Some Aspects of Holographic Entanglement of Purification

We consider the entanglement of purification (EoP) in holographic QFTs, which admit a dual Einstein gravity description. It is proposed that this quantity is equal to the minimal area of the entanglement wedge cross section. Using this prescription, we examine in detail the low and high temperature corrections to this quantity and show that it obeys the area law even in the finite temperature. We also study EoP in nonrelativistic field theories with nontrivial Lifshitz and hyperscaling violating exponents. The resultant EoP is an increasing function of the dynamical exponent due to the enhancement of spatial correlations between subregions for larger values of $z$. We find that EoP is monotonically decreasing as the hyperscaling violating exponent increases. We also obtain EoP for an entangling region with singular boundary in a three dimensional field theory and find a universal contribution where the coefficient depends on the central charge. Finally, we verify that for higher dimensional singular regions the corresponding EoP obeys the area law.


Introduction
In recent years, the study of quantum information concepts such as entanglement using gauge/gravity correspondence has been an active area of research. In particular, improving the duality by adding an explicit relation between a measure of entanglement in the boundary theory and a geometric entity which lives in the bulk spacetime is of great interest. In this context, the Ryu- Takayanagi (RT) proposal is a remarkably simple prescription to compute entanglement entropy (EE) for the QFTs dual to Einstein gravity. Consider a spatial region A in the boundary field theory, the corresponding entanglement entropy between A and its complement is given by [1] A∪B Γ dis. Figure 1: Schematic configurations for computing S A (left) and S A∪B (right). Note that in the latter case we have two different extremal configurations denoted by Γ con. A∪B and Γ dis. A∪B corresponding to connected and disconnected RT surfaces respectively. information (HMI) given as follows 1

A∪B
(1. 2) The mutual information is free from UV divergences and subadditivity guarantees that I(A, B) ≥ 0.
In [7] it was shown that HMI exhibits a phase transition which is due to the competition between two different configurations for computing S AB . Indeed, in order to find this contribution we should consider two minima corresponding to a connected configuration and to a disconnected one (see figure 1). At small distances the connected configuration has the minimal area, while for large separations the RT surface changes topology and the disconnected configuration is favored. Hence using eq.(1.2) the HMI vanishes in the latter case. It is worth to mention that HMI phase transition is a feature of large N limit of quantum field theory and considering O( 1 N ) corrections changes this picture [8]. Besides the already mentioned case of the HEE and HMI, there are many attempts to construct a holographic prescription for other information measures, e.g., relative entropy [9], quantum information metric [10] and computational complexity [11,12]. However, in this paper, we focus on another concept that has recently entered this discussion which is entanglement of purification (EoP) and on its conjectured holographic dual [13,14]. Consider a bipartite system with Hilbert space equal to the direct product of two factors, i.e., H = H A ⊗ H B and let ρ AB be a density matrix corresponding to a mixed state on H. It is a well known fact that by adding auxiliary degrees of freedom to H one can construct a pure state |ψ out of ρ AB such that ρ AB = tr A B |ψ ψ| and |ψ ∈ H AA ⊗ H BB . Indeed, this procedure is not unique and one can find different purifications for a given mixed state. Now following [15], the EoP is defined as where ρ AA = tr BB |ψ ψ| and the minimization is taken over any |ψ . The EoP is a measure of correlation between A and B and reduces to EE for pure states. Considering a general quantum system, the EoP is subject to the following inequalities 2 Recently [13,14] made a proposal for the calculation of EoP of the boundary field theory in the context of holography. Given a particular spatial region composed of two components in the boundary theory, their prescription for the EoP is as follows where Σ min AB is the minimal cross sectional area of the corresponding entanglement wedge (see Fig.2). 3 Using this prescription, it was shown that, the resultant quantity obeys all the properties of EoP [13,14]. Also it was shown that, keeping the geometry of A and B fixed while their separation increases, the holographic EoP has a phase transition such that E P = 0 when the two regions are distant enough. This behavior which is similar to the phase transition of HMI is due to the competition between two different configurations for the entanglement wedge. Despite the fact that in large distance limit, HMI vanishes continuously, the EoP experiences a discontinuous transition.
In [19], EoP for a 2-dimensional scalar theory was studied where its behavior qualitatively agrees with the conjectured holographic proposal. Also using a generalization of the above proposal to time dependent backgrounds, [14,20] studied the evolution of EoP after a quantum quench in the dual field theory. Related investigations attempting to better understand EoP both in the field theory and holography have also appeared in [21][22][23][24][25][26][27][28][29][30][31].
The aim of this paper is to more investigate the holographic aspects of EoP in field theories dual to Einstein gravity. We will study the phase transition of EoP in a relativistic theory at finite temperature and find the low and high temperature expansion of this quantity. We show explicitly that EoP obeys an area law scaling even in finite temperature. We also study the transition of EoP in a non-relativistic QFT with nontrivial dynamical and hyperscaling violating exponents, i.e., z and θ. Moreover, we investigate the properties of EoP for nonsmooth entangling regions where the boundary contains conical singularity. Considering a simple configuration for the subregions, we find a universal contribution to EoP due to the presence of corner in a four dimensional field theory when the subregions coincide. This paper is organized as follows. In section 2, after a short review on HEE and HMI for a strip entangling region at finite temperature, we investigate the corresponding EoP in different dimensions and obtain analytical results at low and high temperature limits. The role of dynamical Figure 2: Left: Schematic configuration for computing EoP where the entanglement wedge (shaded region) is connected. In this case EoP is proportional to the area of Σ min AB . Right: For small entangling regions where the entanglement wedge is disconnected and Σ min AB becomes empty the corresponding EoP vanishes. and hyperscaling violating exponents in the phase transition of EoP is discussed in section 3. In section 4, we study the corner contributions to EoP considering a union of kinks and creases in four and higher dimensions. We conclude with a discussion of our results, as well as possible future directions, in section 5. We relegate some details of the computations to the appendix.

