Van der Waals-Like Phase Transition from Holographic ...2.2. Van der Waals-Like Phase Transition of...

Research ArticleVan der Waals-Like Phase Transition from HolographicEntanglement Entropy in Lorentz Breaking Massive Gravity

Xian-Ming Liu1 Hong-Bo Shao2 and Xiao-Xiong Zeng3

1School of Science Hubei University for Nationalities Enshi 445000 China2College of Science Agricultural University of Hebei Baoding 071000 China3School of Material Science and Engineering Chongqing Jiaotong University Chongqing 400074 China

Correspondence should be addressed to Xiao-Xiong Zeng xxzengphysics163com

Received 12 June 2017 Revised 15 August 2017 Accepted 24 August 2017 Published 2 October 2017

Academic Editor Li Li

Copyright copy 2017 Xian-Ming Liu et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

Phase transition of AdS black holes in Lorentz breaking massive gravity has been studied in the framework of holography We findthat there is a first-order phase transition (FPT) and second-order phase transition (SPT) both in Bekenstein-Hawking entropy-(BHE-) temperature plane and in holographic entanglement entropy- (HEE-) temperature plane Furthermore for the FPT theequal area law is checked and for the SPT the critical exponent of the heat capacity is also computed Our results confirm that thephase structure of HEE is similar to that of BHE in Lorentz breaking massive gravity which implies that HEE and BHE have somepotential underlying relationship

1 Introduction

The study of HEE and quantum phase transitions of blackholes has attracted a lot of interest in recent years On onehand HEE can be used as a perfect probe to study quantuminformation science [1ndash3] strongly correlated quantum sys-tems [4ndash13] and Many-Body Systems [14 15] On the otherhand investigation on HEE of black holes may shed somelight on understanding the nature of BHE [16 17]

Nearly ten years ago a holographic derivation of the HEEin conformal quantum field theories was proposed by Ryuand Takayanagi using the famous AdSCFT correspondence[18 19] Recently the HEE has been used as a probe toinvestigate the phase structure of the Reissner-NordstromAdS black hole [20] The results showed that there is aVan der Waals-like (VDW) phase transition at the samecritical temperature in both the fixed charge ensemble andchemical potential ensemble in the HEE-temperature planeThey also found that the SPT occurs for the HEE at thesame critical point as the BHE with nearly the same criticalexponent This work was soon generalized to the extendedphase space where the cosmological constant is considered

as a thermodynamical variable [21] Very recently the equalarea law of HEE was proved to hold for the FPT in the HEE-temperature plane [22] Based on [20] VDWphase transitionof HEE in various AdS black holes has been studied in [23ndash32] All of these works showed that the HEE undergoes thesame VDW phase transition as that of the BHE

Massive gravity theories have attracted considerableinterest recently One of these reasons is that these alternativetheories of gravity could explain the accelerated expansionof the universe without dark energy The graviton behaveslike a lattice excitation and exhibits a Drude peak in thistheory Current experimental data from the observation ofgravitational waves by advanced LIGO require the gravitonmass to be smaller than the inverse period of orbital motionof the binary system that is 119898119892 lt 12 times 10minus22 eVc2[33] Another important reason for the interest in massivegravity is that the possibility of the mass graviton couldhelp to understand the quantum gravity effect The firstto introduce a mass to the graviton is in [34] Howeverthis primitive linear massive gravity theory contains the so-called Boulware-Deser ghosts problem [35] that was solvedby a nonlinear massive gravity theory [36 37] where the

HindawiAdvances in High Energy PhysicsVolume 2017 Article ID 6402101 8 pageshttpsdoiorg10115520176402101

2 Advances in High Energy Physics

mass terms are obtained by introducing a reference metricRecently Vegh proposed a new reference metric to describea class of strongly interacting quantum field theories withbroken translational symmetry in the holographic framework[38] The recent progress in massive gravity can be found in[39 40]

Here we consider AdS black holes in Lorentz breakingmassive gravity In themassive gravity the graviton acquires amass by Lorentz symmetry breaking which is very similar tothe Higgs mechanism A review of Lorentz-violating massivegravity theory can be found in [41 42] In this paper we focuson the study of the VDW phase transition of AdS black holesin Lorentz breakingmassive gravity using the HEEThemainmotivation of this paper is to explore whether the BHE phasetransition can also be described by HEE in Lorentz breakingmassive gravity Firstly we would like to extend proposalsin [20] to study VDW phase transitions in AdS black holewith a spherical horizon in Lorentz-violating massive gravitywith the HEE as a probe What is more we also would liketo checkMaxwellrsquos equal area law and critical exponent of theheat capacity which are two characteristic quantities in VDWphase transition

The organization of this paper is as follows In thenext section we shall provide a brief review of the blackhole solution in Lorentz breaking massive gravity firstlyThen we will study the VDW phase transitions and criticalphenomena for the AdS black hole in the BHE-temperatureplane In Section 3 wemainly concentrate on theVDWphasetransition and critical phenomena in the framework of HEEThe last section is devoted to our discussions and conclusions

2 Phase Transition and CriticalPhenomena of AdS Black Holes in LorentzBreaking Massive Gravity

21 Review of AdS Black Holes in Lorentz Breaking MassiveGravity The four-dimensional Lorentz breaking massivegravity can be obtained by adding nonderivative couplingscalar fields to the standard Einstein gravity theory As amatter field is considered the theory can be described by thefollowing action [41 42]

119878 = int1198894119909radicminus119892 [minus1198722Pl119877 + 119871119898 + ℓ4Ψ(119883Π119894119895)] (1)

here the first two terms are the curvature and ordinarymatterminimally coupled to gravity respectively and the third termΨ contains two functions 119883 and Π119894119895 which relate to the fourscalar fields Ξ0 and Ξ119894 as

119883 = 120597120583Ξ0120597120583Ξ0ℓ4 Π119894119895 = 120597120583Ξ119894120597120583Ξ119895ℓ4 minus 120597120583Ξ119894120597120583Ξ0120597]Ξ119895120597]Ξ0ℓ8119883

(2)

When the four scalar fields get a space-time dependingvacuum expectation value the system will break the Lorentzsymmetry What is more the action can also be taken as a

low-energy effective theory with the ultraviolet cutoff scale ℓHere the scale parameter ℓ has the dimension of mass and isin the order of radic119898119892119872Pl where 119898119892 and 119872Pl are the gravitonmass and the Plank mass respectively

The AdS black hole solutions can be obtained from theabove theory [43 44] The metric corresponding to the AdSblack holes is given by

1198891199042 = minus119891 (119903) 1198891199052 + 119891 (119903)minus1 1198891199032 + 1199032 (1198891205932 + sin21205931198891206012) (3)

with

119891 (119903) = 1 minus 2119872119903 minus 1205741198762119903120582 minus Λ11990323 (4)

