the SU(2) gauge field play the role of charged fields dual to some vector operators that break the U(1) symmetry, leading to a phase transition in the dual field theory.Motived by all the ideas described above, although the Yang-Mills fields are confined to acting inside nuclei while the Maxwell field dominates outside, the consideration of such theory where the two kinds of field live is encouraged by the existence of exotic and highly dense matter in our universe. In this work, we try to contribute to this rich area by revisiting the phase transition of Anti-de-Sitter black holes in Einstein-Maxwell-Yang-Mills (EMYM) gravity. More especially, we investigate the first- and second-order phase transition by different approaches including the holographic one and in different canonical ensembles.This work is organized as follow: First, we present some thermodynamic properties and phase structure of the EMYM-AdS black holes in (temperature, entropy)-plane in canonical and grand canonical ensemble. Next, we show in Section3 that the holographic approach exhibits the same behavior; in other words we recall the entanglement entropy and two-point correlation function to check the Maxwell’s equal area law and calculate the critical exponent of the specific heat capacity which is consistent with that of the mean field theory of the Van der Waals in the canonical ensemble near the critical point. In the grand canonical one, a new phase structure arises where the critical behavior disappears in the thermal as well as the holographic framework. The last section is devoted to a conclusion.


## 2. Critical Behavior of Einstein-Maxwell-Yang-Mills-AdS Black Holes in Thermal Picture

### 2.1. Canonical Ensemble

We start this section by writing theN-dimensional for Einstein-Maxwell-Yang-Mills gravity with a cosmological constant Λ described by the following action [46, 47](1)I=12∫MdxN-gR-N-1N-23Λ-FM-FYM,where R is the Ricci scalar while Λ is the cosmological constant. Also FM=FμνFμν and FYM=TrFμνaFaμν are the Maxwell invariant and the Yang-Mills invariant, respectively; the trace element stands for Tr.=∑a=1(N-1)(N-2)/2.. Varying the action (1) with respect to the metric tensor gμν, the Faraday tensor Fμν, and the YM tensor Fμν(a), one can obtain the following field equations(2)Gμν+Λgμν=TμνM+TμνYM(3)∇Fμν=0(4)D^μFμνa=∇μFaμν+1ΘεbcaAμbFcμν=0where Gμν is the Einstein tensor, the quantity ε(b)(c)(a)’s stands for the structure constants of the N-1N-2/2-parameters Lie group G, Θ is coupling constant, and Aμ(a) denotes the SO(N-1) gauge groupe YM potential. We also note that the internal indices {a,b,c,⋯} do not differ whether in covariant or contravariant form. In addition, TμνM and TμνYM are the energy momentum tensor of Maxwell and YM fields with the following formula(5)TμνM=-12gμνFρσFρσ+2FμλFνλ(6)TμνYM=∑a=1N-1N-2/2-12gμνFρσaFaρσ+2FμλaFνaλ(7)Fμνa=∂μAνa-∂νAμa+12ΘεbcaAμbAνc,(8)Fμν=∂μAν-∂νAμ.where Aμ is the usual Maxwell potential. The metric for such N dimensional spherical black hole may be chosen to be [47](9)ds2=-frdt2+1frdr2+r2dΩn2in which dΩn2 represents the volume of the unit n-sphere which can be expressed in the standard spherical form(10)dΩN-22=dθ12+∑i=2N-3Πj=1i-1sin2⁡θjdθi2where0≤θ1≤π,0≤θi≤2π.In order to find the electromagnetic field, we recall the following radial gauge potential ansatzAμ=h(r)δμ0 which obeys the Maxwell field equations (3) with the following solution(11)dhrdr=crN-2where c is an integration constant related to electric charge C of the solutions. To solve the YM field, (4), we use the magnetic Wu–Yang ansatz of the gauge potential [48, 49] given by(12)Aa=Qr2xidxj-xjdxi2≤i≤N-1,1≤j≤i-11≤a≤N-1N-22where we imply (to have a systematic process) that the super indices a are chosen according to the values of i and j in order. For instance, we present some of them(13)A1=Qr2x2dx1-x1dx2A2=Qr2x3dx1-x1dx3A3=Qr2x3dx2-x2dx3A4=Qr2x4dx1-x1dx4A5=Qr2x4dx2-x2dx4A6=Qr2x4dx3-x3dx4A7=Qr2x5dx1-x1dx5A8=Qr2x5dx2-x2dx5A9=Qr2x5dx3-x3dx5A10=Qr2x5dx4-x4dx5⋯in which r2=∑i=1N-1xi2. The YM field 2-forms are defined by the expression(14)Fa=dAa+12QεbcaAb∧Ac.In general forN we must have N-1N-2/2 gauge potentials. The integrability conditions(15)dFa+1QεbcaAb∧Fc=0are easily satisfied by using (28). The YM equations(16)d∗Fa+1QεbcaAb∧∗Fc=0also are all satisfied. The energy-momentum tensor (4) becomes after(17)∑a=1N-1N-2/2FλσaFaλσ=N-3N-2Q2r4with the nonzero components(18)T00=N-3N-2Q2fr2r4T11=-N-3N-2Q22r4frT22=-N-3N-6Q22r2TAA=-N-3N-6Q22r2Πi=1A-2sin2⁡θi2<A≤N-1.Using (2) and after some simplifications, one can find that the metric function f(r) has the following form given by(19)fr=1-2mrn-1-Λ3r2+2n-1C2nr2n-2-n-1Q2n-3r2,where N=n+2; one can note in the particular case for n=3 that the last term of (19) diverges, involving an unusual logarithmic term in Yang-Mills charge [46]. For this gravity background the parameter m is related to the mass of such black hole, while C and Q are the charges of Maxwell field and Yang-Mills field, respectively. Following previous literature [8, 9], one can find a close connection between the cosmological constant and pressure as P=-Λ/8π, leading to the following expressions of Hawking temperature, mass, and entropy of such black hole in terms of the horizon radius r+(20)T=f′r+4π=n-14πr++2n+1P3r+-n-1Q24πr3-n-12C22πnr+2n-1,(21)M=nωn48π8πPr+n+1+3n-1Q2rn-33-n+3r+n-1+6n-1C2nr+n-1,(22)S=ωnr+n4,The Yang-Mills potentialΦQ and the electromagnetic one ΦC can be written as(23)ΦQ=∂M∂QC,P,r+=ωnn-1nQ8π3-nr+n-3,(24)ΦC=∂M∂CQ,P,r+=ωnn-1C4πr+n-1,where ωn=2π(n+1)/2/Γ(n+1/2) is the volume of the unit n-sphere. In fact, according to the interpretation of the black hole mass M as an enthalpy [8] in the extended phase space context, the free energy F of black hole can be written as(25)F=M-T·Sand the heat capacity is given by(26)CQ=T∂S∂TQIt is straightforward to show that obtained quantities (20), (21), and (22) obey the first law of black hole thermodynamics in the extended phase space(27)dM=TdS+ΦQdQ+ΦCdC+VdP,where V is the Legendre transform of the pressure, which denotes the thermodynamic volume with(28)V=∂M∂PS,ΦQ,ΦC=nωnπ3rn+1. In addition to this, using scaling argument, the corresponding Smarr formula is(29)M=nn-1TS+ΦCC+1n-1ΦQQ-2n-1VP. Without loss of generality, inserting (22) into (20), we can get the entropy Hawking temperature relation of such black hole, namely,(30)T=112πn6Cn-1241/nSωn1/n1-2n+π22/n+3nn+1PSωn1/n-3n-1nQ24-3/nSωn-3/n+34-1/nn-1nSωn-1/nThis relation is depicted in Figure1; indeed it has been shown that there is a Van der Waals-like phase transition; furthermore a direct confirmation comes from the solution of the following system(31)∂T∂SQ=∂2T∂S2Q=0.which reveal the existence of a critical point. The critical charge, entropy, and temperature are given in Table 1; for all the rest of the paper we keep n=2.Table 1
Coordinates of the critical points for different values ofC in (T,S)-diagram.
Q c S c T c C = 0 1 2 3 π 2 2 3 π   C ≠ 0 1 6 3 - 36 C 2 π 2 2 3 πFigure 1
The temperature as a function of the entropy for different values of chargeC.(Left)C=0.(Right)C≠0.An important remark that can be observed here is that both quantitiesTc and Sc are insensible with the charge C. The behavior of the free energy F with respect to the temperature may be investigated by plotting in Figure 2 the graph F-T for a fixed value of charge Q under the critical one. From this plot, we can observe the characteristic swallow-tail which guarantees the existence of the Van der Waals-like phase transition.Figure 2
The Helmholtz free energy in function of the entropy for EMYM-AdS black holes for different values of the chargeC.(Left)C=0.(Right)C≠0.Using Figure2, we can derive numerically the coexistence temperature T⋆ needed in the Maxwell’s equal area law construction(32)A1≡∫S1S2TSdS-T⋆S2-S1=T⋆S3-S2-∫S2S3TSdS≡A2,where S3, S2, and S1 are the solutions of T(S)=T⋆ in descending order, in addition to the numerical values of these points, and we report in Table 2 the areas A1 and A2 in Maxwell’s law (32)Table 2
Check of the equal area law in theT-S plane for different C.
C T ⋆ S 1 S 2 S 3 A 1 A 2 0 0.15376 0.638237 1.52994 3.13241 0.383499 0.38350  14 150965 1.05043 11.54616 2.25932 00.18248 0.1825Obviously, the areaA1 equals area A2 for different C, so the equal area law does not break. For the second phase transition, we know that, near the critical point, there is always a linear relation with slope equal to 3 [47, 50](33)log⁡T-Tc=3log⁡S-Sc+constantin this context, the heat capacity behaves like(34)CQ~T-Tc-2/3 where the critical exponent of the second-order phase is -2/3, which is consistent with the mean field theory.