EoP at Finite Temperature in Relativistic Theories
In this section we study the finite temperature contribution to the Eop for holographic theories dual to Einstein gravity. We begin by reviewing the calculation of the finite temperature corrections to HEE and HMI using a systematic expansion, which was originally performed in [32,33]. 4 Then applying this method allows us to evaluate the thermal corrections to EoP for a straight belt entangling region in section 2.2.
The bulk geometry will be a (d + 2)-dimensional AdS black brane in Poincare coordinates where r 0 is the horizon radius and L is the AdS radius. In the following without loss of generality we set L = 1. From eq.(2.1), one obtains that the temperature and thermal entropy density are given by (2.2) Figure 3 shows the entangling regions that we consider for computing HEE and HMI. The straight 4 On the CFT side thermal corrections to EE is computed in [34,35]. In the latter case, we just demonstrate the connected configuration where the HMI is non-zero.
belt entangling region can be parametrized as where we assume H .

Low and High Temperature Behavior of HEE and HMI
In this section we review the computation of HEE for strip entangling region in the low and high temperature limit. This analysis has been done in [32] for HEE and generalized to HMI in [33].
Here we present the main steps and fix our notations. 5 Employing the RT prescription and using eq.(2.1), the corresponding HEE functional is given by Extremizing the above expression yields the equation of motion for x(r), however, since there is no explicit x(r) dependence, the corresponding momentum is a conserved quantity. Therefore we find the following first integral where r t is the turning point of the minimal hypersurface. Using the above expression the relation 5 The following analysis does not apply to three dimensional bulk geometry, i.e, d = 1. We will come back to this case in section 2.2.1. between and r t is given by On the other hand plugging eq.(2.5) back into eq.(2.4), we find (2.7) The above integrals can be carried out analytically in d = 1 case. Hence before going to general dimensions, let us first consider this special case which corresponds to a three dimensional bulk geometry.

