Holographic entanglement negativity for a single subsystem in conformal field theories with a conserved charge
aa r X i v : . [ h e p - t h ] F e b Holographic entanglement negativity for a singlesubsystem in conformal field theories with a conservedcharge
Sayid Mondal ∗ , Boudhayan Paul † , Gautam Sengupta ‡ and Punit Sharma § Center for High Energy Physics and Department of Physics,Chung-Yuan Christian University, Chung-Li 320, Taiwan Department of Physics,Indian Institute of Technology,Kanpur 208 016, India Department of Physics and Astronomy,The University of Iowa,Iowa City, IA 52242, USA
Abstract
We investigate the application of a holographic entanglement negativity construction tobipartite states of single subsystems in
CF T d s with a conserved charge dual to bulk AdS d +1 geometries. In this context, we obtain the holographic entanglement negativity for singlesubsystems with long rectangular strip geometry in CF T d s dual to bulk extremal andnon-extremal Reissner-Nordstr¨om (RN)- AdS d +1 black holes. Our results demonstrate thatthe holographic entanglement negativity involves the subtraction of the thermal entropy fromthe entanglement entropy confirming earlier results. This conforms to the characterizationof entanglement negativity as the upper bound on the distillable entanglement in quantuminformation theory and constitutes an important consistency check for our higher dimensionalconstruction. ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected] ontents
CF T dual to AdS -RN 8 AdS -RN . . . . . . . . . . . . . . . . . . . . . . . . 83.2 The geometrical configuration for a single subsystem . . . . . . . . . . . . . . . . 103.3 Non extremal AdS -RN black holes . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3.1 Small charge - low temperature . . . . . . . . . . . . . . . . . . . . . . . . 113.3.2 Small charge - high temperature . . . . . . . . . . . . . . . . . . . . . . . 133.3.3 Large charge - high temperature . . . . . . . . . . . . . . . . . . . . . . . 143.4 Extremal AdS -RN black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.1 Small charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.2 Large charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 CF T d dual to AdS d +1 -RN 18 AdS d +1 -RN . . . . . . . . . . . . . . . . . . . . . . . 184.2 Non extremal AdS d +1 -RN black holes . . . . . . . . . . . . . . . . . . . . . . . . 214.2.1 Small chemical potential - low temperature . . . . . . . . . . . . . . . . . 214.2.2 Small chemical potential - high temperature . . . . . . . . . . . . . . . . . 244.2.3 Large chemical potential - low temperature . . . . . . . . . . . . . . . . . 244.3 Extremal AdS d +1 -RN black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3.1 Small chemical potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3.2 Large chemical potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 AdS -RN 31 A.1 Non extremal
AdS -RN (Small charge - low temperature) . . . . . . . . . . . . . 31A.2 Non extremal AdS -RN (Small charge - high temperature) . . . . . . . . . . . . . 31A.3 Extremal AdS -RN (Small charge) . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Appendix B Non extremal and extremal
AdS d +1 -RN 33 – 2 – Introduction
The recent past has witnessed the emergence of quantum entanglement as a central theme in thestudy of many body systems relevant to diverse fields from condensed matter physics to quantumgravity and black holes. For bipartite pure states, entanglement entropy turns out to be a viablemeasure for the characterization of entanglement, and is defined as the von Neumann entropy of the reduced density matrix for the subsystem under consideration. Though computable forfinite dimensional systems, it becomes difficult to obtain for extended many body systems likequantum field theories. For such extended systems, the entanglement entropy may be formallydefined through a replica technique, however an explicit computation is intractable in general.Remarkably for 1 + 1 dimensional conformal field theories (
CF T s), the authors in [1–5] couldexplicitly compute the entanglement entropy for bipartite states utilizing the replica technique.For bipartite mixed state configurations however the entanglement entropy fails to correctlycapture the entanglement of the corresponding subsystem, as it incorporates contributions fromcorrelations irrelevant to the entanglement of the subsystem in question. This crucial issue inquantum information theory was addressed by Vidal and Werner in [6], where they advanced acomputable measure which characterized an upper bound on the distillable entanglement for thebipartite mixed state under consideration. This measure, termed entanglement negativity , wasdefined as the logarithm of the trace norm for the partially transposed reduced density matrixwith respect to one of the subsystems of the bipartite system in question. Despite being nonconvex, it was shown to be an entanglement monotone in [7] .For a CF T , the entanglement negativity could be obtained for various bipartite statesthrough a suitable variant of the replica technique mentioned earlier, as described in [9–11]. Itwas shown that the entanglement negativity for mixed state configurations involving two or moredisjoint intervals contained non universal functions which depend on the full operator contentthe
