Holographic entanglement entropy and generalized entanglement temperature

In this work we study the flow of holographic entanglement entropy in dimensions $d \geq 3$ in the gauge/gravity duality set up. We observe that a generalized entanglement temperature $T_g$ can be defined which gives the Hawking temperature $T_H$ in the infrared region and leads to a generalized thermodynamics like law $E= \left(\frac{d-1}{d}\right)T_g~S_{REE}$, which becomes an exact relation in the entire region of the subsystem size $l$, including both the infrared ($l\rightarrow\infty$) as well as the ultraviolet ($l\rightarrow 0$) regions. Furthermore, in the IR limit, $T_g$ produces the Hawking temperature $T_H$ along with some correction terms which bears the signature of short distance correlations along the entangling surface. Moreover, for $d\geq 3$, the IR limit of the renormalized holographic entanglement entropy gives the thermal entropy of the black hole as the leading term, however, does not have a logarithmic correction to the leading term unlike the BTZ black hole ($d=2$) case. The generalized entanglement temperature $T_g$ also firmly captures the quantum mechanical to thermal crossover in the dual field theory at a critical value $l_c$ of the subsystem size in the boundary which we graphically represent for $AdS_{3+1}$ and $AdS_{4+1}$ black holes. We observe that this critical value $l_c$ where the crossover takes place decreases with increase in the dimension of the spacetime.


Introduction
The von Neumann entropy or the entanglement entropy (EE) is one of the fundamental and wellstudied entities of quantum physics [1]. It is non-local in nature and represents how the degrees of freedom of two subsystems are correlated. This bipartite scenario can be extended further to a multipartite scenario [2]. In the bipartite case, there exists two subsystems namely A and B. The mathematical definition of the EE is then given by where ρ A is the reduced density matrix of the subsystem A constructed by tracing out the degrees of freedom of the subsystem B from the total density matrix (ρ total ) of the system as The EE has been a matter of great interest in quantum information theory as it captures the amount of information loss suffered due to spatial division of the concerned system. It has also been shown a strong connection exists relating the quantum information with the thermal entropy of a system [3]. This led to the computation of EE in the field theoretical set up. However, The computation of EE in field theoretical scenario is notoriously difficult as it is divergent in nature and demands a number of symmetries for analytical computations. In a field theory with conformal group symmetry, the EE has been computed in 1+1-dimensions for various topologies of the subsystems [4]. However, the computation of EE in a interacting field theory or a higher dimensional free field theory is not quite clear [5]. This problem was resolved by the AdS/CF T correspondence in a satisfactory way. [6], [7]. The interesting connection between a weakly coupled gravitational theory and a strongly coupled quantum field theory (QFT), provided by the AdS/CFT correspondence has been a matter of great interest for the last two decades [8], [9]. This correspondence has led to the concept of holographic computation of EE via the Ryu-Takayanagi prescription [10]. The holographic computation of EE states that the holographic entanglement entropy (HEE) of a asymptotically AdS d+1 spacetime is equal to EE of a d-dimensional CFT which lives at the boundary of the classical bulk theory. The formula for the HEE is given by [11] where γ A represents the static minimal surface extending into the bulk due to the presence of the subsystem A at the boundary of the assymptotically AdS spacetime. Interestingly, this is quite similar to the famous Bekenstein-Hawking area law for black hole entropy [12]. The computation of HEE under different circumstances has been a well studied subject thereafter [13]- [20]. The computation of HEE has also been extended to time-dependent scenarios. This is the covariant HEE prescription [21]. Interestingly in the UV limit, the HEE gives a thermodynamics like law. This has been named as entanglement thermodynamics