Here the four scalar fields Ξ0 and Ξ119894 for this particularsolution are given by

Ξ0 = ℓ2 (119905 + 120578 (119903)) Ξ119894 = ℓ2120572119909119894 (5)

in which

120578 (119903) = plusmnint 119889119903119891 (119903)sdot [[1 minus 119891 (119903)(1205741198762120582 (120582 minus 1)1211989821198921205726 1119903120582+1 + 1)minus1]]

12

(6)

in which the scalar charge 119876 is related to massive gravityand the constant 120572 which is determined by the cosmologicalconstant Λ and the graviton mass 119898119892 with the relation Λ =21198982119892(1 minus 1205726) In this paper we will set 120572 gt 1 such thatΛ lt 0 leading to Anti-de Sitter black holes The constant 120582 isa positive integration constant When 120582 lt 1 the ADM massof the black hole solution diverges For 120582 gt 1 the metricapproaches the Schwarzschild-AdS black holes with a finitemass 119872 as 119903 rarr infin Thus we set 120582 gt 1 in this paper Theconstant 120574 = plusmn1 When 120574 = 1 the black hole only has asingle horizon 119903ℎ which is the root of the equation 119891(119903ℎ) = 0The function 119891(119903) for this case is given in Figure 1 whichis similar to the Schwarzschild-AdS black hole For 120574 = minus1the black hole is very similar to the Reissner-Nordstrom-AdSblack holeThe function119891(119903) for this case is given in Figure 2The black hole event horizon 119903ℎ is the largest root of theequation 119891(119903ℎ) = 0

At the event horizon the Hawking temperature and BHEcan be written as

119879 = 141205871198911015840 (119903ℎ) = 14120587 ( 1119903ℎ minus 119903ℎΛ + 120574 (120582 minus 1)1198762119903120582+1ℎ

) (7)

119878 = 1205871199032ℎ (8)

The chemical potential in this black hole is

Φ = minus120574 119876119903120582minus1ℎ

(9)

Advances in High Energy Physics 3

10 15 20 25 3005

r

minus20

minus15

minus10

minus5

5

10

f(r)

Q = 05

Q = 0

Q = 18

Q = 12

Figure 1 The figure shows 119891(119903) versus 119903 for 120574 = 1 for varying 119876Here 120582 = 24 Λ = minus3 and 119872 = 15

Q = 18

Q = 12

Q = 05

Q = 0minus4

minus2

2

4

f(r)

10 15 20 2505r

Figure 2 The figure shows 119891(119903) versus 119903 for 120574 = minus1 for varying 119876Here 120582 = 24 Λ = minus3 and 119872 = 15We can check the first law of the black hole which is given by

119889119872 = 119879119889119878 + Φ119889119876 (10)

There have been some works to study the thermodynamicsand phase transitions of black holes in Lorentz breakingmassive gravity [45ndash48]

22 Van der Waals-Like Phase Transition of Bekenstein-Hawking Entropy In this subsection we focus on the VDWphase transition of BHE Substituting (8) into (7) and elimi-nating the parameter 119903ℎ one can get the relation between theHawking temperature 119879 and BHE 119878 of the AdS black holes inmassive gravity as

119879= 119878(12)(minus120582minus1) (1205741205871205822+1 (120582 minus 1)119876 minus Λ1198781205822+1 + 1205871198781205822)412058732 (11)

This is the state equation of the AdS black hole thermody-namics system in massive gravity Using (11) we investigatethe phase diagram of the AdS black holes in massive gravity

035

040

045

050

055

T

2 4 6 80S

TＧＣＨ

Figure 3 The figure shows 119879 versus 119878 for 120574 = 1 Here 120582 = 24Λ = minus3 and 119876 = 03

05 10 15 2000S

Q = 0

Q = 06Qc

Q = 12Qc

024

025

026T

027

028

029

Q = Qc

Figure 4The figure shows119879 versus 119878 for 120574 = minus1 for varying119876 Here120582 = 24 Λ = minus3 The top dashed curve is at 119876 = 0 and the rest have119876 = 06119876119888 119876119888 12119876119888 where 119876119888 = 012346

The temperature119879 is plotted as a function of the BHE 119878 inFigures 3 and 4 for 120574 = 1 and 120574 = minus1 respectively In Figure 3the temperature is plotted for 120574 = 1 where only one eventhorizon existsThis behavior of temperature is very similar tothe behavior in the Schwarzschild-AdS black hole That is tosay there is a minimum temperature 119879min which divides thethermodynamics systems into small and large black holes It isshown that above theminimum temperature small and largeblack holes coexist In fact this behavior will break for there isa first-order transition which is similar to the Hawking-Pagethermodynamic transition in [49]

In Figure 4 the temperature is plotted for 120574 = minus1 whereevent horizon and inner horizon exist Various values of 119876are used to plot the relations between the temperature andhorizons The top curve corresponds to 119876 = 0 The systemof this case is similar to the case 120574 = 1 described aboveWhen the scalar charge 119876 increases the temperature hastwo turning points Further increasing of the scalar charge119876 makes these two turning points merge to one It is shownthat there exists a critical point 119876119888 Above the critical pointthe curve does not have any turning points Thus we findthat the phase structure is very similar to that of the Van derWaals gas-fluid phase transition It should be noted that we


0250

0255

0260

0265

25 3020

040

045

050

055

060

20 25 3015

005

010

015

020

25 3020

Tc S c

Qc

Figure 5 The figure shows the critical temperature 119879119888 the critical entropy 119878119888 and the critical charge 119876119888 versus 120582 for 120574 = minus1 Here Λ = minus3

minus30 minus20 minus15minus25

Λ

040

045

050

055

060

minus20 minus15 minus10 minus05minus25

Λ

05

10

15

20

005

010

015

020

025

minus30 minus20 minus15 minus10 minus05minus25

Λ

Tc S c Qc

Figure 6 The figure shows the critical temperature 119879119888 the critical entropy 119878119888 and the critical charge 119876119888 versus Λ for 120574 = minus1 Here 120582 = 12mainly concentrate on this type of phase structure in thispaper Furthermore using the definition of the specific heatcapacity 119862119876 that is

119862119876 = 119879( 120597119878120597119879)119876

(12)

one can see that the specific heat capacity is divergent at thecritical point and it is obvious that this phase transition is aSPT At this critical point the critical charge 119876119888 and criticalentropy 119878119888 can be obtained by the following equations

(120597119879120597119878 )119876119888 119878119888

= (12059721198791205971198782 )119876119888 119878119888

= 0 (13)

After some calculation and using (11) 119876119888 119878119888 and thecorresponding 119879119888 can be also got as