CF T . Interestingly in [12] it was demonstrated through a monodromy technique that forbipartite configurations involving two disjoint intervals in proximity, a universal contribution forthe entanglement negativity may be extracted in the large central charge limit.In a different context a holographic characterization of entanglement for conformal fieldtheories in arbitrary dimensions (
CF T d ) through the AdS d +1 /CF T d correspondence wasinitiated by Ryu and Takayanagi (RT) in a seminal work [13, 14]. In this work they showed thatthe universal part of the entanglement entropy for bipartite states in a CF T d was proportionalto the area of a co dimension two static minimal surface in the dual bulk AdS d +1 geometry,homologous to the subsystem. A proof of the RT conjecture was developed from the bulkperspective initially for the AdS /CF T scenario, and was extended to higher dimensionsin [15–19]. A covariant generalization of the RT conjecture for CF T d dual non static bulk AdS d +1 geometries was proposed by Hubeny, Rangamani and Takayanagi (HRT) in [20] witha corresponding proof in [21]. The RT/HRT conjectures inspired the remarkable developmentof the field of holographic quantum entanglement in dual CF T d s summarized in [22–27] andreferences therein .Note that the above developments involved the holographic characterization of entanglemententropy which as mentioned earlier was not a valid measure for mixed state entanglement inquantum information theory. This naturally leads to the interesting issue of the holographiccharacterization of mixed state entanglement in dual CF T d s. In this context the authors in [8]advanced a holographic computation of the entanglement negativity for a zero temperature For an excellent review see [8]. Note that the field is extremely diverse and the list of references are not meant to be exhaustive. – 3 –ipartite pure vacuum state in a
CF T d dual to a bulk pure AdS d +1 geometry. However aholographic construction for the entanglement negativity of bipartite mixed state configurationsin a holographic CF T d remained an outstanding open issue. Interestingly, the authors in [28]advanced a holographic construction for the entanglement negativity of bipartite states of asingle interval in CF T s dual to bulk pure
AdS geometries and Euclidean BTZ black holes.Their construction involved a specific algebraic sum of the lengths of bulk space like geodesicshomologous to certain appropriate combinations of the single interval and required auxiliaryintervals. Interestingly the holographic entanglement negativity for the pure state configurationdescribed by the vacuum state of the CF T dual to the bulk pure
AdS geometry was foundto be proportional to the holographic entanglement entropy and equal to the correspondingR´enyi entropy of order half consistent with quantum information theory results. For the finitetemperature mixed state, the holographic entanglement negativity was given by the differenceof the entanglement entropy and the thermal entropy consistent with its characterization as anupper bound on the distillable entanglement. Remarkably in both the cases, the results werein exact agreement with the corresponding replica technique results in the large central chargelimit [9–11]. Subsequently a covariant extension of this holographic construction was establishedin [29]. Following this the proposal described in [28] was substantiated through a large centralcharge analysis of the entanglement negativity of the configuration of a single interval in aholographic CF T employing a monodromy technique in [30].Extension of the above construction to a generic higher dimensional
