Ratio of critical quantities related to Hawking temperature-entanglement entropy criticality

We revisit the Hawking temperature$-$entanglement entropy criticality of the $d$-dimensional charged AdS black hole with our attention concentrated on the ratio $\frac{T_c \delta S_c}{Q_c}$. Comparing the results of this paper with those of the ratio $\frac{T_c S_c}{Q_c}$, one can find both the similarities and differences. These two ratios are independent of the characteristic length scale $l$ and dependent on the dimension $d$. These similarities further enhance the relation between the entanglement entropy and the Bekenstein-Hawking entropy. However, the ratio $\frac{T_c \delta S_c}{Q_c}$ also relies on the size of the spherical entangling region. Moreover, these two ratios take different values even under the same choices of parameters. The differences between these two ratios can be attributed to the peculiar property of the entanglement entropy since the research in this paper is far from the regime where the behavior of the entanglement entropy is dominated by the thermal entropy.

framework of d-dimensional charged AdS black hole spacetime. The motivations are as follows. Firstly, we are curious about whether the analogous ratio of critical quantities related to the Hawking temperature-entanglement entropy criticality is also universal. In other words, does it depend on the parameters? Probing the universal ratios of critical quantities of charged AdS black holes has its own right. The analogy between charged AdS black holes and van der Waals liquid-gas system has gained extensive attention ever since the famous work [20,21]. The ratio Pcvc kTc is a universal number for all van der Waals fluids in classical thermodynamics, motivating us to searching for the universal ratios for charged AdS black holes. Secondly, we are interested in both the similarities and differences (if any) between the ratio for the Hawking temperature-entanglement entropy criticality and that for the Hawking temperature-thermal entropy criticality. On the one hand, the similarities will help further understand the close relation between the entanglement entropy and the Bekenstein-Hawking entropy. On the other hand, the differences may shed light on some yet unknown physics.
The organization of this paper is as follows. Sec.2 devotes to a short review of three different kinds of criticality of d-dimensional charged AdS black holes. In Sec.3, we will revisit the Hawking temperature−entanglement entropy criticality of the d-dimensional charged AdS black hole and investigate the analogous ratio of critical quantities for various cases where different parameters are chosen as the variable respectively. In the end, Sec.4 devotes to conclusions.

