Van der Waals-like phase transition from holographic entanglement entropy in Lorentz breaking massive gravity

In this paper, phase transition of AdS black holes in lorentz breaking massive gravity has been studied in the framework of holography. We find that there is a first order phase transition(FPT) and second order phase transition(SPT) both in Bekenstein-Hawking entropy(BHE)-temperature plane and holographic entanglement entropy(HEE)-temperature plane. Furthermore, for the FPT, the equal area law is checked and for the SPT, the critical exponent of the heat capacity is also computed. Our results confirm that the phase structure of HEE is similar to that of BHE in lorentz breaking massive gravity, which implies that HEE and BHE have some potential underlying relationship.


Introduction
The study of HEE and quantum phase transitions of black holes have gained a lot of interest recent years. On one hand, HEE can be used as a perfect probe to study quantum information science [1,2,3], strongly correlated quantum systems [4,5,6,7,8,9,10,11,12,13] and Many-Body Systems [14,15]. On the other hand, investigation on HEE of black holes may shed some light on understanding the nature of BHE [16,17].
Nearly ten years ago, a holographic derivation of the HEE in conformal quantum field theories was proposed by Ryu and Takayanagi using the famous AdS/CFT correspondence [18,19]. Recently the HEE has been used as a probe to investigate the phase structure of the Reissner-Nordstrom AdS black hole [20]. The results showed that there is a Van der Waals-like(VDP) phase transition at the same critical temperature in both the fixed charge ensemble and chemical potential ensemble in the holographic HEE-temperature plane. They also found that the SPT occurring for the HEE at the same critical point as the BHE with nearly the same critical exponent. This work was soon generalized to the extended phase space where the cosmological constant is considered as a thermodynamical variable [21]. Very recently, the equal area law of HEE was proved to hold for the FPT in the HEE-temperature plane [22]. Based on [20], VDP phase transition of HEE in various AdS black holes have been studied in [23,24,25,26,27,28,29,30,31,32]. All of these works showed that the HEE undergoes the same VDP phase transition as that of the BHE.
Massive gravity theories have been considerable interest recently. One of these reasons is that this alternative theories of gravity could explain the accelerated expansion of the universe without dark energy. The graviton behaves like a lattice excitation and exhibits a Drude peak in this theory. Current experimental data from the observation of gravitational waves by advanced LIGO require the graviton mass to be smaller than the inverse period of orbital motion of the binary system, that is m g < 1.2 × 10 −22 eV /c 2 [33]. Another important reason for the interest in massive gravity is that the possibility of the mass graviton could be help to understand the quantum gravity effect. The first to introduce a mass to the graviton is in [34]. However this primitive linear massive gravity theory contains the so called Boulware-Deser ghosts problem [35] that was solved by a nonlinear massive gravity theory [36,37], where the mass terms are obtained by introducing a reference metric. Recently Vegh proposed a new reference metric to describe a class of strongly interacting quantum field theories with broken 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 a class of Lorentz breaking massive gravity. In the massive gravity, the graviton acquires a mass by Lorentz symmetry breaking, which is very similar to the Higgs mechanism. A review of Lorentz violating massive gravity theory can be found in [41,42]. In this paper, we focus on the study of the VDP phase transition of AdS black holes in Lorentz breaking massive gravity using the holographic HEE. The main motivation of this paper is to explore whether the BHE phase transition can also be described by HEE in Lorentz breaking massive gravity. Firstly, we would like to extend proposals in [20] to study VDP phase transitions in AdS black hole with a spherical horizon in Lorentz-violating massive gravity with the HEE as a probe. What's more, we also would like to check the Maxwell's equal area law and critical exponent of the heat capacity, which are two characteristic quantities in VDP phase transition.
The organization of this paper is as follows. In the next section, we shall provide a brief review of the black hole solution in Lorentz breaking massive gravity firstly. Then we will study the VDP phase transitions and critical phenomena for the AdS black hole in the BHE-temperature plane. In Section 3, we mainly concentrate on the VDP phase transition and critical phenomena in the framework of HEE. The last section is devoted to our discussions and conclusions.