in the literature which in turn gives a entanglement temperature T ent [22]. However, to define the entanglement temperature one has to neglect the higher order terms in the temperature expansion of the HEE [23]. This restricts the definition of T ent only in the UV domain ( l z h 1) of the theory. One must note that the definition of T ent is also not quite clear as it is quite different from our known notion of a thermodynamic temperature and it is inversely proportional to the the subsystem size at the boundary. This invokes the thrust to look out for a real thermodynamical notion of a temperature in the theory which shall have a scale independent, global defintion unlike T ent , as this would provide a deep understanding about the microscopic origin of thermodynamics. Moreover, this raises another question about a possible connection existing between the entanglement temperature and a real thermodynamical temperature. The usual approach to the computation of the HEE has been heavily influenced by the UV and IR domain of the theory determined by the size of the boundary subsystem. However, to observe the evolution of the HEE with the subsystem size correctly one has to work with the entire domain of subsystem size l. In case of a (1 + 1)-dimensional conformal field theory (CFT) this is quite simple as the computed results of EE has a well defined closed analytical form. This leads to some fascinating results by providing us a connection between EE and Hawking entropy along with the Hawking temperature [24], but the story drastically changes in higher dimensions due to technical computations. In this work we observe the evolution of HEE with the subsystem size in dimensions d ≥ 3 in both UV and IR domain of the theory by incorporating all the terms in the expansion. Such a study was carried out earlier in [24] in the case of the 2 + 1-dimensional BTZ black hole which corresponds to a 1 + 1-dimensional CFT. We define a generalized entanglement temperature T g using the first law of black hole thermodynamics (E = d−1 d T H S BH ) and we show that T g produces the exact Hawking temperature in the IR domain of the dual field theory. We specifically represent this for AdS 3+1 and AdS 4+1 black holes. These interesting results indicate that the entanglement temperature can evolve to a real thermodynamic temperature satisfying a real thermodynamical law. In the ultraviolet domain, T g gives rise to an entanglement temperature T e that once again satisfies the thermodynamics like law REE . It should be noted that the generalized entanglement temperature defined in this paper should not be identified as the thermodynamic temperature of the boundary CFT. The thermodynamic temperature of the CFT (which does not flow with the subsystem size l) corresponds to the Hawking temperature of the black hole and arises from the IR limit of generalized entanglement temperature T g . We would like to emphasize that the definition of the generalized entanglement temperature given in this paper is different from the definition of the entanglement temperature given in [22] upto a constant factor. It should be noted that the definition given in [22] agrees with the Hawking temperature only upto a constant multiplicative factor. We have demonstrated this explicitly for the AdS 3+1 and BTZ black holes. The paper is organized as follows. In section 1, we holographically compute the EE of both the ground state and the excited state of a d-dimensional field theory by incorporating the Ryu-Takayanagi conjecture for a strip like subsystem. We then produce a divergence free entanglement entropy, namely, renormalized entanglement entropy S REE . In section 2, we define a generalized entanglement temperature T g by utilizing the first law of black hole thermodynamics and the result of S REE . We also study the behavior of T g in the UV and IR domain in arbitrary dimensions. In section 3, we concentrate on AdS 3+1 and AdS 4+1 black holes to graphically represent the variation of the inverse generalized entanglement temperature β g with respect to the subsystem size l. We then observe the variation of the dimensionless quantity Finally, we discuss the results that we have obtained in section 4. The paper ends with an appendix.