AdS d +1 /CF T d wasdescribed in [31] which involved a specific algebraic sum of the areas of co dimension twobulk static minimal surfaces homologous to the appropriate combinations of the subsystemunder consideration and certain auxiliary subsystems similar to that described in [28]. Thisconstruction was then utilized to obtain the holographic entanglement negativity for the pure andmixed state configuration of a single subsystem with rectangular strip geometry in CF T d s dualto bulk pure AdS d +1 and AdS d +1 -Schwarzschild black hole geometries. For the zero temperaturepure state configuration the holographic entanglement negativity was found to be proportionalto the corresponding holographic entanglement entropy consistent with quantum informationtheory expectations. For the corresponding finite temperature mixed state configuration theholographic entanglement negativity involved the subtraction of the thermal entropy from theentanglement entropy of the subsystem conforming to its characterization as an upper boundon the distillable entanglement in quantum information theory. Interestingly these higherdimensional results exactly reproduced the universal features of entanglement negativity forthe AdS /CF T scenario described in [28]. However we should mention here that a bulk proofof this holographic proposal along the lines of [18, 19] remains a non trivial outstanding issue.The holographic construction described above in [28, 29] concerning a single subsystemin the context of the AdS /CF T scenario was later extended to the bipartite mixed stateconfigurations of adjacent intervals in dual CF T s in [32], with the corresponding covariantextension described in [33]. A higher dimensional generalization of the construction for theadjacent intervals in [32] was advanced in [34], which was employed to examine the holographicentanglement negativity for various bipartite states in
CF T d s dual to the bulk pure AdS d +1 geometry, AdS d +1 -Schwarzschild black hole and the AdS d +1 -Reissner-Nordstr¨om (RN) blackhole in [34, 35]. Recently a holographic construction for the entanglement negativity for themixed state configurations of two disjoint intervals in proximity in a CF T dual to static bulk
AdS geometries was proposed in [36], with its covariant generalization described in [37]. Ahigher dimensional extension of the construction was subsequently advanced in [38], and wasemployed to compute the holographic entanglement negativity for mixed state configurations– 4 –ith long rectangular strip geometry in CF T d s dual to bulk pure AdS d +1 geometry and AdS d +1 -Schwarzschild black hole. Furthermore the time evolution of holographic entanglementnegativity following a global quench in a CF T dual to a bulk eternal black hole sliced inhalf by an end of the world (ETW) brane was described in [39] . Through the representativeexamples described above, it has been demonstrated that the holographic prescriptions involvinga single subsystem, two adjacent subsystem and two disjoint subsystems in proximity in higherdimensions as proposed in [28, 34, 38] provided a direct and elegant method to compute theholographic entanglement negativity in the general
AdS d +1 /CF T d framework. The resultsobtained from these examples were in conformity with quantum information theory expectationsand served as important consistency checks for the holographic proposals.We should mention here however that recently in [40, 41] the authors advanced an alternateholographic conjecture which states that the holographic entanglement negativity was givenby the back reacted bulk minimal entanglement wedge cross section (EWCS) for sphericalentangling surfaces where the back reaction arises from the bulk cosmic brane for the conicaldefect. It could be shown that in this case the back reaction was given as a dimension dependentoverall factor. The authors utilized the prescription given in [42] to compute the minimalEWCS. Remarkably for the AdS /CF T scenario the results from their proposal match withthe corresponding replica results for the cases of adjacent intervals, and disjoint intervals inproximity. For the case of a single interval however their conjecture had to be coupled with analternate construction for the computation of the minimal EWCS described in [43] to obtain thecorrect replica results with the appropriate subtraction of the thermal entropy. However thisproposal involving the minimal EWCS requires explicit substantiation for the higher dimensional AdS d +1 /CF T d scenario . It was shown in [43] that the results from the alternate holographicconstruction described above, also matched exactly with the results from the holographicproposal involving combinations of bulk geodesics in [31, 32, 36] for the AdS /CF T scenario .This demonstrated that the two proposals were essentially equivalent and it followed that thegeodesic combinations of the former were proportional to the minimal EWCS and this geometricconclusion was expected to extend to the higher dimensional AdS d +1 /CF T d framework. Keepingthe above developments in context, we should mention here that the higher dimensional resultsobtained utilizing the holographic constructions in [28, 34, 38] may also be modified by somedimension dependent proportionality factor arising from the back reaction of the bulk cosmicbrane for the conical defect. However note that the explicit determination of this factor for thesubsystem geometries