### 2.2. Grand Canonical Ensemble

Having described the thermodynamical behavior of the EMYM-AdS black hole with a fixed charge, by showing the occurrence of the first and second phase transition, we will focus this section on the phase structure when the potentials are kept fixed.To facilitate the calculation of relevant quantities, it is convenient to reexpress the Hawking temperature as a function of entropy, Yang-Mills, and electromagnetic potentials, inserting (23) and (24) into (30); one can write(35)T=13π22n+1/nSωn-1/n×96π2-2n+8/nn-32ΦQ241/nS/ωn1/n-2nS/ωn4/n/n-1-nΦc2n2ωn2-161/nΛn+1Sωn2/n+3n-1In our assumptions, wheren=2 and Λ=-1, (35) reduces to(36)T=S-πΦc2+ΦQ2-14π3/2Swhich is in agreement with the result of [51, 52] if we set ΦQ=0. Now we are able to write easily(37)∂T∂SΦQ,Φc=-S-3πΦc2+ΦQ2-116π3/2S5/2(38)∂2T∂S2ΦQ,Φc=S+πΦc2+ΦQ2-18π3/2S3/2The solution of∂T/∂SΦQ,Φc=0 can be derived as(39)S1=-πΦc2+ΦQ2-1where the condition Φc2+ΦQ2<1 should be verified to ensure that the entropy in (39) is positive. In the other case Φc2+ΦQ2>1, one cannot find meaningful root of the equation ∂T/∂SΦQ,Φc=0. Substituting (39) into (38) we obtain the following constraint(40)∂2T∂S2ΦQ,ΦcS=S1=18π3-Φc2-ΦQ2+13/2>0implying that no critical point is observed in theT-S diagram. This observation differs from the result in the previous section where the charges are kept fixed, consolidating the assertion that the thermodynamics in the grand canonical ensemble is quite different from that in the canonical one. In Figure 3, we depict the Hawking temperature for both the cases Φc2+ΦQ2<1 and Φc2+ΦQ2>1 and one can see that there exists minimum temperature for the left panel. Substituting (39) into (36), the minimum temperature can be obtained as(41)Tmin=-Φc2-ΦQ2+12π.However, the Hawking temperature increases monotonically in the right panel.Figure 3
The temperature as a function of the entropy for different potentials.(Left)Φc2+ΦQ2<1.(Right)Φc2+ΦQ2>1.Having obtained the phase picture of the thermal entropy of the AdS-Maxwell-Yang-Mills black hole, the canonical/grand canonical ensemble, we will now revisit the phase structure of the entanglement entropy and two-point correlation function to see whether they have similar phase structure in each thermodynamical ensemble.