Renormalized holographic entanglement entropy
In this section, We shall compute the renormalized HEE for the AdS Schwarzschild black hole in d + 1dimensions. To begin with we write down the metric of the d + 1-dimension AdS Schwarzschild black hole This geometry is then dual to a d-dimensional CFT at the boundary by the AdS/CFT correspondence. The expression for the lapse function f (z) contains the information about the excitation properties of the boundary CFT. The ground state in the d-dimensional boundary CFT corresponds the lapse function f (z) = 1 and the metric represents pure AdS spacetime in (d + 1)-dimensions: Similarly, a thermally excited state in the d-dimensional boundary CFT leads to a deformed AdS spacetime. More precisely, the excited state in the CFT corresponds to an assymptotically AdS space with a black hole with the lapse function where z h is the event horizon radius of the black hole. The metric is the well known AdS d+1 Schwarzschild black hole geometry and is given by The Hawking temperature of the black hole reads We now proceed to holographically calculate the entanglement entropy of the excited state of the ddimensional boundary CFT.

Holographic entanglement entropy of the AdS d+1 Schwarzschild black hole
To begin our analysis, we first make the choice of subsystem A at the boundary. This is a strip whose geometry is specified as − l 2 < x 1 < + l 2 and − L 2 < x 2,3,4,..,d−1 < + L 2 . This specifies the volume of the subsystem at the boundary field theory to be The thermal entropy of the boundary field theory which is the amount of Bekenstein-Hawking entropy contained in the above mentioned boundary subsystem volume is given by [24] where, G d+1 is the Newton's gravitational constant in (d + 1)-dimensions. On the First step, we proceed to compute the HEE and the corresponding subsystem size in terms of the bulk coordinates of the metric (7). Here we keep L, representing the width of the strip is fixed but the length of the strip denoted by l can vary from zero to infinity. We parametrize the static minimal surface The holographic entanglement entropy (S E ) can now be computed from the RT formula [10] z t is the turning point in the bulk satisfying the condition dz dx 1 | z=zt = 0. It is the maximal value of z on minimal surface γ A . The position of the turning point z t with respect to the event horizon radius z h holographically determines the UV ( zt z h 1) and IR ( zt z h ≈ 1) domains of the concerned field theory from the gravity side of the story. The length (l) of the boundary subsystem A in terms of the turning point z t reads In the above expressions we have set the AdS radius R = 1.

Holographic entanglement entropy of the pure AdS d+1 spacetime
We move onto compute the holographic renormalized entanglement entropy of the AdS d+1 Schwarzschild black hole. This we shall do by substracting the holographic entanglement entropy of the pure AdS d+1 spacetime from the holographic entanglement entropy of the AdS d+1 Schwarzschild black hole. The holographic entanglement entropy (S G ) and subsystem size l of the pure AdS d+1 spacetime reads [11] and where z (g) t is the turning point corresponding to ground state of the boundary CFT. The relation between the turning point of the AdS d+1 Schwarzschild geometry and that in the pure AdS d+1 geometry is as follows We now define a divergence free holographic entanglement entropy. This has been named as the renormalized holographic entanglement entropy S REE in the literature [24], in 2 + 1-dimensions. This is obtained by substracting the ground state entanglement entropy from the entanglement entropy for the excited state thereby removing the l independent UV divergence term which arises due to short distance correlation along the entangling surface of the dual field theory. The renormalized holographic entanglement entropy in d + 1-dimensions therefore reads 1 These expressions are valid for any dimension d ≥ 3.

A generalized entanglement temperature T g
In this section we proceed to define a generalized entanglement temperature in order to understand the behaviour of the holographic renormalized entanglement entropy (S REE ) for d-dimensional CFT along the entire domain of l. It has been realized in the case of the BTZ black hole that applying the first law of black hole thermodynamics, the UV behaviour of S REE leads to an entanglement temperature that is quite different from the Hawking temperature of the black hole. Hence, a generalized entanglement temperature was defined in [24] which captured both the UV and the IR regions in the d = 2 case corresponding to the BTZ black hole. We shall do this for the AdS d+1 Schwarzschild black hole in this investigation.
The first law of black hole thermodynamics reads [25] which leads to the following change in the internal energy due to the thermal deformation of the pure AdS d+1 spacetime This internal energy represents the amount of information stored in the boundary volume. This information behaves in a quantum mechanical way in the UV region l z h 1 and in a thermal way in the IR region l → ∞. Note that eq.(s) (8,10,19) yields the relation The expression for the internal energy given in eq.(19) along with the above thermodynamic relation, motivates us to define a generalized entanglement temperature T g as Substituting the expressions for S REE from eq.(17) and E from eq. (19), we obtain the expression of the generalized entanglement temperature T g for the AdS d+1 black hole to be This generalized entanglement temperature T g interpolates between the entanglement temperature in the UV region and a real thermodynamical temperature in the IR region. In the following section we shall verify this statement in spatial dimensions d ≥ 3. Note that we have defined the generalized entanglement temperature by putting a factor of d−1 d in front of S REE E . The factor has been chosen so that T g goes to the Hawking temperature of the black hole in the IR (l → ∞) limit. We also stress that this definition of the generalized entanglement temperature is different from that given in [22], namely, T ent ∆S = ∆E (23) where ∆E is the increased amount of energy in the subsystem, given by The above definition for the entanglement temperature does not lead to the exact Hawking temperature in the IR limit but is equal to the Hawking temperature upto a multiplicative constant factor. This we shall now demonstrate for the SAdS 4 black hole case. For the SAdS 4 black hole spacetime (d = 3), the renormalized entanglement entropy reads Using the definition for the entanglement temperature given in eq.(23), and ∆S ≡ S REE (given in eq.(25)) with the expression for ∆E in d = 3 spatial dimensions we obtain the entanglement temperature to be This in the IR limit yields This clearly shows that the definition in [22] for the entanglement temperature yields the Hawking temperature only upto a multiplicative constant factor in the IR limit. We shall further substantiate our claim by discussing the BTZ black hole case in the appendix of this paper. We shall see in the next subsection that the definition of the generalized entanglement temperature T g given in this paper (eq.(21)) gives the exact Hawking temperature in the IR limit and also makes the entanglement temperature in the UV limit satisfy the thermodynamic like relation where T e = T g in the UV limit. This implies that the generalized thermodynamics like law becomes an exact relation in the entire domain of the subsystem of length l.
The analogue of the above relation for the BTZ black hole case reads [24] Here also we find that there is a factor of 1/2 on the right hand side of the above equation. This factor ensures T g to reproduce the exact Hawking temperature of the BTZ black hole in the IR (l → ∞) limit.