119876119888 = minus2 (minus120582 (120582 + 2) Λ)1205822120574 (120582 + 2) (1205822 minus 1) (14)

119878119888 = minus 120587120582(120582 + 2)Λ (15)

119879119888 = 1205822120587 (120582 + 1)radicminus120582 (120582 + 2) Λ (16)

Obviously these critical parameters depend only on theinternal parameters of the systems 120582 and Λ One can alsosee that 119879119888 and 119878119888 increase with 120582 and 119876119888 decreases with 120582 asshown in Figure 5 119878119888 and119876119888 increase withΛ and119879119888 decreaseswith 120582 as shown in Figure 6 So the results show that theparameter 120582 promotes the thermodynamic system to reachthe stable state

027 028 029026T

0085

0090

0095

0100

0105

0110

F

(0275 01042)

Figure 7 The figure shows 119865 versus 119879 for 120574 = minus1 Here 120582 = 24Λ = minus3 and 119876 = 06119876119888

For the FPT we will also check whether Maxwellrsquos equalarea law holds in this thermodynamic system As is known toall the first-order transition temperature 119879lowast plays a crucialrole for Maxwellrsquos equal area law Thus in order to get 119879lowastwe first plot the curve about the free energy with respectto the temperature 119879 where the free energy is defined by119865 = 119872 minus 119879119878 The relation between 119865 with 119879 is plotted inFigure 7 One can see that there is a swallowtail structurewhich corresponds to the unstable phase in Figure 8 Thenonsmoothness of the junction implies that the phase transi-tion is a FPT The critical temperature 119879lowast is apparently givenby the horizontal coordinate of the junction From Figure 7we get 119879lowast = 02750 Substituting this temperature 119879lowast into(11) one can obtain three values of the entropy 1198781 = 01563


139405277

01563

025

026

027

028

029

T

05 10 15 2000S

S1 =

S2 =S3 =

Figure 8 The figure shows 119879 versus 119878 for 120574 = minus1 Here 120582 = 24Λ = minus3 and 119876 = 061198761198881198782 = 05277 and 1198783 = 1394 With these values we can nowcheck Maxwellrsquos equal area law

119879lowast (1198783 minus 1198781) = int11987831198781

119879 (119878 119876) 119889119878 (17)

After some calculation we find that both the left and rightsides of (17) are equal to 03404 exactlyThus our results showthat Maxwellrsquos equal area law is satisfied in this background

For the SPT we will study the critical exponent associatedwith the heat capacity definition in (12) Near the criticalpoint expanding the temperature as the very small amount119878 minus 119878119888 we find

119879 = 119879119888 + (120597119879120597119878 )119876119888119878119888

(119878 minus 119878119888) + (12059721198791205971198782 )119876119888119878119888

(119878 minus 119878119888)2+ (12059731198791205971198783 )

119876119888 119878119888

(119878 minus 119878119888)3 + 119900 (119878 minus 119878119888)4 (18)

Using (13) the second and third terms vanishThenusing (11)(14) and (15) we get

119879 minus 119879119888 = 120582161205874 (minus120582 (120582 + 2) Λ)72 (119878 minus 119878119888)3 (19)

With the definition of the heat capacity (12) we further get119862119876 sim (119879 minus 119879119888)minus23 So one can find that the critical exponentof the heat capacity is minus23 which is the same as the one fromthe mean field theory in Van der Waals gas-fluid system

3 Van der Waals-Like Phase Transitionand Critical Phenomena of HolographicEntanglement Entropy

In this section our target is to explore whether the HEE hasthe similar VDW phase structure and critical phenomenaas those of the BHE in massive gravity For simplicityhere we only consider the case 120574 = minus1 Now we willinvestigate whether there is VDW phase transition in theHEE-temperature phase plane

Firstly we review some basic knowledge about HEE Fordetailed introduction of HEE one can refer to [18 19] For agiven quantum field theory described by a density matrix 120588HEE for a region 119860 and its complement 119861 are

119878119860 = minus119879119903119860 (120588119860 ln 120588119860) (20)

where 120588119860 is the reduced density matrix However it isusually not easy to get this quantity in quantum field theoryFortunately according to AdSCFT correspondence [18 19]propose a very simple geometric formula for calculating 119878119860for static states with the area of a bulk minimal surface as 120597119860that is

119878119860 = Area (120574119860)4 (21)

where 120574119860 is the codimension-2 minimal surface according toboundary condition 120597120574119860 = 120597119860

Subsequently using definition (21) we will calculate theHEE and study the corresponding phase transition It is notedthat the space on the boundary is spherical in the AdS blackhole in massive gravity and the volume of the space is finiteThus in order to avoid the HEE to be affected by the surfacethat wraps the event horizon we will choose a small region as119860 More precisely as done in [23 25ndash27] we choose region119860to be a spherical cap on the boundary given by 120593 le 1205930 Herethe area can be written as

119860 = 2120587int12059300

Θ(119903 (120593) 120593) 119889120593Θ = 119903 sin120593radic (1199031015840)2119891 (119903) + 1199032

(22)

where 1199031015840 = 119889119903119889120593 Then according to the Euler-Lagrangeequation one can get the equation of motion of 119903(120593) that is

0 = 1199031015840 (120593)2 [sin120593119903 (120593)2 1198911015840 (119903) minus 2 cos1205931199031015840 (120593)]minus 2119903 (120593) 119891 (119903) [119903 (120593) (sin12059311990310158401015840 (120593) + cos1205931199031015840 (120593))minus 3 sin1205931199031015840 (120593)2] + 4 sin120593119903 (120593)3 119891 (119903)2

(23)

After using the boundary conditions 1199031015840(0) = 0 119903(0) = 1199030 wecan get the numeric result of 119903(120593)

It is worth noting that the HEE should be regularized bysubtracting off the HEE in pure AdS because the values ofthe HEE in (21) are divergent at the boundary Now let uslabel the regularized HEE as 120575119878 Here we choose the size ofthe boundary region to be 1205930 = 010 016 and set the UVcutoff in the dual field theory to be 119903(0099) and 119903(0159)respectively The numeric results are shown in Figures 9 and10 One can see that for a given scalar charge 119876 the relationbetween the HEE and temperature is similar to that betweenthe BHE and temperature That is to say the AdS black holesthermodynamic systemwith theHEE undergoes the FPT andSPT one after another as the scalar charge119876 increases step bystep


025026027028029030031

T＜Ｂ

80618

80619

80620

80621

80622

80623

80617

S minus S＞３

(a)

80618

80617

80620

80621

80622

80623

80619

S minus S＞３

022024026028030032

T＜Ｂ

(b)

80617

80618

80619

80620

80621

80622

80623

S minus S＞３

018020022024026028030032

T＜Ｂ

(c)

