PLB30206S0370-2693(14)00341-410.1016/j.physletb.2014.05.032The AuthorsTheory

Fig. 1

(Color online.) The metric functions for a=1.86, Q=6.23 (left), a=64.06, Q=9.76 (middle) and a=1.2i, Q=1/10 (right), with uH=1.

Fig. 2

(Color online.) The Hawking temperature and the entropy density as functions of the inverse horizon radius uH, and the temperature, respectively. For (a) and (b) graphs correspond to “prolate” black brane solution with a2>0, where the initial conditions is chosen as ϕH=−0.22, a=1.46 and Q=0.01. The (c) and (d) graphs plot the “oblate” version of the black brane solution with a2<0, where the initial conditions are chosen as ϕH=0, a=1.2i and Q=0.2.

Fig. 3

(Color online.) (Left) Temperature as a function of the horizon radius for a=1/5 (blue), a=1/2 (red) and a=3/5 (black), where Nc=1 and q=1. (Middle) Temperature vs rH for a=i/14 (blue), a=i/10 (red) and a=i/8 (orange), where Nc=1 and q=1.2. (Right) Entropy density as a function of temperature for “prolate” solution (red) and “oblate” solution (blue), respectively.

Fig. 4

The chemical potential as a function of the charge density for “prolate” and “oblate” solutions, respectively. The values Nc=2, uH=1 and a=0.7,0.2i have been used here. The left graph shows that for the “prolate” solutions, ∂μ/∂ρ<0 at some regions of ρ, signaling an instability of the thermodynamics. On the right, the “oblate” solution is stable as ∂μ/∂ρ>0 is satisfied.

Fig. 5

(Left) The thermodynamic potential G as a function of temperature for fixed charge and anisotropy, where we set Nc=8, a=0.4 and q=1. (Right) The pressure Pz as a function of horizon radius rH for fixed charge and anisotropy, where we set Nc=8, a=0.5 and q=1.8. In both case, we neglect contribution of the reference scale Λ.

Anisotropic plasma with a chemical potential and scheme-independent instabilities

Long

Cheng

Xian-Hui

⁎

gexh@shu.edu.cn

Sang-Jin

Sin

aDepartment of Physics, Shanghai University, Shanghai 200444, ChinaDepartment of PhysicsShanghai UniversityShanghai

200444

ChinabDepartment of Physics, Hanyang University, Seoul 133-791, Republic of KoreaDepartment of PhysicsHanyang UniversitySeoul

133-791

Republic of Korea⁎Corresponding author.

Editor: M. Cvetič

Abstract

Generically, the black brane solution with planar horizons is thermodynamically stable. We find a counter-example to this statement by demonstrating that an anisotropic black brane is unstable. We present a charged black brane solution dual to a spatially anisotropic finite temperature N=4 super Yang–Mills plasma at finite U(1) chemical potential. This static and regular solution is obtained both numerically and analytically. We uncover rich thermodynamic phase structures for this system by considering the cases when the anisotropy constant “a” takes real and imaginary values, respectively. In the case a2>0, the phase structure of this anisotropic black brane is similar to that of Schwarzschild–AdS black hole with S3 horizon topology, yielding a thermodynamical instability at smaller horizon radii. For the condition a2≤0, the thermodynamics is dominated by the black brane phase for all temperatures.