HEE and HMI in d = 1
For d = 1 the minimal surface is a spacelike geodesic whose length can be expressed analytically in closed form [36], which enables us to directly extract its temperature behavior in various regimes.
In this case eqs.(2.6) and (2.7) become It is straightforward to evaluate these quantities and to produce the result where c = 3 2G N denotes the central charge of dual two dimensional CFT. We can make use of the above expression to find the low and high temperature behavior of HEE as follows This shows that the thermal fluctuations increases HEE, as expected. In particular, in high temperature limit the leading finite term in HEE takes precisely the form expected for the volume law contribution to the entanglement entropy in the dual field theory due to thermal fluctuations. That is, the leading thermal contribution is proportional to .
In order to investigate the low and high temperature behavior of HMI, one should keep in mind that we have three different scales, i.e., h, and T . Considering low temperature with respect to the subregion sizes and the separation between them corresponds to hT T 1. One might also Using eq.(2.9) evaluating the above expression is a straightforward exercise, which yields .
Equipped with the above result we can compute HMI in different scaling regimes as follows which demonstrates that HMI is a monotonically decreasing function of temperature. Taking the limit for adjacent subregions h → 0 in the above result, we see that the HMI diverges.

HEE and HMI in d > 1
While the integral in eqs.(2.6) and (2.7) cannot be carried out analytically for general d > 1, to perform an exact estimation, we employ a particular series expansion which is enough to extract the main behavior of HEE at finite temperature. Using this expansion eq.(2.6) can be written as follows (see [32] for details) (2.14) which converges for r t < r 0 . On the other hand using similar expansion in eq.(2.7), we find . Now it is straightforward to find the low and high temperature behavior of HEE using eqs. (2.14) and (2.16).
In this case using eq.(2.2) the T 1 limit can be interpreted in terms of bulk data as r t r 0 . In this limit eq.(2.14) yields . (2.17) Substituting the above expression into eq.(2.16) and expand to leading order in the temperature, we finally obtain According to eq.(2.18), the thermal fluctuations increases the HEE, as expected (note that C 0 and C 1 are both negative).
(ii) HEE at High Temperature Limit T 1 As the length of the subregion becomes large, the turning point of the corresponding hypersurface approaches the horizon and eventually, the minimal hypersurface covers a part of the horizon. In this case the entanglement entropy is determined entirely by the contributions coming from the near horizon part of the minimal hypersurface [36]. Hence we must consider eqs.(2.14) and (2.16) in the limit that r t → r 0 . After some algebra, eq.(2.16) becomes It is easy to see that in the large n limit the above infinite series behaves as 1 , so we can safely consider r t → r 0 limit. Hence the final result for the finite part of the HEE in this case becomes where (2.21) Note that the first term in eq.(2.20) shows a volume law which is a typical property of entanglement entropy at finite temperature and shows that for a mixed thermal state EE measures both classical and quantum correlations.
(iii) HMI at Low and High Temperature Limit Now we are equipped with all we need to calculate the HMI for configuration depicted in figure 3 using eqs. (2.18) and (2.20). As we mentioned before, we are only interested in cases where the HMI is non-zero corresponding to a connected configuration. In this case using eqs. (1.2) and (2.15), we have Considering the hT T 1 limit and using eq.(2.18) for all three distinct entropies appear in the above expression, yields [33,37] where I T =0 is the HMI at zero temperature, which in this case is given by (2.24) We note again that the finite temperature corrections reduce the HMI so the mutual correlations between subregions decrease. Another interesting case to consider is hT 1 T , for which using eqs. (2.18) and (2.20) we find (2.25) The second term in the above expression which is proportional to the area of the entangling region shows that the HMI obeys an area law even in finite temperature.

