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We present an AdS black hole solution with Ricci flat horizon in Einstein-phantom scalar theory. The phantom scalar fields just depend on the transverse coordinates

Due to its confined boundary of anti-de Sitter (AdS) space-time, thermodynamics of black hole in AdS space and those in flat or de Sitter space-time are very different. For example, for a Schwarzschild AdS black hole, it is stable if its horizon is larger than a certain value, while, for a small black hole, the effect of cosmological constant can be neglected, the black hole behaves just like a Schwarzschild black hole in flat space, and it is thermodynamically unstable. In addition, it is found that there is an AdS black hole only when the temperature of black hole is larger than a critical value, while there is only a thermal gas solution when its temperature is less than the value. The first-order phase transition which occurs between the black hole solution and the thermal gas solution in the AdS space-time is named Hawking-Page phase transition [

Another big difference between black holes in AdS space and flat or de Sitter space-time is that the horizon of black holes in flat or de Sitter space must have a spherical structure, while the horizon topology of black holes in AdS space-time could be a zero or negative constant curvature surface, except for the case with a positive constant curvature surface. These kinds of black holes with zero or negative constant curvature horizon have been studied in the literature [

On the other hand, since people discovered the phenomenon of accelerated expansion of the universe at the end of last century, there have been a lot of proposals to explain the cause of this phenomenon. One way is so-called phantom dark energy with state equation,

In this study, we consider Einstein gravity theory with a negative cosmological constant and massless phantom scalar field. The black hole will have a Ricci flat horizon and phantom scalar fields only linearly depend on transverse coordinates. The thermodynamics of AdS phantom black hole are studied, and we find that although its horizon is Ricci flat and infinitely extended, the thermodynamical properties of the black hole and AdS Schwarzschild spherically symmetric black hole are qualitatively the same. In particular, Hawking-Page phase transition can emerge between phantom black hole and the thermal gas solution in AdS space-time in Poincare coordinates. Further, we will calculate holographic entanglement entropy of a strip geometry which is dual to the AdS phantom black hole boundary following the proposal made by Ryu and Takayanagi in [

We first consider a Einstein-Hilbert action with phantom massless scalar fields and a negative cosmological constant as

Now we consider a black hole solution which has a Ricci flat horizon in this system with the metric ansatz as

In this section, the thermodynamics of this AdS black hole solution are discussed. Here we are discussing the case where the transverse coordinates

The horizon of the black hole is given by

The black hole temperature in terms of entropy. The dashed green line represents the Hawking-Page transition temperature, while the solid red line stands for

The Hawking temperature (

With the mass, entropy, and temperature, the Helmholtz free energy is easily calculated as

From (

The Helmholtz free energy of the black hole as a function of temperature

For a quantum mechanical system with many degrees of freedom, we separate the total system into two subsystems

Now we will take formula (

With the help of (

The regularized entanglement entropy versus the black hole temperature. The dashed green line represents the Hawking-Page transition temperature, while the solid red line stands for the minimal temperature.

In Figure

Black hole physics is one of important topics in general relativity and quantum gravity. Thermodynamics, quantum mechanics, statistical physics, and information theory are entangled with each other in black hole thermodynamics. It is generally believed that to completely understand black hole thermodynamics will be greatly helpful to establish a self-consistent quantum gravity theory. In this respect, the AdS black hole might play a central role since the AdS/CFT correspondence relates a quantum gravity in AdS space-time to a conformal field theory in the AdS boundary. In this paper, we have presented an AdS black hole solution with Ricci flat horizon in Einstein-phantom scalar theory. The phantom scalar fields only depend on the transverse coordinates

Indeed we have found that although its horizon structure is Ricci flat, the AdS black hole with phantom scalar has qualitatively same thermodynamic properties as those of AdS Schwarzschild black hole: there exists a minimal temperature and small black hole is thermodynamically unstable, while large black hole is thermodynamically stable. And in particular, the Hawking-Page phase transition can happen between the AdS phantom black hole and the thermal gas solution in AdS space-time in Poincare coordinate.

Further we have calculated the entanglement entropy of dual conformal field theory for a strip geometry in the AdS boundary by the Ryu-Takayanagi proposal. It was found that the behavior of the entanglement entropy is also qualitatively the same as that of the thermodynamical entropy of the black hole. This example further supports that the black hole entropy might indeed be interpreted as entanglement entropy. On the other hand, it also indicates entanglement entropy is indeed a good probe to phase structure and phase transition in black hole thermodynamics.

The authors declare that they have no conflicts of interest.

This work is supported by Meritocracy Research Funds (463508) of China West Normal University and the authors thank R. G. Cai for helpful suggestions and discussions.