Introduction

The AdS/CFT correspondence provides a powerful tool in studying the strongly coupled problems of quantum field theory, ranging from nuclear physics to condensed matter theory [1,2]. This correspondence states the equivalence between type IIB superstring theory in AdS5×S5 and N=4 super Yang–Mills (SYM) gauge theory on the 4-dimensional boundary of AdS5. From gravitational theory on the asymptotically anti-de Sitter view point, we are able to gain profound insights for such strongly coupled field theory. It is thus very crucial to search for the generic asymptotically AdS gravitational solutions, which are dual to interesting phase in the field theory side. The most well-known black brane solutions are the homogeneous and isotropic Schwarzschild–AdS black brane solution and Reissner–Nordström–AdS (RN–AdS) solution. The charged black brane solutions are particularly useful to study quark–gluon plasma (QGP) [3], superconductivity and superfluidity, Fermi surfaces and non-Fermi liquids in condensed matter system [4–8]. Generally, U(1) gauge symmetries in the bulk correspond to conserved number operators in the dual field theory. The gauge field in the AdS space couples to a CFT current Jμ and the CFT states thus containing a plasma of charged quanta.It is well-known that there are many strongly coupled systems which do not satisfy homogeneity and isotropy spontaneously. For example, some systems may have anisotropic Fermi-surface because of the atomic lattice effects and the QGP is anisotropic in a short time after creation. Therefore, the studies on anisotropic and inhomogeneous black brane solutions and their holographic applications have attracted more attention [9–11].In this paper, we will present a charged and spatially anisotropic black brane solution. The neutral anisotropic black brane solution was obtained by Mateos and Trancanelli in their seminal papers [9] and its applications in QCD was discussed. One motivation comes from the fact that the QGP created in RHIC is not only anisotropic but also charged. In the QGP produced in RHIC, the escaped quark is surrounded by high density quark fluid liberated from the heavy ions. Under such conditions, the baryon density of the QGP and the overall U(1) gauge field is relevant. Unlike chargeless case, the introduction of the U(1) gauge field breaks the SO(6) symmetry and thus leads to the excitations of the Kaluza–Klein modes. Another motivation comes from the applications of the anisotropic black brane solutions to condensed matter physics, since the many-body system at a finite U(1) charge density corresponds to the charged black holes in the AdS peace.We will consider the case the anisotropy is introduced through deforming the SYM theory by a θ-parameter of the form θ∝z, which acts as an isotropy-breaking external source that forces the system into an anisotropic equilibrium state [9]. The θ-parameter is dual to the type IIB axion χ with the form χ=az. The constant a has dimensions of mass and is a measure of the anisotropy. From the five-dimensional theory viewpoint, the anisotropy can be interpreted as a non-zero number of dissolved D7-brane wrapped on S5, extending along the xy-direction and distributed along the z-direction with density nD7 [9]. So a can be regarded as “charge density” and should not be imaginary-valued.However, we will consider both cases with a2>0 and a2<0 which have different thermodynamic properties, although imaginary axion field might be unphysical in type IIB supergravity theory. If the axion field merely plays the role of providing the appropriate source to support a spatially anisotropic spacetime, then the imaginary-valued a could be acceptable. Furthermore, we will see later that imaginary a can be understood as a consequence of the tachyon condensate of the dilaton field. A careful analysis in the following will disclose that the anisotropic black brane solution corresponding to a2>0 is actually a “prolate” version of the solution because it has a z-axis longer than the x- and y-axes (i.e. H(uH)>1), while the “oblate” version of the anisotropic black brane solution requires a2<0 (i.e. H(uH)<1) and thus the z-axis is shorter than the x- and y-axes. As what we will uncover, the “prolate” black brane solution suffers thermodynamic instabilities, similar to those of the Schwarzschild–AdS with a spherical horizon, but the “oblate” solution is stable.2