Figure 9 The figure shows 119879 versus 120575119878 for 120574 = minus1 1205930 = 010 Here 120582 = 24 119876 = 06119876119888 119876119888 12119876119888 from (a) to (c)

025026027028029030031

70755

70760

70765

70770

70775

70750

S minus S＞３

T＜Ｂ

(a)

022

024

026

028

030

032

70755

70760

70765

70770

70775

70750

S minus S＞３

T＜Ｂ

(b)

018020022024026028030032

70770

70775

70765

70755

70760

70750

S minus S＞３

T＜Ｂ

(c)


For the FPT of theHEE we now check whetherMaxwellrsquosequal area law is satisfied Firstly we get the interpolatingfunction of the temperature 119879(120575119878) using the data obtainednumerically At the first-order phase transition point we getthe smallest and largest roots for the equation 119879(120575119878) = 119879lowastwhich are 1205751198781 = 806165 1205751198783 = 806191 for 1205930 = 010 and1205751198781 = 707492 1205751198783 = 707599 for 1205930 = 016 Then using thesevalues and the equal area law

119879lowast (1205751198783 minus 1205751198781) = int12057511987831205751198781

119879 (120575119878) 119889119878 (24)

we find the left side equals 000007137 000029425 and theright side equals 000007135 000029446 for 1205930 = 010 016respectively Obviously both the left and the right sides areequal within our numerical accuracy and the relative errorsare less than 0022 0071 for 1205930 = 010 016 respectively

Now let us consider the critical exponent of the SPT inthe HEE-temperature phase plane Here comparing with thedefinition of specific heat capacity 119862119876 in (12) one can alsodefine a specific heat capacity for the HEE as

1198621015840119876 = 119879(120597120575119878120597119879 )119876

(25)

Then providing a similar relation of the critical points thatin (13) is also working and using (25) we can get the criticalexponent of SPT of in the HEE-temperature phase Here weemploy the logarithm of the quantities 119879 minus 119879119888 120575119878 minus 120575119878119888 The

relation between log |119879 minus 119879119888| and log |120575119878 minus 120575119878119888| is plotted inFigure 11The analytical results of these straight lines can alsobe fitted which are for 1205930 = 010

log 1003816100381610038161003816119879 minus 1198791198881003816100381610038161003816 = 23469 + 300654 log 1003816100381610038161003816120575119878 minus 1205751198781198881003816100381610038161003816 (26)

and for 1205930 = 016log 1003816100381610038161003816119879 minus 1198791198881003816100381610038161003816 = 192089 + 300477 log 1003816100381610038161003816120575119878 minus 1205751198781198881003816100381610038161003816 (27)

The results show that the slopes are all around 3 and therelative errors are less than 0218 0159 for 1205930 = 010016 respectively which are consistent with that of the BHEThen one can find that the critical exponent of the specificheat capacity 1198621015840119876 is also approximately minus23 That is to saythe HEE has the same SPT behavior as that of the BHE Bothof them are consistent with the result in themean field theoryof VDW gas-fluid system

4 Conclusions

In this paper we have investigated the VDWphase transitionwith the use of HEE as a probe Firstly we investigated thephase diagrams of the BHE in the 119879-119878 phase plane and foundthat the phase structure depends on the scalar charge 119876 andthe parameter 120574 of the AdS black holes in this massive gravityFor the case that 120574 = 1 or 119876 = 0 we found that therealways exists the Hawking-Page-like phase transition in thisthermodynamic system while for the case 120574 = minus1 we found



ＦＩＡS minus Sc

minus16

minus14

minus12

minus10

ＦＩＡ T

minusTc

(a)

minus22

minus20

minus18

minus16

minus14

minus12

minus10


ＦＩＡS minus Sc

ＦＩＡ T

minusTc

(b)

Figure 11 The figure shows log |119879 minus 119879119888| versus log |120575119878 minus 120575119878119888| for 120574 = minus1 Here 120582 = 24 119876 = 119876119888 where 1205930 = 010 in (a) and 1205930 = 016 in (b)

for the small scalar charge119876 there is always an unstable blackhole thermodynamic system interpolating between the smallstable black hole system and large stable black hole systemThe thermodynamic transition for the small hole to the largehole is a first-order transition and Maxwellrsquos equal area lawis valid As the scalar charge 119876 increases to the critical value119876119888 in this space-time the unstable black hole merges into aninflection point We found there is a SPT at the critical pointWhen the scalar charge is larger than the critical charge theblack hole is stable always That is to say we found that thereexists the VDW gas-fluid phase transition in the 119879-119878 phaseplane of the AdS black hole in massive gravity

The more interesting thing is that we found the HEE alsoexhibits the VDW phase structure in the 119879-120575119878 plane when120574 = minus1 and the scalar charge 119876 = 0 In order to confirmthis result we further showed that Maxwellrsquos equal area law issatisfied and the critical exponent of the specific heat capacityis consistent with that of the mean field theory of the VDWgas-fluid system for the HEE system These results show thatthe phase structure of HEE is similar to that of BHE and theHEE is really a good probe to the phase transition of AdSblack holes in Lorentz breaking massive gravity This alsoimplies that HEE and BHE exhibit some potential underlyingrelationship

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors are grateful to De-Fu Hou and Hong-Bao Zhangfor their instructive discussions This work is supported bythe National Natural Science Foundation of China (Grant no11365008 Grant no 11465007 and Grant no 61364030)

References

[1] C H Bennett G Brassard C Crepeau R Jozsa A Peres andW K Wootters ldquoTeleporting an unknown quantum state viadual classical and Einstein-Podolsky-Rosen channelsrdquo PhysicalReview Letters vol 70 no 13 pp 1895ndash1899 1993

[2] CH Bennett andD P Divincenzo ldquoQuantum information andcomputationrdquo Nature vol 404 no 6775 pp 247ndash255 2000

[3] M A Nielsen and I L Chuang Quantum Computationand Quantum Information Cambridge University Press Cam-bridge UK 1st edition 2000

[4] J Preskill ldquoQuantum information and physics some futuredirectionsrdquo Journal of Modern Optics vol 47 no 2-3 pp 127ndash137 2000

[5] T J Osborne and M A Nielsen ldquoEntanglement quantumphase transitions and density matrix renormalizationrdquo Quan-tum Information Processing vol 1 no 1-2 pp 45ndash53 2002

[6] P Zanardi and X Wang ldquoFermionic entanglement in itinerantsystemsrdquo Journal of Physics A Mathematical and General vol35 no 37 pp 7947ndash7959 2002

[7] X-X Zeng and W-B Liu ldquoHolographic thermalization inGauss-Bonnet gravityrdquo Physics Letters B Particle PhysicsNuclear Physics and Cosmology vol 726 no 1-3 pp 481ndash4872013