Low and High Temperature Behavior of EoP
We turn now to the calculation of EoP, using the recent holographic prescription given in [13,14].
According to this, the Eop of a certain combined region AB is given by eq.(1.5). Since we focus on the case of two intervals which have a reflection symmetry about x = 0, we expect that the corresponding minimal configuration respects this symmetry. Indeed, in this case Σ min AB runs along the radial direction and connects the corresponding turning points of Γ h and Γ 2 +h (see Fig.4). Using eq.(2.1), the area of this hypersurface can be written as Evaluating the above integral is a straightforward exercise both for d = 1 and d > 1. Below we consider these cases separately.

EoP in d = 1
In this case where the boundary theory lives in two dimensions, performing the integral in eq.(2.26), we are left with Using eq.(2.9), the above expression can be rewritten as follows (2.28) Hence using the above result we can find EoP in different scaling regimes. For hT T 1 one where the first term is just the zero temperature EoP. The second term with a negative sign shows that finite temperature reduces EoP and therefore two subsystems becomes more disentangled. On the other hand for hT 1 T we have where we have neglected terms those suppress exponentially with T . Note that the first inequality in eq.(1.4) satisfied in both low and high temperature limits. Also taking the limit for adjacent subregions h → 0 in the above result, we see that the EoP diverges.
The behavior of EoP in a two dimensional field theory can be read off from Further EoP is a monotonically decreasing function of temperature such that in high temperature limit vanishes. It is worth to mention that precisely, the similar situation arose in [32] where the structure of HMI has been investigated, although the HMI transition is continuous.

EoP in d > 1
In this case the integral in eq.(2.26) can be rewritten as follows 6 (2.31) 6 Note that although evaluating this integral gives an exact result [20] using the systematic expansion method is more tractable to find the low and high temperature corrections of EoP. yields and E p (T = 0) is the EoP at zero temperature, which in this case is given by (2.34) On the other hand in hT 1 T limit corresponding to r t (h) r 0 and r t (2 + h) → r 0 we reexpress eq.(2.31) as follows It is easy to see that in the large n limit the first infinite series behaves as 1 , so we can safely consider r t (2 + h) → r 0 limit. Also to consider r t (h) r 0 limit we keep only the leading order terms in the second infinite series, which yields where . The first term in the above expression diverges in h → 0 limit where the subregions coincide. Further, the second term which is proportional to the area of the entangling region shows that the EoP obeys an area law even in finite temperature. As we mentioned before, for a mixed thermal state HEE measures both classical and quantum correlations and as a result scales with the volume. Therefore we may conclude that the EoP carries more relevant content than HEE as far as computing quantum entanglement is concerned.

EoP in Theories with Lifshitz Scaling and Hyperscaling Violation
In this section we study the finite temperature contribution to the entanglement of purification in holographic theories with general dynamical critical exponent z and hyperscaling violation exponent θ. These theories admit a fixed point where the system is invariant under the following anisotropic scaling transformation r → λr, t → λ z t, x → λ x, ds → λ θ d ds. (3.1) Various holographic aspects of these theories have been studied in [38][39][40][41][42][43]. In particular, authors of [33] have studied the HEE and HMI of these theories at finite temperature. 7 In what follows, similar to section 2, we consider an entangling region in the shape of a strip and calculate finite temperature corrections to EoP. For completeness, we also briefly review the main result of [33] about HEE and HMI in theories with Lifshitz scaling and hyperscaling violation at the finite temperature in the appendix A.
Let us consider a (d + 2)-dimensional black brane solution in the Einstein theory of gravity with appropriate matter field (e.g. see [38]) where r 0 is the horizon radius. In addition, r f is a length scale which fixes the dimensions when θ = 0 and in the following without loss of generality we set r f = 1. As mentioned in [38], the null 7 It is worth to mention that various aspects of entanglement measures in QFTs with Lifshitz scaling symmetry are studied in [44][45][46][47]. energy condition implies some bounds on values of θ and z 3) The temperature and thermal entropy density for eq.(3.2) are given by It is easy to see that the entropy density scales as s th ∼ T