Numerical solution

The charged anisotropic black brane solution can be derived from the effective action after S5 reduction of type IIB supergravity [12,13]. In Einstein frame, the type IIB supergravity Lagrangian which have been truncated out NS–NS and R–R 2-form potentials is(1)L=Rˆ⁎1−12dϕˆ∧⁎dϕˆ−12e2ϕˆFˆ1∧⁎Fˆ1−14Fˆ5∧⁎Fˆ5, where ϕˆ and Fˆ1=dχˆ are the dilaton and the axion field-strength in ten-dimensions respectively. The 5-form field Fˆ5 should satisfy the self-duality condition and be imposed at the level of equations of motion. The theory can be reduced on to minimal supergravity and the corresponding five-dimensional axion–dilaton–Maxwell–gravity action is given by(2)S=12κ2∫d5x−g(R+12−12(∂ϕ)2−12e2ϕ(∂χ)2−14FμνFμν)+SGH, where we have set the AdS radius L=1, κ2=4π2/Nc2 and SGH is the Gibbons–Hawking boundary term.In order to obtain an anisotropic D3-brane with an asymmetry between the xy- and z-directions, we assume the Einstein-frame metric takes the form(3)ds52=e−12ϕu2(−FBdt2+dx2+dy2+Hdz2+du2F).(4)A=At(u)dt,andχ=az. The functions ϕ, F, B and H=e−ϕ depend only on the radial coordinate u, which we solved numerically [13]. The electric potential At can be obtained via At(u)=−∫uHuduQBe34ϕu from the Maxwell equations, where Q is an integral constant related to the charge. The horizon locates at u=uH≡1/rH with F(uH)=0 and the boundary is at u=0 where F=B=H=1. The asymptotic AdS5 boundary condition requires the boundary condition ϕ(0)=0. The Hawking temperature is given by T=−F′(uH)BH4π through the Euclidean method.

Fig. 1

depicts the metric functions corresponding to different initial conditions. The first two plots in Fig. 1 reflect that the profile for B is seriously suppressed at the horizon as the charge Q increases. The Hawking temperature depends strongly on Q and the anisotropy a is sensitive to the initial condition ϕ(uH). We can also see from Fig. 1 (right) that even the anisotropy constant a takes imaginary value, the black brane solution is still regular. Note that the metric functions H(uH)>1 and B(uH)<1 for a2>0, corresponding to the “prolate” solution, but H(uH)<1, B(uH)>1 for a2<0, corresponding to the “oblate” solution.

Note that the temperature is determined by the inverse horizon radius uH=1/rH and the charge Q. As can be seen from Fig. 2

(left), for a given temperature there are two branches of allowed black brane solutions, a branch with larger radii and one with smaller. This intriguing behavior is similar to the case of Schwarzschild–AdS black holes with a spherically horizon [15]. The smaller branch of the black brane is unstable with negative specific heat.

It is well-known that for black brane solutions with horizon topology R3, there is only one branch of black brane solutions and the free energy is negative definite, so that the black brane structure is trivial and the thermodynamics is dominated by the black brane for all temperatures [14]. However, the anisotropic black brane solution obtained here provides a counter example to the above statement. We notice that even in the absence of U(1) gauge field, two branches of black brane solution still exist, reflecting that it is mainly caused by the anisotropy. This behavior was not noticed in [9] and all the numerical computation was carried out at the stable black brane branch.As to the “oblate” solution with a2<0, the behavior of the solution differs sharply from the real anisotropy situation, which is qualitatively the same as the planar black brane case [14]: In that situation, there is only one stable branch of black brane solution and the thermodynamics is dominated by this solution for all temperatures (see Fig. 2(c)). The entropy density decreases as temperature goes down so that the specific heat is positive cρ=T(∂s/∂T)ρ>0.

Extremal limit

As shown in Fig. 2(a), for the “prolate” solution with anisotropy parameter a2>0, the temperature T cannot reach zero and thus this charged anisotropic configuration has no extremal limit, which is consistent with [16]. This is further supported by Fig. 2(b), which plots the entropy density as a function of the temperature. Furthermore, Fig. 2(b) shows that the entropy density increases as the temperature goes down, which implies an instability of the black brane, since the heat capacity is then negative. This result further support our previous argument that the “prolate” black brane behaves like the Schwarzschild–AdS black hole with the spherical horizon.In the case of a2<0, it is clear that there exists the extremal black brane solution as shown in Fig. 2(c). Numerical computation implies that even close to zero temperature, the system prefers to be dominated by black brane with non-zero entropy. We will provide a consistent check on the above arguments by using the following analytic study.3