[8] X-X Zeng X-M Liu andW-B Liu ldquoHolographic thermaliza-tion with a chemical potential in Gauss-Bonnet gravityrdquo Journalof High Energy Physics vol 2014 no 3 article 031 2014

[9] X X Zeng D Y Chen and L F Li ldquoHolographic thermal-ization and gravitational collapse in a spacetime dominated byquintessence dark energyrdquo Physical Review D vol 91 no 4Article ID 046005 2015

[10] X-X Zeng X-M Liu andW-B Liu ldquoHolographic thermaliza-tion in noncommutative geometryrdquo Physics Letters B ParticlePhysics Nuclear Physics and Cosmology vol 744 pp 48ndash542015

[11] X X Zeng X YHu and L F Li ldquoEffect of phantomdark energyon holographic thermalizationrdquo Chinese Physics Letters vol 34no 1 Article ID 010401 2017

[12] R Cai X Zeng and H Zhang ldquoInfluence of inhomogeneitieson holographic mutual information and butterfly effectrdquo Jour-nal of High Energy Physics vol 2017 no 7 2017

[13] R Cai L Li L Li and R Yang ldquoIntroduction to holographicsuperconductor modelsrdquo Science China Physics Mechanics ampAstronomy vol 58 no 6 pp 1ndash46 2015

[14] L Amico R Fazio A Osterloh and V Vedral ldquoEntanglementin many-body systemsrdquo Reviews of Modern Physics vol 80 no2 pp 517ndash576 2008


[15] N Laflorencie ldquoQuantum entanglement in condensed mattersystemsrdquo Physics Reports A Review Section of Physics Lettersvol 646 pp 1ndash59 2016

[16] L Susskind and J Uglum ldquoBlack hole entropy in canonicalquantum gravity and superstring theoryrdquo Physical Review DThird Series vol 50 no 4 pp 2700ndash2711 1994

[17] S N Solodukhin ldquoEntanglement entropy of black holesrdquo LivingReviews in Relativity vol 14 article 8 2011

[18] S Ryu and T Takayanagi ldquoHolographic derivation of entan-glement entropy from the anti-de Sitter spaceconformal fieldtheory correspondencerdquo Physical Review Letters vol 96 no 18Article ID 181602 181602 4 pages 2006

[19] S Ryu andT Takayanagi ldquoAspects of holographic entanglemententropyrdquo Journal of High Energy Physics A SISSA Journal no 8045 48 pages 2006

[20] C V Johnson ldquoLarge N phase transitions finite volume andentanglement entropyrdquo Journal of High Energy Physics vol 2014article 47 2014

[21] E Caceres P H Nguyen and J F Pedraza ldquoHolographicentanglement entropy and the extended phase structure of STUblack holesrdquo Journal of High Energy Physics vol 2015 no 9article 184 2015

[22] P H Nguyen ldquoAn equal area law for holographic entanglemententropy of the AdS-RN black holerdquo Journal of High EnergyPhysics vol 15 no 12 article 139 2015

[23] X X Zeng H Zhang and L F Li ldquoPhase transition ofholographic entanglement entropy in massive gravityrdquo PhysicsLetters B vol 756 pp 170ndash179 2016

[24] A Dey S Mahapatra and T Sarkar ldquoThermodynamics andentanglement entropy with Weyl correctionsrdquo Physical ReviewD - Particles Fields Gravitation and Cosmology vol 94 no 2Article ID 026006 2016

[25] X-X Zeng X-M Liu and L-F Li ldquoPhase structure of theBornndashInfeldndashanti-de Sitter black holes probed by non-localobservablesrdquo European Physical Journal C vol 76 no 11 article616 2016

[26] X-X Zeng and L-F Li ldquoHolographic Phase Transition ProbedbyNonlocal ObservablesrdquoAdvances in High Energy Physics vol2016 Article ID 6153435 2016

[27] X X Zeng and L F Li ldquoVan der waals phase transition in theframework of holographyrdquo Physics Letters B vol 764 pp 100ndash108 2017

[28] J-X Mo G-Q Li Z-T Lin and X-X Zeng ldquoRevisiting vanderWaals like behavior of f(R)AdS black holes via the two pointcorrelation functionrdquoNuclear Physics B vol 918 pp 11ndash22 2017

[29] S He L F Li and X X Zeng ldquoHolographic van der waals-likephase transition in the GaussndashBonnet gravityrdquo Nuclear PhysicsB vol 915 pp 243ndash261 2017

[30] X Zeng and Y Han ldquoHolographic van der waals phase transi-tion for a hairy black holerdquoAdvances inHigh Energy Physics vol2017 Article ID 2356174 8 pages 2017

[31] H-L Li S-Z Yang and X-T Zu ldquoHolographic research onphase transitions for a five dimensional AdS black hole withconformally coupled scalar hairrdquo Physics Letters Section BNuclear Elementary Particle and High-Energy Physics vol 764pp 310ndash317 2017

[32] M Cadoni E Franzin and M Tuveri ldquoVan der Waals-likebehaviour of charged black holes and hysteresis in the dualQFTsrdquo Physics Letters B Particle Physics Nuclear Physics andCosmology vol 768 pp 393ndash396 2017

[33] B P Abbott et al ldquoObservation of gravitational waves froma binary black hole mergerrdquo Physical Review Letters vol 116Article ID 061102 2016

[34] M Fierz andW Pauli ldquoOn relativistic wave equations for parti-cles of arbitrary spin in an electromagnetic fieldrdquo Proceedings ofthe Royal Society of London Series A Mathematical and PhysicalSciences vol 173 no 953 pp 211ndash232 1939

[35] D G Boulware and S Deser ldquoCan gravitation have a finiterangerdquo Physical Review D vol 6 no 12 pp 3368ndash3382 1972

[36] C de Rham and G Gabadadze ldquoGeneralization of the Fierz-Pauli actionrdquo Physical Review D vol 82 Article ID 0440202010

[37] C De Rham G Gabadadze and A J Tolley ldquoResummation ofmassive gravityrdquo Physical Review Letters vol 106 no 23 ArticleID 231101 2011

[38] D Vegh ldquoHolography without translational symmetryrdquo 2013httpsarxivorgabs13010537

[39] C de Rham ldquoMassive gravityrdquo Living Reviews in Relativity vol17 no 7 2014

[40] K Hinterbichler ldquoTheoretical aspects of massive gravityrdquoReviews of Modern Physics vol 84 no 2 2012

[41] S L Dubovsky ldquoPhases of massive gravityrdquo Journal of HighEnergy Physics article 076 2004

[42] V A Rubakov and P G Tinyakov ldquoInfrared-modified gravitiesand massive gravitonsrdquo Physics-Uspekhi vol 51 no 8 pp 759ndash792 2008