## 2.1. Canonical Ensemble

We start this section by writing theN-dimensional for Einstein-Maxwell-Yang-Mills gravity with a cosmological constant Λ described by the following action [46, 47](1)I=12∫MdxN-gR-N-1N-23Λ-FM-FYM,where R is the Ricci scalar while Λ is the cosmological constant. Also FM=FμνFμν and FYM=TrFμνaFaμν are the Maxwell invariant and the Yang-Mills invariant, respectively; the trace element stands for Tr.=∑a=1(N-1)(N-2)/2.. Varying the action (1) with respect to the metric tensor gμν, the Faraday tensor Fμν, and the YM tensor Fμν(a), one can obtain the following field equations(2)Gμν+Λgμν=TμνM+TμνYM(3)∇Fμν=0(4)D^μFμνa=∇μFaμν+1ΘεbcaAμbFcμν=0where Gμν is the Einstein tensor, the quantity ε(b)(c)(a)’s stands for the structure constants of the N-1N-2/2-parameters Lie group G, Θ is coupling constant, and Aμ(a) denotes the SO(N-1) gauge groupe YM potential. We also note that the internal indices {a,b,c,⋯} do not differ whether in covariant or contravariant form. In addition, TμνM and TμνYM are the energy momentum tensor of Maxwell and YM fields with the following formula(5)TμνM=-12gμνFρσFρσ+2FμλFνλ(6)TμνYM=∑a=1N-1N-2/2-12gμνFρσaFaρσ+2FμλaFνaλ(7)Fμνa=∂μAνa-∂νAμa+12ΘεbcaAμbAνc,(8)Fμν=∂μAν-∂νAμ.where Aμ is the usual Maxwell potential. The metric for such N dimensional spherical black hole may be chosen to be [47](9)ds2=-frdt2+1frdr2+r2dΩn2in which dΩn2 represents the volume of the unit n-sphere which can be expressed in the standard spherical form(10)dΩN-22=dθ12+∑i=2N-3Πj=1i-1sin2⁡θjdθi2where0≤θ1≤π,0≤θi≤2π.In order to find the electromagnetic field, we recall the following radial gauge potential ansatzAμ=h(r)δμ0 which obeys the Maxwell field equations (3) with the following solution(11)dhrdr=crN-2where c is an integration constant related to electric charge C of the solutions. To solve the YM field, (4), we use the magnetic Wu–Yang ansatz of the gauge potential [48, 49] given by(12)Aa=Qr2xidxj-xjdxi2≤i≤N-1,1≤j≤i-11≤a≤N-1N-22where we imply (to have a systematic process) that the super indices a are chosen according to the values of i and j in order. For instance, we present some of them(13)A1=Qr2x2dx1-x1dx2A2=Qr2x3dx1-x1dx3A3=Qr2x3dx2-x2dx3A4=Qr2x4dx1-x1dx4A5=Qr2x4dx2-x2dx4A6=Qr2x4dx3-x3dx4A7=Qr2x5dx1-x1dx5A8=Qr2x5dx2-x2dx5A9=Qr2x5dx3-x3dx5A10=Qr2x5dx4-x4dx5⋯in which r2=∑i=1N-1xi2. The YM field 2-forms are defined by the expression(14)Fa=dAa+12QεbcaAb∧Ac.In general forN we must have N-1N-2/2 gauge potentials. The integrability conditions(15)dFa+1QεbcaAb∧Fc=0are easily satisfied by using (28). The YM equations(16)d∗Fa+1QεbcaAb∧∗Fc=0also are all satisfied. The energy-momentum tensor (4) becomes after(17)∑a=1N-1N-2/2FλσaFaλσ=N-3N-2Q2r4with the nonzero components(18)T00=N-3N-2Q2fr2r4T11=-N-3N-2Q22r4frT22=-N-3N-6Q22r2TAA=-N-3N-6Q22r2Πi=1A-2sin2⁡θi2<A≤N-1.Using (2) and after some simplifications, one can find that the metric function f(r) has the following form given by(19)fr=1-2mrn-1-Λ3r2+2n-1C2nr2n-2-n-1Q2n-3r2,where N=n+2; one can note in the particular case for n=3 that the last term of (19) diverges, involving an unusual logarithmic term in Yang-Mills charge [46]. For this gravity background the parameter m is related to the mass of such black hole, while C and Q are the charges of Maxwell field and Yang-Mills field, respectively. Following previous literature [8, 9], one can find a close connection between the cosmological constant and pressure as P=-Λ/8π, leading to the following expressions of Hawking temperature, mass, and entropy of such black hole in terms of the horizon radius r+(20)T=f′r+4π=n-14πr++2n+1P3r+-n-1Q24πr3-n-12C22πnr+2n-1,(21)M=nωn48π8πPr+n+1+3n-1Q2rn-33-n+3r+n-1+6n-1C2nr+n-1,(22)S=ωnr+n4,The Yang-Mills potentialΦQ and the electromagnetic one ΦC can be written as(23)ΦQ=∂M∂QC,P,r+=ωnn-1nQ8π3-nr+n-3,(24)ΦC=∂M∂CQ,P,r+=ωnn-1C4πr+n-1,where ωn=2π(n+1)/2/Γ(n+1/2) is the volume of the unit n-sphere. In fact, according to the interpretation of the black hole mass M as an enthalpy [8] in the extended phase space context, the free energy F of black hole can be written as(25)F=M-T·Sand the heat capacity is given by(26)CQ=T∂S∂TQIt is straightforward to show that obtained quantities (20), (21), and (22) obey the first law of black hole thermodynamics in the extended phase space(27)dM=TdS+ΦQdQ+ΦCdC+VdP,where V is the Legendre transform of the pressure, which denotes the thermodynamic volume with(28)V=∂M∂PS,ΦQ,ΦC=nωnπ3rn+1. In addition to this, using scaling argument, the corresponding Smarr formula is(29)M=nn-1TS+ΦCC+1n-1ΦQQ-2n-1VP. Without loss of generality, inserting (22) into (20), we can get the entropy Hawking temperature relation of such black hole, namely,(30)T=112πn6Cn-1241/nSωn1/n1-2n+π22/n+3nn+1PSωn1/n-3n-1nQ24-3/nSωn-3/n+34-1/nn-1nSωn-1/nThis relation is depicted in Figure1; indeed it has been shown that there is a Van der Waals-like phase transition; furthermore a direct confirmation comes from the solution of the following system(31)∂T∂SQ=∂2T∂S2Q=0.which reveal the existence of a critical point. The critical charge, entropy, and temperature are given in Table 1; for all the rest of the paper we keep n=2.Table 1
Coordinates of the critical points for different values ofC in (T,S)-diagram.
Q c S c T c C = 0 1 2 3 π 2 2 3 π   C ≠ 0 1 6 3 - 36 C 2 π 2 2 3 πFigure 1
The temperature as a function of the entropy for different values of chargeC.(Left)C=0.(Right)C≠0.An important remark that can be observed here is that both quantitiesTc and Sc are insensible with the charge C. The behavior of the free energy F with respect to the temperature may be investigated by plotting in Figure 2 the graph F-T for a fixed value of charge Q under the critical one. From this plot, we can observe the characteristic swallow-tail which guarantees the existence of the Van der Waals-like phase transition.Figure 2
The Helmholtz free energy in function of the entropy for EMYM-AdS black holes for different values of the chargeC.(Left)C=0.(Right)C≠0.Using Figure2, we can derive numerically the coexistence temperature T⋆ needed in the Maxwell’s equal area law construction(32)A1≡∫S1S2TSdS-T⋆S2-S1=T⋆S3-S2-∫S2S3TSdS≡A2,where S3, S2, and S1 are the solutions of T(S)=T⋆ in descending order, in addition to the numerical values of these points, and we report in Table 2 the areas A1 and A2 in Maxwell’s law (32)Table 2
Check of the equal area law in theT-S plane for different C.
C T ⋆ S 1 S 2 S 3 A 1 A 2 0 0.15376 0.638237 1.52994 3.13241 0.383499 0.38350  14 150965 1.05043 11.54616 2.25932 00.18248 0.1825Obviously, the areaA1 equals area A2 for different C, so the equal area law does not break. For the second phase transition, we know that, near the critical point, there is always a linear relation with slope equal to 3 [47, 50](33)log⁡T-Tc=3log⁡S-Sc+constantin this context, the heat capacity behaves like(34)CQ~T-Tc-2/3 where the critical exponent of the second-order phase is -2/3, which is consistent with the mean field theory.