Analytic solution in small-anisotropy limit

The analytic black brane solution in small anisotropy limit is obtained to the leading order in a by perturbating the RN–AdS black brane solution. We hope the analytic solution can help us pick out more physics in a straightforward way. The functions F, B and H can be expressed as(5)F=1−(uuH)4+[(uuH)6−(uuH)4]q2+a2F2(u)+O(a4),(6)B=1+a2B2(u)+O(a4),(7)H=e−ϕ(u),with ϕ(u)=a2ϕ2(u)+O(a4), where(8)F2(u)=1241+4q2uH4{3(−4q2u6+uH6)log((1+1+4q2)u2+2uH2(1−1+4q2)u2+2uH2)+u2uH2[81+4q2(−u2+uH2)+u2(3log(−2−2q2+21+4q2))+5(−2+q2)log(−1+2q2+1+4q2)−12q2log(−2−2q2+21+4q2)+7(1+q2)(log((−1+2q2−1+4q2)(2q2u2+(−1+1+4q2)uH2))−log(2q2u2−(1+1+4q2)uH2))]},ϕ2(u)=uH241+4q2log((1+1+4q2)u2+2uH2(1−1+4q2)u2+2uH2),B2(u)=uH224(10u2uH2q2u4−u2uH2−uH4+11+4q2log((1+1+4q2)u2+2uH2(1−1+4q2)u2+2uH2)). The parameter q denotes the dimensionless charge parameter with q=uH3Q23 and the physical range of q2 is 0≤q2<2. The electrical potential is given by(9)At=q8uH33+12q2(241+4q2(uH2−u2)+5a2uH2[uH2log(3−1+4q23+1+4q2)+u2log((1+1+4q2u2)+2uH2(1−1+4q2)u2+2uH2)])+O(a4).By using the Euclidean method, we can easily obtain the Hawking temperature(10)T=−F′(uH)BH4π=2−q22πuH+uH(−41+4q2+5(2+5q2)log(3+1+4q23−1+4q2))96π1+4q2a2+O(a4). The entropy density derived from the Bekenstein–Hawking formula is written as(11)s=AH4GV3=Nc2e−54ϕH2πuH3=Nc22πuH3+5Nc2log(3+1+4q23−1+4q2)32π1+4q2uHa2+O(a4), where V3 is the volume of the black hole horizon. The chemical potential is obtained as(12)μ=q83uH(24+5uH2log(3−1+4q23+1+4q2)1+4q2a2)+O(a4).We notice that for “prolate” solution with a2>0 at finite temperature, the horizon radius rH and the entropy density of the anisotropic black brane are greater than that of isotropic RN–AdS black brane. Thus the chemical potential is less than that of isotropic RN–AdS black brane μ0. The temperature as a function of the horizon radius is shown in Fig. 3

(left and middle) for the cases of real and imaginary anisotropy parameter a respectively. The first graph in Fig. 3 shows that there are two branches for the small-anisotropy but a2>0 case. The smaller radii branch corresponds to negative specific heat, which implies a Hawking–Page transition from a black brane set up to a thermal AdS space. This can be seen clearly from the entropy density-temperature plot given in Fig. 3 (third) by the top red line with one branch ∂s/∂T<0 and another ∂s/∂T>0.

In contrast, for the “oblate” case with a2<0, as shown in the center graph of Fig. 3, there is only one stable black brane configuration and thus is thermodynamically stable. The specific heat is hence positive for all temperatures (see the blue line in Fig. 3 (third)). At a fixed temperature, the horizon radius and the entropy density are less than those of isotropic RN–AdS black brane. The analytical discussion presented here is consistent with the previous numerical result.