of long rectangular strips is a non trivial open issue.In this article we further extend the higher dimensional construction for the holographicentanglement negativity of bipartite states of a single subsystem described in [28] to CF T d swith a conserved charge dual to bulk extremal and non extremal AdS d +1 -RN black holes.In this context we consider subsystems with long rectangular strip geometries and computethe areas of bulk static minimal surfaces homologous to the relevant subsystems, utilizingcertain perturbative techniques described in [24, 25, 47, 48], and involving non trivial limitsof the relevant parameters. In order to elucidate the perturbative technique we have firstconsidered the relatively simpler AdS /CF T scenario and then extending the same to thegeneric AdS d +1 /CF T d framework. Interestingly as in earlier examples described in the literaturewe are able to exactly reproduce the universal features of the holographic entanglement See also [44]. See also [45, 46]. It should be noted that an exact closed form analytic evaluation of this area has been performed throughthe use of Meijer G-functions in [49]. However the leading order behaviour necessary for our purpose is correctlycaptured through the perturbative techniques used here. – 5 –egativity described in [28, 31] for both zero and finite temperature bipartite states. Ourresults constitute another significant consistency check for the holographic construction forentanglement negativity in higher dimensional
CF T d s for the specific subsystem geometriesunder consideration. Note however that in the context of the developments described in [40, 41]involving the back reacted minimal entanglement wedge our results may be modified by anappropriate overall back reaction factor specific to the subsystem geometry. As mentionedearlier the explicit determination of this back reaction parameter in higher dimensions for thesubsystems under consideration remains a non trivial open issue.This article is organized as follows. In section 2 we describe the holographic entanglementnegativity construction for the bipartite configuration of a single subsystem characterized bylong rectangular strip geometry in CF T d s dual to bulk AdS d +1 configurations. Following thisin section 3 we obtain the holographic entanglement negativity for mixed state configuration ofa single subsystem in a CF T dual to bulk non extremal and extremal RN- AdS black holes.Subsequently in section 4 the construction is extended to compute the holographic entanglementnegativity for mixed state configuration of a single subsystem in a CF T d dual to bulk nonextremal and extremal RN- AdS d +1 black holes. Finally in section 5 we summarize our resultsand present our conclusions. In this section we present a brief review of the holographic entanglement negativity constructionfor the bipartite state of a single subsystem in a
CF T d at a finite temperature dual tobulk AdS d +1 geometries as described in [28]. In this context we first briefly recapitulate thecorresponding issue in the AdS /CF T scenario [28]. For this purpose we consider a bipartitesystem ( A ∪ A c ) which is described by an interval A of length l and its complement A c . Todescribe our construction it is also required to consider two auxiliary large but finite intervals B and B on either side of A and adjacent to it such that ( B = ( B ∪ B ) ⊂ A c ). Theentanglement negativity for the mixed state of the single interval at a finite temperature in thedual CF T may be defined through a replica technique as follows [9] E = lim n e → ln Tr( ρ T A ) n e , (1)where ρ T A is the partial transpose of the reduced density matrix ρ A and the replica limit n e → n e to n e = 1. Note that the quantity Tr( ρ T A ) n e may be expressed as a twist correlator in the CF T appropriate to the configuration. For thecase of the bipartite state of a single interval in the dual
CF T this is given by a four pointtwist correlator on an infinitely long cylinder and the corresponding entanglement negativity isgiven as follows [11] E = lim L →∞ lim n e → ln h(cid:10) T n e ( − L ) T n e ( − l ) T n e (0) T n e ( L ) (cid:11) cyl i , (2)where ( L → ∞ ) is the bipartite limit ( B = B ∪ B ) → A c . It is important to note that theorder of the limits in eq. (2) is crucial where the bipartite limit ( L → ∞ ) should be taken onlyafter the replica limit ( n e →