## 2.2. Grand Canonical Ensemble

Having described the thermodynamical behavior of the EMYM-AdS black hole with a fixed charge, by showing the occurrence of the first and second phase transition, we will focus this section on the phase structure when the potentials are kept fixed.To facilitate the calculation of relevant quantities, it is convenient to reexpress the Hawking temperature as a function of entropy, Yang-Mills, and electromagnetic potentials, inserting (23) and (24) into (30); one can write(35)T=13π22n+1/nSωn-1/n×96π2-2n+8/nn-32ΦQ241/nS/ωn1/n-2nS/ωn4/n/n-1-nΦc2n2ωn2-161/nΛn+1Sωn2/n+3n-1In our assumptions, wheren=2 and Λ=-1, (35) reduces to(36)T=S-πΦc2+ΦQ2-14π3/2Swhich is in agreement with the result of [51, 52] if we set ΦQ=0. Now we are able to write easily(37)∂T∂SΦQ,Φc=-S-3πΦc2+ΦQ2-116π3/2S5/2(38)∂2T∂S2ΦQ,Φc=S+πΦc2+ΦQ2-18π3/2S3/2The solution of∂T/∂SΦQ,Φc=0 can be derived as(39)S1=-πΦc2+ΦQ2-1where the condition Φc2+ΦQ2<1 should be verified to ensure that the entropy in (39) is positive. In the other case Φc2+ΦQ2>1, one cannot find meaningful root of the equation ∂T/∂SΦQ,Φc=0. Substituting (39) into (38) we obtain the following constraint(40)∂2T∂S2ΦQ,ΦcS=S1=18π3-Φc2-ΦQ2+13/2>0implying that no critical point is observed in theT-S diagram. This observation differs from the result in the previous section where the charges are kept fixed, consolidating the assertion that the thermodynamics in the grand canonical ensemble is quite different from that in the canonical one. In Figure 3, we depict the Hawking temperature for both the cases Φc2+ΦQ2<1 and Φc2+ΦQ2>1 and one can see that there exists minimum temperature for the left panel. Substituting (39) into (36), the minimum temperature can be obtained as(41)Tmin=-Φc2-ΦQ2+12π.However, the Hawking temperature increases monotonically in the right panel.Figure 3
The temperature as a function of the entropy for different potentials.(Left)Φc2+ΦQ2<1.(Right)Φc2+ΦQ2>1.Having obtained the phase picture of the thermal entropy of the AdS-Maxwell-Yang-Mills black hole, the canonical/grand canonical ensemble, we will now revisit the phase structure of the entanglement entropy and two-point correlation function to see whether they have similar phase structure in each thermodynamical ensemble.