A short review of criticality of charged AdS black holes
The metric of the d-dimensional (d > 3) charged AdS black hole reads l is the characteristic length scale which is related to the cosmological constant Λ through . Parameters m and q can be identified with the ADM mass and the electric charge of the black hole as follows [20] where the volume of the unit d-sphere ω d can be obtained via 2π The Hawking temperature, the entropy and the electric potential of the d-dimensional (d > 3) charged AdS black hole have been reviewed in Ref. [22] as (2.7) Ref. [22] also investigated its P − V criticality and obtained the following critical quantities where κ = d−2 4 . Note that the cosmological constant has been identified as the thermodynamic pressure through the definition P = −Λ/(8π) and the specific volume v is related to the horizon radius r + through r + = κv. The ratio Pcvc Tc was shown to be 2d−5 4d−8 , which is independent of the parameter q [22].
In our recent work [19], we studied the T − S criticality of these black holes and probed the possible universal ratio for T − S criticality. Here, we listed the results for the d-dimensional (d > 3) charged AdS black hole. (2.11) (2.14) Note that the ratio TcSc Qc (S c denotes the critical thermal entropy) is independent of the parameter l although the relevant critical quantities S c , Q c , T c all depend on l. In this sense, the ratio TcSc Qc can be viewed as a universal ratio for the T − S criticality. And the value of this ratio differs from that of Pcvc Tc .
Ref. [23] studied Q − Φ criticality of these black holes and obtained where n = d − 1. They argued that the ratio ΦcQc Tc depends on both Λ and n and is not universal. In our recent work [19], we construct two ratios for the Q − Φ criticality as follows (2.20) Note that these two ratios only depend on n.
3 Ratio of critical quantities for Hawking temperature−entanglement entropy criticality In the former section, we review the ratios of critical quantities for three different kinds of criticality. Namely, P − V criticality, T − S criticality and Q − Φ criticality. In all these cases, there exist universal ratios. Considering the close relation between the entanglement entropy and the Bekenstein-Hawking entropy, we will revisit the Hawking temperature−entanglement entropy criticality of the d-dimensional charged AdS black hole and probe the analogous ratio of critical quantities. Suppose Σ is the codimension-2 minimal surface with boundary condition ∂Σ = ∂A, the entanglement entropy S A between the region A and its complement can be defined holographically as [1][2][3] where G N is the Newton's constant and Area(Σ) denotes the area of Σ (a minimal surface anchored on ∂A).
To avoid dealing with the phase transition between connected and disconnected minimal surfaces, one can consider a spherical cap on the boundary delimited by θ ≤ θ 0 as Ref. [8] did. Then the minimal surface can be parametrized by r(θ). Utilizing Eq.(3.1), the entanglement entropy can be derived as where L can be read from Eq.(3.2) and the boundary condition can be chosen as r(0) = r 0 , r ′ (0) = 0.
With the numerical solution of r(θ), one can calculate the holographic entanglement entropy by utilizing Eq. (3.2). Note that the result should be regularized by subtracting the entanglement entropy in pure AdS S 0 with the same boundary region to avoid the divergence. And the regularized entanglement entropy δS reads S A − S 0 . The analytic result for r AdS (θ) corresponding to S 0 was presented as [24,25] r AdS (θ) = l cos θ cos θ 0 Here, we are interested in the ratio TcδSc Qc , which is analogous to the ratio TcSc Qc . Note that δS c denotes the regularized entanglement entropy at the critical point. Specifically, we will study the cases where l, d and θ 0 are chosen as the variable respectively to probe whether the ratio TcδSc Qc is universal. Firstly, we fix d = 4 and θ 0 = 0.2 and let l vary from 0.1 to 2. The cutoff θ c is chosen as 0.199. Since the focus of our research is the critical quantities, we focus on the case Q = Q c and ignore the cases Q < Q c and Q > Q c . The T − δS curves corresponding to different choices of l are depicted in Fig. 1(a)-1(f) while the relevant critical physical quantities are listed in Table 1. With the increasing of l, the critical quantities Q c , S c and δS c increase while T c decreases. The ratio TcSc Qc is almost unchanged, just as the analytic result we obtained before showed [19]. The ratio TcδSc Qc changes slightly. The mean value is 0.00271188 while the standard deviation turns out to be 0.00015822. The standard deviation is so small that one can also conclude that the ratio TcδSc Qc does not depend on the characteristic length scale l.  Table 2. It can be clearly witnessed that both the ratio TcSc Qc and TcδSc Qc change with n. With the increasing of n, the ratio TcSc Qc increases while the ratio TcδSc Qc decreases. So these ratios are dimensionality dependent.   Table 3. With the increasing of θ 0 , the ratio TcSc Qc remains unchanged while TcδSc Qc increases. It is not difficult to explain this result considering the observation that δS c increases with θ 0 while S c is not affected.

Concluding Remarks
To summarize, we revisit the Hawking temperature−entanglement entropy criticality of the d-dimensional charged AdS black hole and concentrate our attention on the ratio of critical quantities. Specifically, we calculated numerically the ratio TcδSc Qc for the cases where l, d and θ 0 are chosen as the variable respectively.
Comparing the results of this paper with those of the ratio TcSc Qc [19], one can find that both the similarities and differences exist. These two ratios are independent of the characteristic length scale l and dependent on the dimension d. These similarities further enhance the relation between the entanglement entropy and the Bekenstein-Hawking entropy.
Contrary to the ratio TcSc Qc , the ratio TcδSc Qc relies on the size of the spherical entangling region. Moreover, these two ratios take different values under the same choices of parameters. Note that we focus on the entanglement for a subsystem whose volume is very small.
In this sense, the research in this paper is far from the regime where the behavior of the entanglement entropy is dominated by the thermal entropy. So the differences between these two ratios may be attributed to the peculiar property of the entanglement entropy. And the deep physics behind it certainly deserves more attention in the future research.