Behaviour of β g in the IR region
In this subsection, we shall study the behaviour of the generalized entanglement temprerature T g in the IR region of the theory. Before going to the IR regime, we shall simplify the expression of S REE given in eq. (17). This gives where we have used the identity Γ[p + 1] = pΓ[p]. By using eq.(13) we obtain Substituting this in eq.(32) gives Now we shall take the IR limit in the above expression by taking the limit z t → z h . In this limit, the infinite series in the above equation goes as ≈ 1 n 2 ( zt z h ) nd for large n, which implies that the series converges in the z t → z h limit. The expression for S REE in the IR limit reads The above result can be recast in the form is the area of the of the near horizon part of the static minimal surface in the IR limit and ∆ 1 and ∆ 2 is given by .
Substituting eq.(36) in eq.(21), we obtain the behaviour of T g in the IR region to be It is interesting to notice that in the IR regime the leading term of the generalized entanglement temperature β −1 g is equal to the inverse of the Hawking temperature. The rest of the terms are correction terms and subsystem size dependent. In the large l limit, these correction terms are smaller in amplitude compared to the Hawking temperature T H which is l independent and the correction terms are inversely proportional to l. For d = 3 corresponding to the AdS 3+1 black hole eq(s).(36), 39 read where Similarly, for d = 4 corresponding to AdS 4+1 black hole eq(s).(36), 39 read where S th = 1 4G Note that although in the IR limit, we use the fact z t ≈ z h , however it is known that in the high temperature limit (IR limit), the static minimal surface approaches the event horizon but always stays at a finite distance behind the event horizon z h [26]. The above expressions show that in the IR region the leading term of S REE is the amount of Bekenstein-hawking entropy contained in the boundary subsystem volume given in eq.(10).

Behaviour of β g in the UV region
In this subsection, we look at the UV behaviour of the generalized entanglement temperature T g . In the UV region ( zt z h 1, l z h 1) the renormalized holographic entanglement entropy given in eq.(17) reads The subsystem size in the UV limit reads Inverting eq.(47), we obtain Similarly, for d = 4 corresponding to AdS 4+1 black hole, eq(s).(49), 51 read in the boundary N = 4 SYM theory starts to thermalize. This plot shows that in higher dimension the ther-boundary of the AdS 4+1 black hole spacetime. The flow of β g firmly shows quantum mechanical to thermal crossover of the theory. This crossover takes place at a particular size l c of the subsystem. Interestingly, we observe that l c decreases with increase in the spacetime dimensions. We find that l c ≈ 2.031z h for d = 3 and l c ≈ 1.45z h for d = 4 which is less than the value obtained for the d = 2 case, namely, l c ≈ 7.019 [24]. This clearly indicates that the critical length decreases with the increase in the spacetime dimensions which in turn implies that the thermalization at the boundary CFT takes place much earlier in the subsystem size l. Furthermore, we also observe that the purely microscopic entanglement entropy evolves to a macroscopic thermal entropy revealing the microscopic origin of the real thermodynamic law in the context of black hole thermodynamics. We believe this connection between non-classical, microscopic entanglement thermodynamics and real thermodynamics is quite fascinating as it directly helps us to understand the microscopic origins of a macroscopic entity. The scenario discussed in this paper in principle can be extended for hyperscale violating spacetimes and also for a boundary field theory with a chemical potential. We leave these as future works.