Zero temperature limit

From Eq. (10), we learn that the black brane temperature can approach zero only when the inverse horizon radius uH takes the formuH=43[(2−q2)1+4q2]1/2[a2(41+4q2−5(2+5q2)log(3+1+4q23−1+4q2))]−1/2. The positiveness of the horizon radius requires that the axion field parameter a must be imaginary-valued. If the anisotropy constant a takes a real number, Eq. (13) cannot be satisfied for any physical q and uH. This is in agreement with the previous numerical analysis that “prolate” black brane solution yields no extremal configurations. Fig. 3 (middle) plots the temperature as a function of the horizon radius. The temperature becomes a monotonic function of the horizon radius and zero temperature is available for imaginary-valued anisotropy.The appearance of the imaginary valued a can be interpreted as follows: The coupling of the dilatonic field to the axion field induces an effective negative mass term for the dilatonic field. In our case, in the dilatonic equation of motion ∇μ∇μϕ−e2ϕ(∂χ)2=0, e2ϕ(∂χ)2 corresponds to the mass term. As the temperature is lowered, this mass term eventually drives the dilatonic field tachyonic. So that the dilatonic field could condensate in the IR, similar to that observed in the condensation of neutral scalar field in holographic superconductors [5]. Note that although the mass squared term is negative, it is above the Breitenlohner–Freedman (BF) bound.4

Holographic stress tensor

A concrete calculation on holographic renormalization for this charged system is presented in [13] and it is proved that the presence of the U(1) gauge field contributes no additional logarithmic divergences. The counter terms to action (2) are the same as those of axion–dilaton–gravity system discussed in [9](13)Sct=1κ2∫d4xγ(3−18e2ϕ∂iχ∂iχ)−logv∫d4xγA, where v is the Ferrerman–Graham (FG) coordinate, γ is the induced metric on a v=v0 surface. A detailed analysis reveals that the U(1) gauge field does not change the value of the conformal anomaly A=〈Tii〉=Nc2a448π2 obtained in [9]. Note that the stress tensor is diagonal 〈Tij〉=diag(E,Pxy,Pxy,Pz) and obeys ∂i〈Tij〉=0. Due to the presence of conformal anomaly, the transformation of the stress tensor under a rescaling of a, T and μ contains an inhomogeneous term(14)〈Tij(ka,kT,kμ)〉=k4〈Tij(a,T,μ)〉+k4logk Ahij, where hij=diag(1,−1,−1,3). In turn, the stress tensor has the form(15)〈Tij(a,T,μ)〉=a4tij(aT,aμ)+log(aΛ)Nc2a448π2hij, where Λ is an arbitrary reference scale, a remnant of the renormalization process like the substraction point in QCD. Different choices of Λ correspond to different choices of renormalization scheme. This means the physics depends on three dimensionless ratio T/Λ, a/Λ and μ/Λ. The phase diagram of the thermodynamics is then deeply influenced by this reference scale Λ, because the energy density and the pressure are dependent on Λ i.e. scheme-dependent. However, we stress that the U(1) chemical potential μ is scheme-independence. Under a rescaling of the coordinates of the form xi=kxi′, v=kv′ this would not shift μ and there is no such logarithmic term as logv in the expression for μ. The scheme-independence of μ is also implied by its thermodynamic definition [13].One can see from the original papers [9] that the reference scale Λ plays a crucial role in the phase diagram. The introduce of a reference scale Λ aiming to define the theory with a≠0 is a direct consequence of the conformal anomaly, in analogy with the situation in QCD with one quark flavor with Mq≠0. The aim of this paper is that even without considering the renormalization scale Λ, there exists an alternative type of instability.5