## 3. Phase Transitions of Einstein-Maxwell-Yang-Mills-AdS Black Holes in Holographic Picture

### 3.1. Holographic Entanglement Entropy

First, let us provide a concise review of some generalities about the holographic entanglement entropy. For a given quantum field theory described by a density matrixρ, with A being some region of a Cauchy surface of spacetime and Ac standing for its complement, the von Neumann entropy traduces the entanglement between these two regions(42)SA=-TrAρAlog⁡ρA, with ρA being the reduced density matrix of given by AρA=TrAc(ρ). Ryu and Takayanagi propose a simple geometric way to evaluate the entanglement entropy as [53, 54](43)SA=AreaΓA4GN,in which ΓA denotes a codimension-2 minimal surface with boundary condition ∂ΓA=∂A and GN stands for the gravitational Newton’s constant. In our black hole model we choose the region A to be a spherical cap on the boundary delimited by θ≤θ0, and the minimal surface can be parametrized by the function r(θ). According to definition of the area and (9) and (43), one can show that the holographic entanglement entropy is governed by(44)SA=ωn-24∫0θ0rn-2sinn-2⁡θr′2fr+r2dθ,where the notation prime denotes the derivative with respect to θ; e.g., r′≡dr/dθ. Treat (44) as a Lagrangian and solve the equation of motion given by(45)r′θ2sin⁡θrθ2f′r-2cos⁡θr′θ-2rθfrrθsin⁡θr′′θ+cos⁡θr′θ-3sin⁡θr′θ2+4sin⁡θrθ3fr2=0.Due to the difficulty to find an analytical form of the solution r(θ), we will perform a numerical calculation with adopting the following boundary conditions(46)r′θ=0,r0=r0.Knowing that the entanglement entropy is divergent at the boundary, we regularized it by subtracting the area of the minimal surface in pure AdS whose boundary is alsoθ=θ0 with(47)rAdSθ=Lcos⁡θcos⁡θ02-1-1/2.We label the regularized entanglement entropy by ΔSA and for our numerical calculation we choose θ0=0.2 and 0.3 while Ultra Violet cutoff is chosen to be θc=0.199 and 0.299, respectively. To compare with the phase structure in the thermal picture, we will study the relation between the entanglement entropy and the Hawking temperature representing the temperature of the dual field theory; this relation is depicted in Figure 4 for different values of Maxwell’s charge with a fixed Yang-Mills one near the critical point and the chosen θ0.Figure 4
Plot of isocharges on the(T,ΔSA)-plan, for different values of θ0,C. For all panels: the values of the charge are Q=.9Qc (red), Q=Qc (dashed black), and Q=2Qc (green). The coexistence phase isotherm T⋆ (dashed magenta line) is obtained from the free energy (Figure 2) and the critical temperature (dashed orange line). For all curves, we also show the data points which are used to create the interpolation.For each panel, the red lines are associated with a charge less than the critical one while the ones equal to the critical charge are depicted in black dashed lines and green lines correspond to a charge upper than the critical one. Particularly, the first-order phase transition temperatureT⋆ and second-order phase transition temperature Tc are plotted by magenta and orange dashed line, respectively. As can be seen in all these plots, the Van der Waals-like phase structure can also be observed in the T-ΔSAdiagram. Particularly, the coexistence temperature T⋆ and second-order phase transition temperature Tc are exactly the same as those in the thermal entropy structure.Adopting the same steps as in the thermal picture, we will also check numerically whether Maxwell’s equal area law holds(48)A1=∫ΔSA1ΔSA2TΔSA,QdΔSA-T∗ΔSA2-ΔSA1(49)A2=T∗ΔSA3-ΔSA2-∫ΔSA2ΔSA3TΔSA,QdΔSAwith the quantitiesΔSA(1), ΔSA(2), and ΔSA(3) being roots of the equation T∗=T(ΔSA,Q) in ascending order. The Maxwell’s equal area law stipulates that(50)A1=A2. We tabulate in Table 3 the values of the both areas A1 and A2 for the chosen θ0 and the charge C as well as the relative error between A1 and A2 taken to be the difference between A1 and A2 divided by their average.Table 3
Comparison ofA1 and A2 for the EMPYM-AdS black hole using entanglement entropy.
C Q Q c θ 0 Δ S A ( 1 ) Δ S A ( 2 ) Δ S A ( 3 ) A 1 A 2 Relative error 0 0.9 0.2 0.00339417 0.00490769 0.00755095 2.271 × 1 0 - 6 2.218 ×10-6 2.36 % 0.3 0.011477 0.0164978 0.0254236 6.563 × 1 0 - 6 6.407 ×10-6 2.4% 0.5 0.2 0.00161291 0.0042458 0.0130777 0.00007432 0.00005560 28.81%  14 0.9 0.2 0.00425941 0.00502275 0.00617356 9.961 × 1 0 - 8 1.023 ×10-7 2.66 % 0.3 0.0151665 0.0169359 0.0208834 1.601 × 1 0 - 7 1.631 ×10-7 1.85% 0.5 0.2 0.00339968 0.00496717 0.00743883 2.12718 × 1 0 - 6 1.76781 ×10-6 18.44%Based on Table3, we can see that, as the pressure approaches the critical one, the relative error which translates the disagreement between Maxwell’s areas decreases. We can claim that the first-order phase transition of the holographic entanglement entropy obeys Maxwell’s equal area law just near the critical point and within our numerical accuracy.The next obvious step in our investigation is the check of the critical exponent of the second-order phase transition by analyzing the slope of the relation betweenlog⁡T-Tc and log⁡ΔSA-ΔSAc, where ΔSAc is the critical entanglement entropy found numerically by an equation T(ΔSA)=Tc. We also introduce the definition of an analog to heat capacity by writing(51)CQ=T∂ΔSA∂TQ.Takingθ0=0.2 and for different charge C, we plot the relationship between log⁡T-Tc and log⁡ΔSA-ΔSAc in Figure 5, and the analytical relation can be fitted as(52)log⁡T-Tc=14.2558+3.09904log⁡ΔSA-ΔSAcC=02.9293+3.031061log⁡ΔSA-ΔSAcC=14Figure 5
The relation betweenlog⁡T-Tc and log⁡ΔSA-ΔSAc for different C.(Left)C=0.(Right)C≠0.The slope of (52) is around 3 indicating that the critical exponent is -2/3 in total concordance with that in (33); therefore the critical exponent for second-order phase transition of the holographic entanglement entropy agrees with that of the thermal entropy in the canonical ensemble [47, 50].Now, we turn our attention to the grand canonical ensemble; we adopt the same analysis and the chosen values of the previous subsection, by writing (19) and (45) as a function of the potentials ΦQ and Φc; taking the same boundary conditions (46), we perform the numerical calculations used in the plot of Hawking temperature as a function of the holographic entanglement entropy with fixed potentials in Figure 6.Figure 6
The relation betweenlog⁡T-Tc and log⁡ΔSA-ΔSAc for different C.(Left)(Φc=0.1,ΦQ=0.2).(Right)(Φc=1,ΦQ=2).Comparing Figure3 with Figure 6, one may find that the thermal picture shares the same behavior of the holographic entanglement entropy; we can also observe the same minimum value of the temperature Tmin=0.155125. Then the holographic framework reproduces the same attitude of the T-S diagram in the grand canonical ensemble.Now, after showing that the holographic entanglement entropy shears the same phase picture as that of the thermal entropy for grand canonical and just near the critical point for the canonical ensemble since the relative disagreement between Maxwell’s areas can become significantly large at low pressure, we attempt in the next section to explore whether the two-point correlation function has similar behavior to that of the entanglement entropy.