Low and High Temperature Behavior of EoP
Now we calculate EoP for holographic theories with hyperscaling violating geometry, using the holographic prescription [13,14]. The EoP for configuration depicted in Fig.4 for a QFT dual to eq.(3.2) can be written as  [38]. In the rest of this section we neglect these two special cases and calculate the EoP at low and high temperature limits.
Employing binomial series for the integrand of eq.(2.26), the EoP integral can be rewritten as follows Let us consider hT T 1 limit where we can use eq. (A.8) for r t (h) and r t (2 + h) to obtain the EoP at low temperature limit whereC 0 andC 1 are defined in eq.(A.10) and In addition, E p (T = 0) corresponds to EoP at zero temperature (3.10) Considering z > 1 andd > 0 case, eq.(3.8) shows that EoP decreases with temperature, as expected.
On the other hand in hT 1 T limit corresponding to r t (h) r 0 and r t (2 + h) → r 0 we reexpress eq.(3.7) as follows In the large n limit the first infinite series behaves as 1 , so we can safely consider r t (2 + h) → r 0 limit. Also to consider r t (h) r 0 limit, we keep only the first two terms in the second infinite series, which yields wherẽ EoP is an increasing function of the dynamical exponent, i.e., z, and the transition which happens in large separation or high temperature limit is slower in comparing to the relativistic case with z = 1. Such a behavior is not surprising because as discussed in [44] the spatial correlations between subregions become stronger for larger values of z. On the other hand as is clear from the graphs, EoP is monotonically decreasing as the hyperscaling violating exponent increases. As we mentioned before hyperscaling violation leads to an effective dimensiond = d − θ and therefore in a theory with nonvanishing θ, effective spatial dimension decreases (increases) for larger (smaller) values of hyperscaling violating exponent . Hence for larger values of θ we expect that the spatial correlations between subregions decrease and the resultant EoP decreases. 8

Corner Contributions to EoP
In this section we study the corner contribution to the EoP for holographic CFTs dual to Einstein gravity. In the holographic context, considering singular entangling surfaces was first done in [48].
Various features of holographic entanglement entropy for regions with a singular boundary such as cone and crease have been studied, e.g., see [49][50][51][52]. A key feature of these studies is the appearance 8 We would like to thank Ali Mollabashi for useful comments about this point. The coefficient of this universal term depends on the opening angle of the corresponding singular surface such that in the smooth limit where the singularity disappears, vanishes. It is worthwhile to point that, corner contributions to other entanglement/information measures is also studied , e.g., see [53,54]. In particular, for a specific configuration (see Fig.9) the HMI becomes UV divergent when the singular subregions coincide [53]. In the following considering the same setup, we would like to investigate to what extent these singularities in the boundary of entangling regions modify the behavior of EoP. To begin with, we will compute EoP for a union of kinks in d = 2 in the next subsection.