Thermodynamics and phase structure

The grand canonical thermodynamical potential Ω=E−Ts−μρ=−Pxy can be evaluated from the on-shell Euclidean action and the entropy density satisfies (∂Ω∂T)μ=−s [13]. The pressure along the z-direction can be evaluated via Pz=Pxy+(∂Ω∂a)a=−G. Our computations demonstrate that the thermodynamics variables corresponding to the “prolate” and “oblate” solutions are different:prolate:a2>0,s>s0,Ω<Ω0,Pz<P0,μ<μ0,oblate:a2<0,s<s0,Ω>Ω0,Pz>P0,μ>μ0, where we have not included the contribution of Λ, and s0, Ω0, P0 and μ0 denote the entropy density, thermodynamical potential, pressure and chemical potential of the isotropic RN-AdS black brane. Ω and Pz are Λ-dependent, but s and μ are Λ-independent.We emphasize that the presence of the U(1) chemical potential significantly changes the phase structure of the whole system. Note that a charged black brane in AdS space can be considered as a system with an infinite charge reservoir and the chemical potential eventually equilibrate to the same value everywhere, then the chemical potential at phase equilibrium should be the same in the isotropic and anisotropic regions of QGP. For the “prolate” solution, the chemical potential of the isotropic phase is higher than that of the anisotropic phase. This means that the anisotropic phase is more stable than the isotropic phase. As a consequence, charges or baryons would immigrate from the isotropic phase to the anisotropic phase. While for the “oblate” solution, the anisotropic chemical potential is greater than the isotropic case, implying a metastable state of the anisotropic plasma. That is to say, charges or baryons would escape from the anisotropic region to the isotropic region.The necessary and sufficient condition for local thermodynamic stability are written as(16)cρ≡T(∂s∂T)ρ>0,μ′≡(∂μ∂ρ)T>0. The heat capacity cρ at constant charges ρ should be positive and regular. The second condition (16) states that the system is stable against infinitesimal charge fluctuations. For “prolate” solution, we have already shown that the specific heat takes a negative value at the smaller horizon radius, while it is positive for “oblate” solution. Further evidence can be found in the μ–ρ diagram (see Fig. 4

First, consider the “prolate” black brane solution, the thermodynamic potential G as a function of the temperature has a “cusp” shape, which is qualitatively the same as the Schwarzschild–AdS black holes. The thermodynamic potential is positive for some range of T, and it is only above the critical temperature Tc that the thermodynamics is dominated by the black brane phase. This can be seen clearly from Fig. 5

. The pressure along the z-direction is negative at smaller horizon radii. As to the “oblate” solutions, the thermodynamic potential is strictly negative for all the temperatures and thus is thermodynamically stable.

Conclusion and discussion

In the present paper, we have obtained the new charged anisotropic black brane solutions, which might be “prolate” or “oblate”. We have mainly compared the thermal properties of these two solutions, and uncovered a new instabilities by using the scheme-independent parameters s, μ and T for the “prolate” solutions. Generally, for the black holes in equilibrium with the heat bath, the increase in the temperature leads to the increase in the black hole radius and mass for stable black holes. However, from Fig. 2(a), we can see that the local slope of the 1/T curve is positive for the smaller radii branch, meaning that the temperature decreases as rH increases, which is quite similar to the familiar case of the uncharged Schwarzschild–AdS black bole with S3 horizon topology. Therefore, the smaller branch with smaller radii is unstable, having negative specific heat. The smaller branch solution is unphysical and should not be applied to studying the dual CFT.Note that the anisotropy constant “a”, the temperature and the horizon radius rH have the same dimension of mass. The instability uncovered here is due to a competing effect between the scale set by the anisotropy and the scale set by the temperature. The instability revealed here is independent of the reference scale Λ i.e. scheme-independent. If we include the effects of the conformal anomaly Λ, we can reproduce the similar results discussed in [9].On the other hand, for the “oblate” case with a2<0, the specific heat is positive everywhere. In this case, for a given temperature T, the horizon radii rH and the entropy density are less than those of RN–AdS black brane. We also note that μ>μ0 in this case. Ignoring the reference scale Λ, the pressure along the z-direction satisfies Pz>P0, inferring mechanical stability of the black brane.The potential Ω reduces to the free energy F when the U(1) charge is absent. In [9], Mateos and Trancanelli have investigated the “prolate” solution by exploring the scheme-dependent free energy F. Specially, through inspecting F″=(∂2F/∂a2)T, they clarified the phases into three zones, namely, unstable zones, metastable zones and stable zones determined by the ratio a/Λ. The unstable zones correspond to unstable thermal equilibriums against infinite charge-“a” fluctuations, while the stable zones correspond to metastable thermal equilibriums against finite charge-“a” fluctuations. The unstable and metastable states they uncovered will fall apart into a mixed phase similar to the high-density anisotropic ‘droplet’ or ‘filaments’ surrounded by isotropic regions [9]. However, the instabilities uncovered here cannot be rescued by adding the conformal anomaly term, because they are scheme-independent.