### 3.2. Two-Point Correlation Function

According to the Anti-de-Sitter/conformal fields theory correspondence, the time two-point correlation function can be written under the saddle-point assumption and in the large limit ofΔ as [55](53)Ot0,xiOt0,xj≈e-ΔL,where Δ is the conformal dimension of the scalar operator O in the dual field theory and the quantity L stands for the length of the bulk geodesic between the points (t0,xi) and (t0,xj) on the AdS boundary. Taking into account the symmetry of the considered black hole spacetime, we can simply use xi=θ with the boundary θ0 and employ it to parameterize the trajectory. In this case, the proper length can be expressed as (54)L=∫0θ0Lrθ,θdθ,L=r′θ2frθ+rθ2,wherer′=drdθTreating L as Lagrangian and θ as time, one can write the equation of motion for r(θ) as(55)r′θ2f′rθ-2frθr′′θ+2rθfrθ2=0.Recalling the boundary conditions of (46) we attempt to solve this equation by choosing the same background of the previous section, in other words the same values of the parameter θ0 with the same UV cutoff values in the dual field theory. The regularized two-point correlation function is labeled as ΔLA=L-L0, where L0 denotes the geodesic length in pure AdS under the same boundary region. In Figure 7 we depict the behavior of the temperature T in function of ΔLA; all plots show the Van der Waals-like phase transition as in the case of the thermal and the holographic entanglement entropy portrait.Figure 7
Plot of isocharges on the(T,ΔLA)-plan, for C=0 (left) and C≠0 (right). For all panels: the values of the charge are Q=0.9Qc (red), Q=Qc (dashed black), and Q=2Qc (green).As in the case of the holographic entanglement entropy, the relevant calculated results are listed in Table4 which are the θ0 values, the ΔLA1,2,3, and the areas A1 and A2.Table 4
Comparison ofA1 and A2 for the EMYM-AdS black hole using two-point correlation function.
C Q Q c θ 0 Δ L A ( 1 ) Δ L A ( 2 ) Δ L A ( 3 ) A 1 A 2 relative error 0 0.9 0.2 1.34978 1.34987 1.35003 0.000287 0.000300 4.42% 0.3 0.881705 0.881911 0.882274 0.000197 0.000208 5.43% 0.4 0.2 1.34967 1.34983 1.35043 0.001431 0.001812 23.49%  1 4 0.9 0.2 1.34983 1.34988 1.34995 0.000142 0.000138 2.85% 0.3 0.881822 0.881926 0.882085 0.000209 0.000216 3.29% 0.4 0.2 1.34978 1.34987 1.35003 0.0003907 0.0004653 17.42%The results of Table4 tell us that under our numerical accuracy and just near the critical point Maxwell’s equal area law still is verified implying that A1 and A2 are equal in the proximity to the critical pressure. These remarks consolidate the behavior of all panels of Figure 7. At this point, one can conclude that, like the entanglement entropy, the two-point correlation function also exhibits apparently a first-order phase transition as that of the thermal entropy. However, exploring broader ranges of the pressure has revealed the fact that the equal area law does not also hold.For the second phase transition, we will be interested in the quantitieslog⁡T-Tc and log⁡ΔLA-ΔLAc in which ΔLAc is obtained numerically by the equation T(ΔLA)=Tc. The relations between the logarithms of T-Tc and ΔLA-ΔLAc are shown in Figure 8Figure 8
The relation betweenlog⁡T-Tc and log⁡ΔLA-ΔLAc for different charge C.The straight blue line in each panel of Figure8 is fitted following the linear equations(56)log⁡T-Tc=19.056+3.03594log⁡ΔLA-ΔLAcC=014.25487+3.04967log⁡ΔLA-ΔLAcC=14Again, we found a slope around 3; then the critical exponent of the specific heat capacity is consistent with that of the mean field theory of the Van der Waals as in the thermal and entanglement entropy portraits [47, 50]. Therefore, we conclude that the two-point correlation function of the Anti-de-Sitter-Maxwell-Yang-Mills black hole exists in a second-order phase transition at the critical temperature Tc.For the grand canonical ensemble, we also plot the temperatureT in function of ΔLA in Figure 9, from which we can see that thermodynamical behavior is held as the thermal and the holographic entanglement entropy frameworks.Figure 9
The relation betweenlog⁡T-Tc and log⁡ΔSA-ΔSAc for different C.(Left)(Φc=0.1,ΦQ=0.2).(Right)(Φc=1,ΦQ=2).At this level we remark that radical rupture appears when we change the thermodynamical ensemble (canonical/grand canonical). The complete comprehension of such different behaviors is not yet completely understood. We believe that it is typical of such system [41].