EoP for a Union of Kinks in AdS 4
In this section, in order to find the EoP for a union of kinks, we use the following 4-dimensional bulk geometry where we have written the boundary spatial directions in polar coordinates. A kink entangling region is specified by where H is the IR cut-off (see Fig.9). Due to the scaling symmetry of the AdS background and the fact that ρ is the only scale in our setup, for the RT surface we choose the parametrization r(ρ, φ) = ρ Φ(φ). In this case the corresponding HEE functional becomes where Φ t ≡ Φ(0) is the turning point given by Φ (0) = 0 and = ρ Φ( Ω 2 − δ) is the UV cut off. Since there is no explicit φ dependence, the corresponding Hamiltonian is a conserved quantity. Therefore, we find the following first integral (4.4) Using the above equation and the boundary condition eq.(4.2), the opening angle is . (4.5) Substituting eq.(4.4) back into the expression for the HEE eq.(4.3), we finally obtain where . (4.7) In particular in Ω → 0 limit where we have a sharp corner, one finds [48,50] s(Ω → 0) ∼ κ Ω + · · · , κ = π 2 6 Γ 3 4 where C T ≡ 3 π 3 G N is the central charge appearing in two-point function of the stress tensor for the underlying CFT. 9 These results are easily extended to general multipartite subregions, to compute other entanglement measures, e.g., mutual and tripartite information (see [53] for a complete discussion).
In order to compute EoP we consider the configuration depicted in the right panel of Fig.9.
Once again we focus our attention on the connected configuration for RT surfaces where both the HMI and EoP are nonzero. Note that due to the axial symmetry we expect that the minimal cross section of entanglement wedge locates at φ = 0. Using eq.(4.1), the EoP functional is given by where Φ t (ω) and Φ t (2Ω + ω) denote the turning points of the corresponding minimal surfaces. The above integral can be evaluated explicitly yielding 11) which demonstrates that Eop is divergent when the subregions coincide. It is worth to mention that precisely, the similar situation arose in [53] in investigating the structure of HMI, although the transition is continuous. Considering the case where we have two adjacent sharp corners, i.e., ω Ω 1 and using eq.(4.8) we may further simplify the result to It is important to mention that the above result can be obtained using a conformal map relating the corner geometry to a strip in four dimensions. A similar derivation to the one presented for HMI in [53] holds in the present case which shows that the above expression reduces to eq.(2.34) for d = 2. As another consistency check, we note that the EoP should satisfy eq.(1.4). Indeed, as shown in [53] the HMI in this particular limit is given by (4.13) 9 The explicit expression for the corresponding two-point function is Tµν (r)T αβ (0) = CT r 2d I µν,αβ (r), (4.9) where I µν,αβ is a tensor fixed by symmetry.
In comparing the above expression with eq.(4.12), we see that the constraint on the lower bound of EoP satisfied.