[43] M V Bebronne and P G Tinyakov ldquoBlack hole solutions inmassive gravityrdquo Journal of High Energy Physics vol 09 no 04article 100 2008

[44] D Comelli F Nesti and L Pilo ldquoStars and (furry) black holes inLorentz breaking massive gravityrdquo Physical Review D - ParticlesFields Gravitation and Cosmology vol 83 no 8 Article ID084042 2011

[45] S Fernando ldquoPhase transitions of black holes in massive grav-ityrdquo Modern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 31 no 16 Article ID 16500961650096 19 pages 2016

[46] F Capela and G Nardini ldquoHairy black holes in massive gravitythermodynamics and phase structurerdquo Physical Review D vol86 no 2 Article ID 024030 12 pages 2012

[47] B Mirza and Z Sherkatghanad ldquoPhase transitions of hairyblack holes in massive gravity and thermodynamic behavior ofcharged AdS black holes in an extended phase spacerdquo PhysicalReview D vol 90 no 8 Article ID 084006 2014

[48] S Fernando ldquoP-V criticality in AdS black holes of massivegravityrdquo Physical Review D vol 94 no 12 Article ID 1240492016

[49] S W Hawking and D N Page ldquoThermodynamics of blackholes in anti-de Sitter spacerdquo Communications in MathematicalPhysics vol 87 no 4 pp 577ndash588 198283

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


mass terms are obtained by introducing a reference metricRecently Vegh proposed a new reference metric to describea class of strongly interacting quantum field theories withbroken translational symmetry in the holographic framework[38] The recent progress in massive gravity can be found in[39 40]

Here we consider AdS black holes in Lorentz breakingmassive gravity In themassive gravity the graviton acquires amass by Lorentz symmetry breaking which is very similar tothe Higgs mechanism A review of Lorentz-violating massivegravity theory can be found in [41 42] In this paper we focuson the study of the VDW phase transition of AdS black holesin Lorentz breakingmassive gravity using the HEEThemainmotivation of this paper is to explore whether the BHE phasetransition can also be described by HEE in Lorentz breakingmassive gravity Firstly we would like to extend proposalsin [20] to study VDW phase transitions in AdS black holewith a spherical horizon in Lorentz-violating massive gravitywith the HEE as a probe What is more we also would liketo checkMaxwellrsquos equal area law and critical exponent of theheat capacity which are two characteristic quantities in VDWphase transition

The organization of this paper is as follows In thenext section we shall provide a brief review of the blackhole solution in Lorentz breaking massive gravity firstlyThen we will study the VDW phase transitions and criticalphenomena for the AdS black hole in the BHE-temperatureplane In Section 3 wemainly concentrate on theVDWphasetransition and critical phenomena in the framework of HEEThe last section is devoted to our discussions and conclusions

2 Phase Transition and CriticalPhenomena of AdS Black Holes in LorentzBreaking Massive Gravity

21 Review of AdS Black Holes in Lorentz Breaking MassiveGravity The four-dimensional Lorentz breaking massivegravity can be obtained by adding nonderivative couplingscalar fields to the standard Einstein gravity theory As amatter field is considered the theory can be described by thefollowing action [41 42]

119878 = int1198894119909radicminus119892 [minus1198722Pl119877 + 119871119898 + ℓ4Ψ(119883Π119894119895)] (1)

here the first two terms are the curvature and ordinarymatterminimally coupled to gravity respectively and the third termΨ contains two functions 119883 and Π119894119895 which relate to the fourscalar fields Ξ0 and Ξ119894 as

119883 = 120597120583Ξ0120597120583Ξ0ℓ4 Π119894119895 = 120597120583Ξ119894120597120583Ξ119895ℓ4 minus 120597120583Ξ119894120597120583Ξ0120597]Ξ119895120597]Ξ0ℓ8119883

(2)

When the four scalar fields get a space-time dependingvacuum expectation value the system will break the Lorentzsymmetry What is more the action can also be taken as a

low-energy effective theory with the ultraviolet cutoff scale ℓHere the scale parameter ℓ has the dimension of mass and isin the order of radic119898119892119872Pl where 119898119892 and 119872Pl are the gravitonmass and the Plank mass respectively

The AdS black hole solutions can be obtained from theabove theory [43 44] The metric corresponding to the AdSblack holes is given by

1198891199042 = minus119891 (119903) 1198891199052 + 119891 (119903)minus1 1198891199032 + 1199032 (1198891205932 + sin21205931198891206012) (3)

with

119891 (119903) = 1 minus 2119872119903 minus 1205741198762119903120582 minus Λ11990323 (4)

Here the four scalar fields Ξ0 and Ξ119894 for this particularsolution are given by

Ξ0 = ℓ2 (119905 + 120578 (119903)) Ξ119894 = ℓ2120572119909119894 (5)

in which

120578 (119903) = plusmnint 119889119903119891 (119903)sdot [[1 minus 119891 (119903)(1205741198762120582 (120582 minus 1)1211989821198921205726 1119903120582+1 + 1)minus1]]

12

(6)

in which the scalar charge 119876 is related to massive gravityand the constant 120572 which is determined by the cosmologicalconstant Λ and the graviton mass 119898119892 with the relation Λ =21198982119892(1 minus 1205726) In this paper we will set 120572 gt 1 such thatΛ lt 0 leading to Anti-de Sitter black holes The constant 120582 isa positive integration constant When 120582 lt 1 the ADM massof the black hole solution diverges For 120582 gt 1 the metricapproaches the Schwarzschild-AdS black holes with a finitemass 119872 as 119903 rarr infin Thus we set 120582 gt 1 in this paper Theconstant 120574 = plusmn1 When 120574 = 1 the black hole only has asingle horizon 119903ℎ which is the root of the equation 119891(119903ℎ) = 0The function 119891(119903) for this case is given in Figure 1 whichis similar to the Schwarzschild-AdS black hole For 120574 = minus1the black hole is very similar to the Reissner-Nordstrom-AdSblack holeThe function119891(119903) for this case is given in Figure 2The black hole event horizon 119903ℎ is the largest root of theequation 119891(119903ℎ) = 0

At the event horizon the Hawking temperature and BHEcan be written as

119879 = 141205871198911015840 (119903ℎ) = 14120587 ( 1119903ℎ minus 119903ℎΛ + 120574 (120582 minus 1)1198762119903120582+1ℎ

) (7)

119878 = 1205871199032ℎ (8)

The chemical potential in this black hole is

Φ = minus120574 119876119903120582minus1ℎ

(9)


10 15 20 25 3005

r

minus20

minus15

minus10

minus5

5

10

f(r)

Q = 05

Q = 0

Q = 18

Q = 12


Q = 18

Q = 12

Q = 05

Q = 0minus4

minus2

2

4

f(r)