## 3.1. Holographic Entanglement Entropy

First, let us provide a concise review of some generalities about the holographic entanglement entropy. For a given quantum field theory described by a density matrixρ, with A being some region of a Cauchy surface of spacetime and Ac standing for its complement, the von Neumann entropy traduces the entanglement between these two regions(42)SA=-TrAρAlog⁡ρA, with ρA being the reduced density matrix of given by AρA=TrAc(ρ). Ryu and Takayanagi propose a simple geometric way to evaluate the entanglement entropy as [53, 54](43)SA=AreaΓA4GN,in which ΓA denotes a codimension-2 minimal surface with boundary condition ∂ΓA=∂A and GN stands for the gravitational Newton’s constant. In our black hole model we choose the region A to be a spherical cap on the boundary delimited by θ≤θ0, and the minimal surface can be parametrized by the function r(θ). According to definition of the area and (9) and (43), one can show that the holographic entanglement entropy is governed by(44)SA=ωn-24∫0θ0rn-2sinn-2⁡θr′2fr+r2dθ,where the notation prime denotes the derivative with respect to θ; e.g., r′≡dr/dθ. Treat (44) as a Lagrangian and solve the equation of motion given by(45)r′θ2sin⁡θrθ2f′r-2cos⁡θr′θ-2rθfrrθsin⁡θr′′θ+cos⁡θr′θ-3sin⁡θr′θ2+4sin⁡θrθ3fr2=0.Due to the difficulty to find an analytical form of the solution r(θ), we will perform a numerical calculation with adopting the following boundary conditions(46)r′θ=0,r0=r0.Knowing that the entanglement entropy is divergent at the boundary, we regularized it by subtracting the area of the minimal surface in pure AdS whose boundary is alsoθ=θ0 with(47)rAdSθ=Lcos⁡θcos⁡θ02-1-1/2.We label the regularized entanglement entropy by ΔSA and for our numerical calculation we choose θ0=0.2 and 0.3 while Ultra Violet cutoff is chosen to be θc=0.199 and 0.299, respectively. To compare with the phase structure in the thermal picture, we will study the relation between the entanglement entropy and the Hawking temperature representing the temperature of the dual field theory; this relation is depicted in Figure 4 for different values of Maxwell’s charge with a fixed Yang-Mills one near the critical point and the chosen θ0.Figure 4
Plot of isocharges on the(T,ΔSA)-plan, for different values of θ0,C. For all panels: the values of the charge are Q=.9Qc (red), Q=Qc (dashed black), and Q=2Qc (green). The coexistence phase isotherm T⋆ (dashed magenta line) is obtained from the free energy (Figure 2) and the critical temperature (dashed orange line). For all curves, we also show the data points which are used to create the interpolation.For each panel, the red lines are associated with a charge less than the critical one while the ones equal to the critical charge are depicted in black dashed lines and green lines correspond to a charge upper than the critical one. Particularly, the first-order phase transition temperatureT⋆ and second-order phase transition temperature Tc are plotted by magenta and orange dashed line, respectively. As can be seen in all these plots, the Van der Waals-like phase structure can also be observed in the T-ΔSAdiagram. Particularly, the coexistence temperature T⋆ and second-order phase transition temperature Tc are exactly the same as those in the thermal entropy structure.Adopting the same steps as in the thermal picture, we will also check numerically whether Maxwell’s equal area law holds(48)A1=∫ΔSA1ΔSA2TΔSA,QdΔSA-T∗ΔSA2-ΔSA1(49)A2=T∗ΔSA3-ΔSA2-∫ΔSA2ΔSA3TΔSA,QdΔSAwith the quantitiesΔSA(1), ΔSA(2), and ΔSA(3) being roots of the equation T∗=T(ΔSA,Q) in ascending order. The Maxwell’s equal area law stipulates that(50)A1=A2. We tabulate in Table 3 the values of the both areas A1 and A2 for the chosen θ0 and the charge C as well as the relative error between A1 and A2 taken to be the difference between A1 and A2 divided by their average.Table 3
Comparison ofA1 and A2 for the EMPYM-AdS black hole using entanglement entropy.
C Q Q c θ 0 Δ S A ( 1 ) Δ S A ( 2 ) Δ S A ( 3 ) A 1 A 2 Relative error 0 0.9 0.2 0.00339417 0.00490769 0.00755095 2.271 × 1 0 - 6 2.218 ×10-6 2.36 % 0.3 0.011477 0.0164978 0.0254236 6.563 × 1 0 - 6 6.407 ×10-6 2.4% 0.5 0.2 0.00161291 0.0042458 0.0130777 0.00007432 0.00005560 28.81%  14 0.9 0.2 0.00425941 0.00502275 0.00617356 9.961 × 1 0 - 8 1.023 ×10-7 2.66 % 0.3 0.0151665 0.0169359 0.0208834 1.601 × 1 0 - 7 1.631 ×10-7 1.85% 0.5 0.2 0.00339968 0.00496717 0.00743883 2.12718 × 1 0 - 6 1.76781 ×10-6 18.44%Based on Table3, we can see that, as the pressure approaches the critical one, the relative error which translates the disagreement between Maxwell’s areas decreases. We can claim that the first-order phase transition of the holographic entanglement entropy obeys Maxwell’s equal area law just near the critical point and within our numerical accuracy.The next obvious step in our investigation is the check of the critical exponent of the second-order phase transition by analyzing the slope of the relation betweenlog⁡T-Tc and log⁡ΔSA-ΔSAc, where ΔSAc is the critical entanglement entropy found numerically by an equation T(ΔSA)=Tc. We also introduce the definition of an analog to heat capacity by writing(51)CQ=T∂ΔSA∂TQ.Takingθ0=0.2 and for different charge C, we plot the relationship between log⁡T-Tc and log⁡ΔSA-ΔSAc in Figure 5, and the analytical relation can be fitted as(52)log⁡T-Tc=14.2558+3.09904log⁡ΔSA-ΔSAcC=02.9293+3.031061log⁡ΔSA-ΔSAcC=14Figure 5
The relation betweenlog⁡T-Tc and log⁡ΔSA-ΔSAc for different C.(Left)C=0.(Right)C≠0.The slope of (52) is around 3 indicating that the critical exponent is -2/3 in total concordance with that in (33); therefore the critical exponent for second-order phase transition of the holographic entanglement entropy agrees with that of the thermal entropy in the canonical ensemble [47, 50].Now, we turn our attention to the grand canonical ensemble; we adopt the same analysis and the chosen values of the previous subsection, by writing (19) and (45) as a function of the potentials ΦQ and Φc; taking the same boundary conditions (46), we perform the numerical calculations used in the plot of Hawking temperature as a function of the holographic entanglement entropy with fixed potentials in Figure 6.Figure 6
The relation betweenlog⁡T-Tc and log⁡ΔSA-ΔSAc for different C.(Left)(Φc=0.1,ΦQ=0.2).(Right)(Φc=1,ΦQ=2).Comparing Figure3 with Figure 6, one may find that the thermal picture shares the same behavior of the holographic entanglement entropy; we can also observe the same minimum value of the temperature Tmin=0.155125. Then the holographic framework reproduces the same attitude of the T-S diagram in the grand canonical ensemble.Now, after showing that the holographic entanglement entropy shears the same phase picture as that of the thermal entropy for grand canonical and just near the critical point for the canonical ensemble since the relative disagreement between Maxwell’s areas can become significantly large at low pressure, we attempt in the next section to explore whether the two-point correlation function has similar behavior to that of the entanglement entropy.