EoP for a Union of Creases in AdS d+2
In this section we will compute the EoP in the presence of singular regions in higher dimensions.
The calculations are analogous to those for three dimensions. Consider the following bulk geometry (4.14) In this case the entangling region is specified by where H andH are the IR regulators where in the following we setH = H. Using the scaling symmetry of the background and assuming r(ρ, φ) = ρ Φ(φ), the corresponding HEE functional Once again, since there is no explicit φ dependence, we have a first integral [49] (4.17) This eventually leads to the following expression for the opening angle and HEE where we have defined It is worth to mention that the second divergent term in eq.(4.18) is produced by the singularity in the entangling surface and vanish when the surface is smooth. We would like to stress that this contribution is due to adding a flat locus to the kink [49]. Further in d = 2 this term modified and we recover a universal logarithmic contribution.
Turning to EoP, we expect that the minimal cross section of entanglement wedge locates at (4.20) Evaluating the above integral, we are left with which is divergent and obeys the area law. The final result for Eop in various dimensions agree at a qualitative level, so we just consider d = 3 case. In Fig.10 we demonstrate the EoP as a function of ω for Ω = π 4 . We observe that EoP is a monotonically decreasing function of the angular separation between the two subregions and for distant enough regions E P = 0. Further, as a consistency check in this figure we also plot I 2 to see weather eq.(1.4) satisfied or not. Here, it is worth mentioning that, it was shown in [53] that HMI for a union of creases in ω Ω limit is given by the following which is divergent.
In this paper, we explored the general behavior of holographic entanglement of purification (EoP) in various geometries and for different entangling regions. We used the recent proposal for computing this quantity which states that EoP is given by the minimal cross section of the entanglement wedge, as in eq.(1.5). In the following we would like to first summarize our main results and then continue with discussing some further problems.
• In a two dimensional relativistic quantum field theory EoP is a monotonically decreasing function of temperature such that at higher temperature the two subsystems becomes more disentangled. Also keeping the length and separation of the subregions fixed while temperature increases, the EoP shows a discontinuous phase transition, such that E P = 0 when T is high enough. In the bulk, the vanishing of EoP results because of the disconnected configuration for the RT surfaces and the fact that in this case the corresponding entanglement wedge is disconnected, e.g., see Fig.2.
• In higher dimensions considering a strip entangling region, the qualitative behaviors of Eop at finite temperature are very similar to d = 1 case. A key observation is that the EoP obeys an area law even in finite temperature where the HEE shows a volume law. Therefore one may regard the EoP as a more appropriate measure of quantum entanglement for thermal mixed states. 11 • In a nonrelativistic field theory with nontrivial dynamical and hyperscaling exponents, EoP is a monotonically decreasing function of temperature and the separation between subregions.
In this case the transition point after that the EoP vanishes, depends on the value of z and θ. In particular EoP is an increasing function of the dynamical exponent and the transition which happens in large separation or high temperature limit is slower in comparing to the relativistic case with z = 1. As we mentioned, the physical reason behind this is that the quantum correlations between subregions increase as one increases z. On the other hand, EoP is monotonically decreasing as the hyperscaling violating exponent increases. In a field theory with nonvanishing θ, effective spatial dimension decreases for larger values of hyperscaling violating exponent and therefore for larger values of θ we expect that the spatial correlations between subregions decrease and the resultant EoP decreases.
• Considering an entangling region with singular boundary, we demonstrated that Eop is divergent when the subregions coincide. In particular for a three dimensional boundary theory we found a universal contribution to EoP due to the presence of corner where the coefficient is proportional to the central charge for the underlying CFT. Moreover, considering a singular region in higher dimensions we verified that the corresponding EoP obeys area law.
We can extend this study to different interesting directions. A key feature of EoP is the discontinuous phase transition which happens at large separation or high temperature. As we mentioned before, the corresponding (continuous) phase transition of HMI is a reminiscent of the large N limit of the dual field theory and it disappears if one considers quantum corrections. It will be an important future problem to study the quantum corrections to EoP using the prescription proposed in [8]. We expect that considering this quantum corrections change the phase diagram of EoP and especially one find a smooth behavior near the critical point.
Finally it would be interesting to study EoP in more general holographic setups, e.g., field theories dual to higher curvature gravities. It is known that for such theories the RT prescription for computing HEE fails and one should use other proposals [3][4][5][6]. At present, our preliminary analysis suggests that in this case one should replace eq.(1.5) with another functional which contains higher curvature corrections. We leave the details of this interesting problem for future study [55].
between and r t as = 2r t One may consider two special cases d = θ and θ = d − 1, z = 1. In the former, the RT surface lies on the boundary slice r = and we have an extensive violation of area law [38]. The latter, up to an overall factor H θ , is exactly same as d = 1 relativistic theory so we do not mention it again (see section 2.1.1). Therefore, in the rest, we neglect these two special cases.
By employing method of section 2 we can extract the behavior of HEE and HMI at low and high temperature. Using binomial series and performing the integral we obtain the relation between length of entangling region and turning point r t as = 2r t and we have definec = . Using these results we can find the low and high temperature behavior of HEE and HMI. . (A.10) For z > −1 −d the thermal correction increases the HEE. However, for z < −1 −d the HEE decreases by thermal correction, but as we mentioned, the negative value of z (for θ > d) has been excluded by thermodynamic stability of black brane solution.
(ii) HEE at High Temperature Limit T 1 In the high temperature the near horizon part of RT surface has the main contribution to HEE.
Therefore, to obtain HEE we can consider eqs.(A.5) and (A.7) in the limit that r t → r 0 . By manipulating eq.(A.7) we get .
In the large n limit the above series behaves as 1 n 2 rt r 0 nd+z , so we can take r t → r 0 limit. Now using eq.(3.4), we obtain the finite part of HEE at high temperature