10 15 20 2505r


119889119872 = 119879119889119878 + Φ119889119876 (10)





035

040

045

050

055

T

2 4 6 80S

TＧＣＨ


05 10 15 2000S

Q = 0

Q = 06Qc

Q = 12Qc

024

025

026T

027

028

029

Q = Qc





0250

0255

0260

0265

25 3020

040

045

050

055

060

20 25 3015

005

010

015

020

25 3020

Tc S c

Qc



Λ

040

045

050

055

060


Λ

05

10

15

20

005

010

015

020

025


Λ

Tc S c Qc


119862119876 = 119879( 120597119878120597119879)119876

(12)


(120597119879120597119878 )119876119888 119878119888

= (12059721198791205971198782 )119876119888 119878119888

= 0 (13)



119878119888 = minus 120587120582(120582 + 2)Λ (15)

119879119888 = 1205822120587 (120582 + 1)radicminus120582 (120582 + 2) Λ (16)


027 028 029026T

0085

0090

0095

0100

0105

0110

F

(0275 01042)




139405277

01563

025

026

027

028

029

T

05 10 15 2000S

S1 =

S2 =S3 =


119879lowast (1198783 minus 1198781) = int11987831198781

119879 (119878 119876) 119889119878 (17)



119879 = 119879119888 + (120597119879120597119878 )119876119888119878119888

(119878 minus 119878119888) + (12059721198791205971198782 )119876119888119878119888

(119878 minus 119878119888)2+ (12059731198791205971198783 )

119876119888 119878119888

(119878 minus 119878119888)3 + 119900 (119878 minus 119878119888)4 (18)







119878119860 = minus119879119903119860 (120588119860 ln 120588119860) (20)


119878119860 = Area (120574119860)4 (21)



119860 = 2120587int12059300

Θ(119903 (120593) 120593) 119889120593Θ = 119903 sin120593radic (1199031015840)2119891 (119903) + 1199032

(22)



(23)




025026027028029030031

T＜Ｂ

80618

80619

80620

80621

80622

80623

80617

S minus S＞３

(a)

80618

80617

80620

80621

80622

80623

80619

S minus S＞３

022024026028030032

T＜Ｂ

(b)

80617

80618

80619

80620

80621

80622

80623

S minus S＞３

018020022024026028030032

T＜Ｂ

(c)


025026027028029030031

70755

70760

70765

70770

70775

70750

S minus S＞３

T＜Ｂ

(a)

022

024

026

028

030

032

70755

70760

70765

70770

70775

70750

S minus S＞３

T＜Ｂ

(b)

018020022024026028030032

70770

70775

70765

70755

70760

70750

S minus S＞３

T＜Ｂ

(c)



119879lowast (1205751198783 minus 1205751198781) = int12057511987831205751198781

119879 (120575119878) 119889119878 (24)



1198621015840119876 = 119879(120597120575119878120597119879 )119876

(25)






4 Conclusions




ＦＩＡS minus Sc

minus16

minus14

minus12

minus10

ＦＩＡ T

minusTc

(a)

minus22

minus20

minus18

minus16

minus14

minus12

minus10


ＦＩＡS minus Sc

ＦＩＡ T

minusTc

(b)






Acknowledgments


References
























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




10 15 20 25 3005

r

minus20

minus15

minus10

minus5

5

10

f(r)

Q = 05

Q = 0

Q = 18

Q = 12


Q = 18

Q = 12

Q = 05

Q = 0minus4

minus2

2

4

f(r)

10 15 20 2505r


119889119872 = 119879119889119878 + Φ119889119876 (10)





035

040

045

050

055

T

2 4 6 80S

TＧＣＨ


05 10 15 2000S

Q = 0

Q = 06Qc

Q = 12Qc

024

025

026T

027

028

029

Q = Qc





0250

0255

0260

0265

25 3020

040

045

050

055

060

20 25 3015

005

010

015

020

25 3020

Tc S c

Qc



Λ

040

045

050

055

060


Λ

05

10

15

20

005

010

015

020

025


Λ

Tc S c Qc


119862119876 = 119879( 120597119878120597119879)119876

(12)


(120597119879120597119878 )119876119888 119878119888

= (12059721198791205971198782 )119876119888 119878119888

= 0 (13)



119878119888 = minus 120587120582(120582 + 2)Λ (15)

119879119888 = 1205822120587 (120582 + 1)radicminus120582 (120582 + 2) Λ (16)


027 028 029026T

0085

0090

0095

0100

0105

0110

F

(0275 01042)




139405277

01563

025

026

027

028

029

T

05 10 15 2000S

S1 =

S2 =S3 =


119879lowast (1198783 minus 1198781) = int11987831198781

119879 (119878 119876) 119889119878 (17)



119879 = 119879119888 + (120597119879120597119878 )119876119888119878119888

(119878 minus 119878119888) + (12059721198791205971198782 )119876119888119878119888

(119878 minus 119878119888)2+ (12059731198791205971198783 )

119876119888 119878119888

(119878 minus 119878119888)3 + 119900 (119878 minus 119878119888)4 (18)







119878119860 = minus119879119903119860 (120588119860 ln 120588119860) (20)


119878119860 = Area (120574119860)4 (21)



119860 = 2120587int12059300

Θ(119903 (120593) 120593) 119889120593Θ = 119903 sin120593radic (1199031015840)2119891 (119903) + 1199032

(22)



(23)




025026027028029030031

T＜Ｂ

80618

80619

80620

80621

80622

80623

80617

S minus S＞３

(a)

80618

80617

80620

80621

80622

80623

80619

S minus S＞３

022024026028030032

T＜Ｂ

(b)

80617

80618

80619

80620

80621

80622

80623

S minus S＞３

018020022024026028030032

T＜Ｂ

(c)


025026027028029030031

70755

70760

70765

70770

70775

70750

S minus S＞３

T＜Ｂ

(a)

022

024

026

028

030

032

70755

70760

70765

70770

70775

70750

S minus S＞３

T＜Ｂ

(b)

018020022024026028030032

70770

70775

70765

70755

70760

70750

S minus S＞３

T＜Ｂ

(c)



119879lowast (1205751198783 minus 1205751198781) = int12057511987831205751198781

119879 (120575119878) 119889119878 (24)



1198621015840119876 = 119879(120597120575119878120597119879 )119876

(25)






4 Conclusions




ＦＩＡS minus Sc

minus16

minus14

minus12

minus10

ＦＩＡ T

minusTc

(a)

minus22

minus20

minus18

minus16

minus14

minus12

minus10


ＦＩＡS minus Sc

ＦＩＡ T

minusTc

(b)