## 3.2. Two-Point Correlation Function

According to the Anti-de-Sitter/conformal fields theory correspondence, the time two-point correlation function can be written under the saddle-point assumption and in the large limit ofΔ as [55](53)Ot0,xiOt0,xj≈e-ΔL,where Δ is the conformal dimension of the scalar operator O in the dual field theory and the quantity L stands for the length of the bulk geodesic between the points (t0,xi) and (t0,xj) on the AdS boundary. Taking into account the symmetry of the considered black hole spacetime, we can simply use xi=θ with the boundary θ0 and employ it to parameterize the trajectory. In this case, the proper length can be expressed as (54)L=∫0θ0Lrθ,θdθ,L=r′θ2frθ+rθ2,wherer′=drdθTreating L as Lagrangian and θ as time, one can write the equation of motion for r(θ) as(55)r′θ2f′rθ-2frθr′′θ+2rθfrθ2=0.Recalling the boundary conditions of (46) we attempt to solve this equation by choosing the same background of the previous section, in other words the same values of the parameter θ0 with the same UV cutoff values in the dual field theory. The regularized two-point correlation function is labeled as ΔLA=L-L0, where L0 denotes the geodesic length in pure AdS under the same boundary region. In Figure 7 we depict the behavior of the temperature T in function of ΔLA; all plots show the Van der Waals-like phase transition as in the case of the thermal and the holographic entanglement entropy portrait.Figure 7
Plot of isocharges on the(T,ΔLA)-plan, for C=0 (left) and C≠0 (right). For all panels: the values of the charge are Q=0.9Qc (red), Q=Qc (dashed black), and Q=2Qc (green).As in the case of the holographic entanglement entropy, the relevant calculated results are listed in Table4 which are the θ0 values, the ΔLA1,2,3, and the areas A1 and A2.Table 4
Comparison ofA1 and A2 for the EMYM-AdS black hole using two-point correlation function.
C Q Q c θ 0 Δ L A ( 1 ) Δ L A ( 2 ) Δ L A ( 3 ) A 1 A 2 relative error 0 0.9 0.2 1.34978 1.34987 1.35003 0.000287 0.000300 4.42% 0.3 0.881705 0.881911 0.882274 0.000197 0.000208 5.43% 0.4 0.2 1.34967 1.34983 1.35043 0.001431 0.001812 23.49%  1 4 0.9 0.2 1.34983 1.34988 1.34995 0.000142 0.000138 2.85% 0.3 0.881822 0.881926 0.882085 0.000209 0.000216 3.29% 0.4 0.2 1.34978 1.34987 1.35003 0.0003907 0.0004653 17.42%The results of Table4 tell us that under our numerical accuracy and just near the critical point Maxwell’s equal area law still is verified implying that A1 and A2 are equal in the proximity to the critical pressure. These remarks consolidate the behavior of all panels of Figure 7. At this point, one can conclude that, like the entanglement entropy, the two-point correlation function also exhibits apparently a first-order phase transition as that of the thermal entropy. However, exploring broader ranges of the pressure has revealed the fact that the equal area law does not also hold.For the second phase transition, we will be interested in the quantitieslog⁡T-Tc and log⁡ΔLA-ΔLAc in which ΔLAc is obtained numerically by the equation T(ΔLA)=Tc. The relations between the logarithms of T-Tc and ΔLA-ΔLAc are shown in Figure 8Figure 8
The relation betweenlog⁡T-Tc and log⁡ΔLA-ΔLAc for different charge C.The straight blue line in each panel of Figure8 is fitted following the linear equations(56)log⁡T-Tc=19.056+3.03594log⁡ΔLA-ΔLAcC=014.25487+3.04967log⁡ΔLA-ΔLAcC=14Again, we found a slope around 3; then the critical exponent of the specific heat capacity is consistent with that of the mean field theory of the Van der Waals as in the thermal and entanglement entropy portraits [47, 50]. Therefore, we conclude that the two-point correlation function of the Anti-de-Sitter-Maxwell-Yang-Mills black hole exists in a second-order phase transition at the critical temperature Tc.For the grand canonical ensemble, we also plot the temperatureT in function of ΔLA in Figure 9, from which we can see that thermodynamical behavior is held as the thermal and the holographic entanglement entropy frameworks.Figure 9
The relation betweenlog⁡T-Tc and log⁡ΔSA-ΔSAc for different C.(Left)(Φc=0.1,ΦQ=0.2).(Right)(Φc=1,ΦQ=2).At this level we remark that radical rupture appears when we change the thermodynamical ensemble (canonical/grand canonical). The complete comprehension of such different behaviors is not yet completely understood. We believe that it is typical of such system [41].


## 4. Conclusion

In this work We have investigated the phase transition of Anti-de-Sitter black hole in the Einstein-Maxwell-Yang-Mills gravity considering the canonical and the grand canonical ensemble. We first studied the phase structure of the thermal entropy in the(T,S)-plane for fixed charges and found that the phase structure agrees with the study made in [47] when the electrodynamics is linear. The authors consider the thermodynamics of such black hole in the (P,V)-plane, notably the critical behavior and the analogy with the Van der Waals gas. We also have shown that this behavior disappears in the grand canonical ensemble where the potentials ΦQ and Φc are kept fixed.Then, we found that this phase structure of the EMYM-AdS black hole can be probed by the two-point correlation function and holographic entanglement entropy in each thermodynamical ensemble, which reproduce the same thermodynamical behavior of the thermal portrait just for a range of the pressure near the critical one where the equal area law holds within our numerical accuracy; for broader ranges the disagreement between Maxwell’s areas becomes significant. These remarks remain open questions while this approach provides a new step in our understanding of the black hole phase structure from the point of view of holography. Considering the high dimensional solutions or additional hairs by adding Yang-Mills fields and taking into account their confinement can be the object of a future publication.



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*Source: 6870793-2019-01-20.xml*