Acknowledgements

We are indebted to Hong Lu and Jian-Xin Lu for help on the Type IIB supergravity action. We would like also to thank LiQing Fang, Xiao-Mei Kuang, Jia-Rui Sun, Shang-Yu Wu, Yi Yang and Shao-Jun Zhang for helpful discussions. X.H.G. was supported by NSFC, China (No. 11375110), and Shanghai Rising-Star Program (No. 10QA1402300). S.J.S. was supported by the NRF, Korea (NRF-2013R1A2A2A05004846).

References

[1]

J.M.

MaldacenaAdv. Theor. Math. Phys.

1998

231

[2]

WittenAdv. Theor. Math. Phys.

1998

253

S.S.

Gubser

I.R.

Klebanov

A.M.

PolyakovPhys. Lett. B

428

1998

105

[3]

D.T.

Son

A.O.

StarinetsJ. High Energy Phys.

2006052

F.L.

Lin

MatsuoPhys. Lett. B

641

2006

Bigazzi

A.L.

Cotrone

Mas

Mayerson

TarrioJ. High Energy Phys.

1104

2011060

[4]

S.S.

GubserPhys. Rev. D

2008065034

[5]

S.A.

Hartnoll

C.P.

Herzog

G.T.

HorowitzPhys. Rev. Lett.

101

2008031601

S.A.

Hartnoll

C.P.

Herzog

G.T.

HorowitzJ. High Energy Phys.

0812

2008015

[6]

S.A.

HartnollClass. Quantum Gravity

2009224002

[7]

C.P.

HerzogJ. Phys. A

2009343001

McGreevyAdv. High Energy Phys.

2010

2010723105

[8]

S.S.

LeePhys. Rev. D

2009086006

Liu

McGreevy

VeghPhys. Rev. D

2011065029

Faulkner

Liu

McGreevy

VeghPhys. Rev. D

2011125002

Cubrovic

Zaanen

SchalmScience

325

2009

439

[9]

Mateos

TrancanelliPhys. Rev. Lett.

107

2011101601

Mateos

TrancanelliJ. High Energy Phys.

1107

2011054

[10]

Ryblewski

FlorkowskiActa Phys. Pol. B

2011

115

R.A.

Janik

WitaszczykJ. High Energy Phys.

0809

2008026

Rebhan

SteinederJ. High Energy Phys.

1208

2012020

Jahnke

Luna

Patino

TrancanelliJ. High Energy Phys.

1401

2014149

arXiv:1311.5513 [hep-th]

Gynther

Rebhan

SteinederJ. High Energy Phys.

1210

2012012

arXiv:1207.6283

[11]

G.T.

Horowitz

J.E.

Santos

TongJ. High Energy Phys.

1207

2012168

G.T.

Horowitz

J.E.

Santos

TongJ. High Energy Phys.

1211

2012102

Donos

J.P.

Gauntlett

PantelidouJ. High Energy Phys.

1201

2012061

Donos

J.P.

GauntlettPhys. Rev. Lett.

108

2012211601

[12]

M.S.

Bremer

M.J.

Duff

Lü

C.N.

Pope

K.S.

StelleNucl. Phys. B

543

1999

321

Cvetic

Duff

Hoxha

J.T.

Liu

Lü

Nucl. Phys. B

558

1999

Lü

C.N.

Pope

T.A.

TranPhys. Lett. B

475

2000

261

[13]

Cheng

X.H.

S.J.

Sin

arXiv:1404.5027

[14]

Chamblin

Emparan

C.V.

Johnson

R.C.

MyersPhys. Rev. D

1999064018

R.G.

Cai

OhtaPhys. Rev. D

2000124012

J.X.

Ran

WeiJ. High Energy Phys.

1304

2013100

[15]

S.W.

Hawking

D.N.

PageCommun. Math. Phys.

1983

577

[16]

Iizuka

MaedaJ. High Energy Phys.

1207

2012129