2
Phase transition and critical phenomena of AdS Black Holes in Lorentz breaking massive gravity

Review of AdS Black Holes in Lorentz breaking massive gravity
The four dimensional Lorentz breaking massive gravity can be obtained by adding nonderivative coupling scalar fields to the standard Einstein gravity theory. As a matter field is considered, the theory can be described by the following action [41,42] here the first two terms are the curvature and ordinary matter minimally coupled to gravity respectively, the third term Ψ contains two functions X and Π ij which relate to the four scalar fields, Ξ 0 and Ξ i as When the four scalar fields get a space-time depending vacuum expectation value, the system will break the Lorentz symmetry. What's more, the action can also be taken as a low-energy effective theory with the ultraviolet cutoff scale . Here the scale parameter has the dimensions of mass and is in the order of m g M P l , where m g and M P l are the graviton mass and the Plank mass respectively. The AdS black holes solution can be obtained from the above theory [43,44]. The metric corresponding to the AdS black holes is given by with Here, the four scalar fields, Ξ 0 and Ξ i , for this particular solution are given by in which in which the scalar charge Q is related to massive gravity and the constant α which is determined by the cosmological constant Λ and the graviton mass m g with the relation Λ = 2m 2 g (1 − α 6 ). In this paper we will set α > 1 such that Λ < 0 leading to Anti-de Sitter black holes. The constant λ is a positive integration constant. When λ < 1, the ADM mass of the black hole solution diverges. For λ > 1, the metric approaches the Schwarzschild-AdS black holes with a finite mass M as r → ∞. Thus we set λ > 1 in this paper. The constant γ = ±1. When γ = 1, the black hole only has a single horizon r h , which is the root of the equation f (r h ) = 0. The function f (r) for this case is given in Fig.(1), which is similar to the Schwarzschild AdS black hole. For γ = −1, the black hole is very similar to the Reissner-Nordstrom-AdS black hole. The function f (r) for this case is given in Fig.(2). The black hole event horizon r h is the largest root of the equation At the event horizon, the Hawking temperature and BHE can be written as   The chemical potential in this black hole is We can check the first law of the black hole, which is given by There have been some works to study the thermodynamics and phase transitions of black holes in Lorentz breaking massive gravity [45,46,47,48].

Van der Waals-like Phase transition of Bekenstein Hawking entropy
In this subsection, we focus on the VDP phase transition of BHE. Substituting (10) into (9) and eliminating the parameter r h , one can get the relation between the Hawking temperature T and BHE S of the AdS black holes in massive gravity as This is the state equation of the AdS black hole thermodynamics system in massive gravity. Using (13), we investigate the phase diagram of the AdS black holes in massive gravity.
The temperature T is plotted as a function of the BHE S in Fig.(3) and Fig.(4) for γ = 1 and γ = −1 respectively. In Fig.(3), the temperature is plotted for γ = 1 where only one event horizon exists. This behavior of temperature is very similar to the behavior in the Schwazschild-AdS black hole. That is to say there is a minimum temperature T min which divides the thermodynamics systems into small and large black holes. It is shown that above the minimum temperature, small and large black holes coexist. In fact, this behavior will break for there is a first order transition, which is similar to the Hawking-Page thermodynamic transition in Ref. [49].
In Fig.(4), the temperature is plotted for γ = −1 where event horizon and inner horizon exist. Various values of Q is used to plot the relations between the temperature and horizons. The top curve corresponds to Q = 0. The system of this case is similar to the case γ = 1 described above. When the scalar charge Q increases, the temperature has two turning points. Further increasing of the scalar charge Q brings these two turning points merge to one. It is shown that there exist a critical point Q c . Above the critical point, the curve does not have any turning points. Thus we find that the phase structure is very similar to that of the Van der Waals gas-fluid phase transition. It should be noted that we mainly concentrate on this type of phase structure in this paper. Furthermore using the definition of the specific heat capacity C Q , that is one can see that the specific heat capacity is divergent at the critical point and it is obvious that this phase transition is a SPT. At this critical point, the critical charge Q c and critical entropy S c can be obtained by the following equations After some calculation and using (13), Q c , S c and the corresponding T c can be also got as Obviously, these critical parameters depend only on the internal parameters of the system λ and Λ. One can also see that T c , S c increase with λ and Q c decrease with λ as shown in Figure (5), S c , Q c increase with Λ and T c decrease with λ as shown in Figure (6). So the results show that the parameter λ promote the thermodynamic system to reach the stable state.
For the FPT, we will also check whether Maxwell's equal area law holds in this thermodynamic system. As is known to all, the first order transition temperature T * plays an crucial role for the Maxwell's equal area law. Thus in order to get T * , we first plot the curve about the free energy with respect to the temperature T , where the free energy is defined by F = M − T S. The relation between F with T is plotted in Figure (7). One can see that there is a swallowtail structure, which corresponds to the unstable phase in Figure  (8). The non-smoothness of the junction implies that the phase transition is a FPT. The critical temperature T * is apparently given by the horizontal coordinate of the junction. From Figure (7), we get T * = 0.2750. Substituting this temperature T * into (13), one can obtain three values of the entropy, S 1 = 0.1563, S 2 = 0.5277 and S 3 = 1.394. With these values, we can now check the Maxwell's equal area law After some calculation, we find that both the left and right of (19) equal to 0.3404 exactly. Thus, our results show that the Maxwell's equal area law is satisfied in this background.
For the SPT, we will study the critical exponent associated with the heat capacity definition in (14). Near the critical point, expanding the temperature as the very small amount S − S c , we find Using (15), the second and third terms vanish. Then using (13) , (16) and (17), we get With the definition of the heat capacity (14), we further get C Q ∼ (T − T c ) −2/3 . So one can find the critical exponent of the heat capacity is −2/3, which is the same as the one from the mean field theory in Van der Waals gas-fluid system.
In this section, our target is to explore whether the HEE have the similar VDP phase structure and critical phenomena as that of the BHE in massive gravity. For simplicity, here we only consider the case γ = −1. Now we will investigate whether there is VDP phase transition in the HEE -temperature phase plane. Firstly, we review some basic knowledge about holographic HEE. Detailed introduction of HEE can refer to Refs. [18,19]. For a given quantum field theory described by a density matrix ρ, HEE for a region A and its complement B is where ρ A is the reduced density matrix. However,it is usually not easy to get this quantity in quantum field theory. Fortunately according to AdS/CFT correspondence, Refs. [18,19] propose a very simple geometric formula for calculating S A for static states with the area of a bulk minimal surface as ∂A, that is where γ A is the codimension-2 minimal surface according to boundary condition ∂γ A = ∂A. Subsequently using the definition (23), we will calculate the HEE and study the corresponding phase transition. It is noted that the space on the boundary is a spherical in the AdS black hole in massive gravity and the volume of the space is finite. Thus in order to avoid the HEE to be affected by the surface that wraps the event horizon, we will choose a small region as A. More precisely, as done in [23,25,26,27], we choose the region A to be a spherical cap on the boundary given by ϕ ≤ ϕ 0 . Here the area can be written as where r = dr dϕ . Then according to the Euler-Lagrange equation, one can get the equation of motion of r(ϕ), that is After using the boundary conditions r (0) = 0, r(0) = r 0 , we can get the numeric result of r(ϕ).
It is worth noting that the HEE should be regularized by subtracting off the HEE in pure AdS, because of the values of the HEE in the equation (23) is divergent at the boundary. Now let's label the regularized HEE as δS. Here we choose the size of the boundary region to be ϕ 0 = 0.16 and set the UV cutoff in the dual field theory to be r(0.159). The numeric results for λ = 2.4 are shown in Figure(9). One can see that for a given scalar charge Q, the relation between the HEE and temperature is similar to that between the black hole BHE and temperature. That is to say , the AdS black holes thermodynamic system with the HEE undergos the FPT and SPT one after another as the scalar charge Q increase step by step.
For the FPT of the HEE, we now check whether the Maxwell's equal area law satisfies. Firstly, for the case λ = 2.4, ϕ 0 = 0.16, we get an interpolating function of the temperature T (δS) using the data obtained numerically. At the first order phase transition point, we get the smallest and largest roots for the equation T (δS) = 0, that are δS 1 = 7.07492, δS 1 = 7.07599. Then using these values and the equal area law we find the left equals 0.00029486 and the right equals 0.00029476. Obviously both the left and the right are equal within our numerical accuracy, where the relative error is less than 0.034%   (14), one can also define an specific heat capacity for the HEE as Then providing a similar relation of the critical points that in (15) is also working and using (27) The result show that the slope is around 3 with less than relative error 4%, which is consistent with that of the BHE. Then one can find the critical exponent of the specific heat capacity C Q is also approximately −2/3. That is to say,the HEE has the same SPT behavior as that of the BHE. Both of them are consistent with the result in the mean field theory of Van der Waals gas-fluid system.