Acknowledgments


References
























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0250

0255

0260

0265

25 3020

040

045

050

055

060

20 25 3015

005

010

015

020

25 3020

Tc S c

Qc



Λ

040

045

050

055

060


Λ

05

10

15

20

005

010

015

020

025


Λ

Tc S c Qc


119862119876 = 119879( 120597119878120597119879)119876

(12)


(120597119879120597119878 )119876119888 119878119888

= (12059721198791205971198782 )119876119888 119878119888

= 0 (13)



119878119888 = minus 120587120582(120582 + 2)Λ (15)

119879119888 = 1205822120587 (120582 + 1)radicminus120582 (120582 + 2) Λ (16)


027 028 029026T

0085

0090

0095

0100

0105

0110

F

(0275 01042)




139405277

01563

025

026

027

028

029

T

05 10 15 2000S

S1 =

S2 =S3 =


119879lowast (1198783 minus 1198781) = int11987831198781

119879 (119878 119876) 119889119878 (17)



119879 = 119879119888 + (120597119879120597119878 )119876119888119878119888

(119878 minus 119878119888) + (12059721198791205971198782 )119876119888119878119888

(119878 minus 119878119888)2+ (12059731198791205971198783 )

119876119888 119878119888

(119878 minus 119878119888)3 + 119900 (119878 minus 119878119888)4 (18)







119878119860 = minus119879119903119860 (120588119860 ln 120588119860) (20)


119878119860 = Area (120574119860)4 (21)



119860 = 2120587int12059300

Θ(119903 (120593) 120593) 119889120593Θ = 119903 sin120593radic (1199031015840)2119891 (119903) + 1199032

(22)



(23)




025026027028029030031

T＜Ｂ

80618

80619

80620

80621

80622

80623

80617

S minus S＞３

(a)

80618

80617

80620

80621

80622

80623

80619

S minus S＞３

022024026028030032

T＜Ｂ

(b)

80617

80618

80619

80620

80621

80622

80623

S minus S＞３

018020022024026028030032

T＜Ｂ

(c)


025026027028029030031

70755

70760

70765

70770

70775

70750

S minus S＞３

T＜Ｂ

(a)

022

024

026

028

030

032

70755

70760

70765

70770

70775

70750

S minus S＞３

T＜Ｂ

(b)

018020022024026028030032

70770

70775

70765

70755

70760

70750

S minus S＞３

T＜Ｂ

(c)



119879lowast (1205751198783 minus 1205751198781) = int12057511987831205751198781

119879 (120575119878) 119889119878 (24)



1198621015840119876 = 119879(120597120575119878120597119879 )119876

(25)






4 Conclusions




ＦＩＡS minus Sc

minus16

minus14

minus12

minus10

ＦＩＡ T

minusTc

(a)

minus22

minus20

minus18

minus16

minus14

minus12

minus10


ＦＩＡS minus Sc

ＦＩＡ T

minusTc

(b)






Acknowledgments


References
























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




139405277

01563

025

026

027

028

029

T

05 10 15 2000S

S1 =

S2 =S3 =


119879lowast (1198783 minus 1198781) = int11987831198781

119879 (119878 119876) 119889119878 (17)



119879 = 119879119888 + (120597119879120597119878 )119876119888119878119888

(119878 minus 119878119888) + (12059721198791205971198782 )119876119888119878119888

(119878 minus 119878119888)2+ (12059731198791205971198783 )

119876119888 119878119888

(119878 minus 119878119888)3 + 119900 (119878 minus 119878119888)4 (18)







119878119860 = minus119879119903119860 (120588119860 ln 120588119860) (20)


119878119860 = Area (120574119860)4 (21)



119860 = 2120587int12059300

Θ(119903 (120593) 120593) 119889120593Θ = 119903 sin120593radic (1199031015840)2119891 (119903) + 1199032

(22)



(23)




025026027028029030031

T＜Ｂ

80618

80619

80620

80621

80622

80623

80617

S minus S＞３

(a)

80618

80617

80620

80621

80622

80623

80619

S minus S＞３

022024026028030032

T＜Ｂ

(b)

80617

80618

80619

80620

80621

80622

80623

S minus S＞３

018020022024026028030032

T＜Ｂ

(c)


025026027028029030031

70755

70760

70765

70770

70775

70750

S minus S＞３

T＜Ｂ

(a)

022

024

026

028

030

032

70755

70760

70765

70770

70775

70750

S minus S＞３

T＜Ｂ

(b)

018020022024026028030032

70770

70775

70765

70755

70760

70750

S minus S＞３

T＜Ｂ

(c)



119879lowast (1205751198783 minus 1205751198781) = int12057511987831205751198781

119879 (120575119878) 119889119878 (24)



1198621015840119876 = 119879(120597120575119878120597119879 )119876

(25)






4 Conclusions




ＦＩＡS minus Sc

minus16

minus14

minus12

minus10

ＦＩＡ T

minusTc

(a)

minus22

minus20

minus18

minus16

minus14

minus12

minus10


ＦＩＡS minus Sc

ＦＩＡ T

minusTc

(b)






Acknowledgments


References
























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




025026027028029030031

T＜Ｂ

80618

80619

80620

80621

80622

80623

80617

S minus S＞３

(a)

80618

80617

80620

80621

80622

80623

80619

S minus S＞３

022024026028030032

T＜Ｂ

(b)

80617

80618

80619

80620

80621

80622

80623

S minus S＞３

018020022024026028030032

T＜Ｂ

(c)


025026027028029030031

70755

70760

70765

70770

70775

70750

S minus S＞３

T＜Ｂ

(a)

022

024

026

028

030

032

70755

70760

70765

70770

70775

70750

S minus S＞３

T＜Ｂ

(b)

018020022024026028030032

70770

70775

70765

70755

70760

70750

S minus S＞３

T＜Ｂ

(c)



119879lowast (1205751198783 minus 1205751198781) = int12057511987831205751198781

119879 (120575119878) 119889119878 (24)



1198621015840119876 = 119879(120597120575119878120597119879 )119876

(25)






4 Conclusions




ＦＩＡS minus Sc

minus16

minus14

minus12

minus10

ＦＩＡ T

minusTc

(a)

minus22

minus20

minus18

minus16

minus14

minus12

minus10


ＦＩＡS minus Sc

ＦＩＡ T

minusTc

(b)






Acknowledgments


References
























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





ＦＩＡS minus Sc

minus16

minus14

minus12

minus10

ＦＩＡ T

minusTc

(a)

minus22

minus20

minus18

minus16

minus14

minus12

minus10


ＦＩＡS minus Sc

ＦＩＡ T

minusTc

(b)






Acknowledgments


References
























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics












































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Van der Waals-Like Phase Transition from Holographic ...2.2. Van der Waals-Like Phase Transition of...

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