Conclusions
In this paper, we have investigated the VDP phase transition with the use of HEE as a probe. Firstly, we investigated the phase diagrams of the BHE in the T − S phase plane and find that the phase structure depends on the scalar charge Q and the parameter γ of the AdS black holes in this massive gravity. For the case that γ = 1 or Q = 0, we find there always exists the Hawking-Page like phase transition in this thermodynamic system. While for the case γ = −1, we found for the small scalar charge Q, there is always an unstable black hole thermodynamic system interpolating between the small stable black hole system and large stable black hole system. The thermodynamic transition for the small hole to the large hole is first order transition and the Maxwell's equal area law is valid. As the scalar charge Q increases to the critical value Q c in this space time, the unstable black hole merges into an inflection point. We found there is a SPT at the critical point. When the scalar charge is larger than the critical charge, the black hole is stable always. That is to say, we found that there exists the VDP gas-fluid phase transition in the T − S phase plane of the AdS black hole in massive gravity. The more interesting thing is that we found the HEE also exhibits the VDP phase structure in the T − δS plane when γ = −1 and the scalar charge Q = 0. In order to confirm this results, we further showed that the Maxwell's equal area law is satisfied and the critical exponent of the specific heat capacity is consistent with that of the mean field theory of the VDP gas-fluid system for the HEE system. These results show that the phase structure of HEE is similar to that of BHE and the HEE is really a good probe to the phase transition of AdS black holes in Lorentz breaking massive gravity. This also implies that HEE and BHE exhibit some potential underlying relationship.