1) .The form of the four point twist correlator on the complex plane C is fixed upto a function– 6 –f the cross ratio as (cid:10) T n e ( z ) T n e ( z ) T n e ( z ) T n e ( z ) (cid:11) C = c n e c n e / z ne z (2) ne F n e ( x ) x ∆ (2) ne , (3)where the function F n e ( x ) is non universal and depends on the cross ratio x = z z z z , z ij = | z i − z j | , and c n e and c n e / are the normalization constants.Note that the universal part of the four point twist correlator on the complex plane C in eq.(3) is dominant in the large central charge limit, which factorizes into two point twistcorrelators as follows (cid:10) T n e ( z ) T n e ( z ) T n e ( z ) T n e ( z ) (cid:11) C = (cid:10) T ne ( z ) T ne ( z ) (cid:11) (cid:10) T n e ( z ) T n e ( z ) (cid:11) × (cid:10) T ne ( z ) T ne ( z ) (cid:11)(cid:10) T ne ( z ) T ne ( z ) (cid:11)(cid:10) T ne ( z ) T ne ( z ) (cid:11)(cid:10) T ne ( z ) T ne ( z ) (cid:11) + O [ c ] , (4)where the form of the two point twist correlators in eq. (4) are as follows (cid:10) T n e ( z k ) T n e ( z l ) (cid:11) C = c n e z ne kl , (5) hT n e ( z i ) T n e ( z j ) i C = (cid:10) T ne ( z i ) T ne ( z j ) (cid:11) C = c n e / z ne ij . (6)The scaling dimension ∆ (2) n e and ∆ n e of the operator T n e and T n e are related to each other asfollows ∆ (2) n e = 2∆ ne = c (cid:18) n e − n e (cid:19) , (7)∆ n e = c (cid:18) n e − n e (cid:19) . (8)It is well known fact from the AdS/CF T correspondence that the two point twist correlator ofthe twist fields located at z i and z j in a CF T is related to the length of the dual bulk spacelike geodesic L ij anchored on the corresponding interval as follows [13, 14] (cid:10) T n e ( z k ) T n e ( z l ) (cid:11) C ∼ e − ∆ ne L klR , (cid:10) T ne ( z i ) T ne ( z j ) (cid:11) C ∼ e − ∆ ne L ijR . (9)Employing the eqs. (4), (9) in eq. (2), the holographic entanglement negativity may then beexpressed as E = lim B → A c G (3) N L A + L B + L B − L A ∪ B − L A ∪ B ! , (10)where we have utilized the Brown-Hennaux formula c = R G (3) N .In the context of the AdS d +1 /CF T d scenario the corresponding holographic entanglementnegativity for the mixed state of a single subsystem A may then given in the bipartite limit– 7 – B → A c ) as E A = lim B → A c G ( d +1) N A A + A B + A B − A A ∪ B − A A ∪ B ! , (11)where A γ is the area of a co dimension two bulk static minimal surface homologous to thesubsystem γ , and G ( d +1) N is the gravitational constant in d + 1 dimensions. In the case when thesubsystems B and B are symmetric with respect to the subsystem A , the eq. (11) simplifiesto E A = lim B → A c G ( d +1) N A A + A B − A A ∪ B ! . (12)Note that the holographic entanglement entropy of a subsystem A in a CF T d dual to an AdS d +1 is given by the area A A of the co dimension two bulk static minimal surface anchored to thesubsystem as [13] S A = A A G ( d +1) N . (13)Having briefly covered the holographic construction to be utilized to obtain the holographicentanglement negativity in the next section 3 we compute the holographic entanglementnegativity for the bipartite states of single subsystems with long rectangular strip geometryin CF T s with a conserved charge dual to bulk non extremal and extremal AdS -RN blackholes. CF T dual to AdS -RN In this section we utilize the construction reviewed in the last section to compute the holographicentanglement negativity for mixed state configurations of a single subsystem in
CF T s with aconserved charge dual to bulk AdS geometries. In this context we compute the holographicentanglement negativity through perturbative techniques for zero and finite temperature mixedstate configurations of a single subsystem with long rectangular strip geometries in CF T swith a conserved charge, dual to bulk non extremal and extremal AdS -RN black holes. Thecomputation involves the perturbative expansion of the area integral for the co dimension twobulk static minimal surface homologous to the subsystem for various limits of the charge andthe temperature of the dual CF T s. AdS -RN In this subsection we briefly review the computation of the area of the co dimension two bulkstatic minimal surface homologous to a subsystem described by a long rectangular strip geometryin a holographic
CF T with a conserved charge, utilizing perturbation techniques as describedin [24, 27, 31, 35].The metric for a Reissner-Nordstr¨om (RN) black hole in AdS space time with a planarhorizon (with the AdS radius R set to unity) is described by ds = − r f ( r ) dt + 1 r f ( r ) dr + r (cid:0) dx + dy (cid:1) , (14)– 8 –ith the lapse function f ( r ) given by f ( r ) = 1 − Mr + Q r , (15)where M and Q are the mass and charge of the black hole respectively. They are related to theradius of the horizon r h through the vanishing of f ( r ) at r = r h as M = r h + Q r h . (16)Utilizing the above relation, the lapse function in eq. (15) may now be expressed in terms of thecharge Q and the horizon radius r h as follows f ( r ) = 1 − r h r − Q r r h + Q r . (17)The Hawking temperature for this black hole is given through T = f ′ ( r )4 π (cid:12)(cid:12)(cid:12)(cid:12) r = r h = 34 πr h (cid:18) − Q r h (cid:19) . (18)Having described the AdS -RN metric, we now proceed to briefly review the computationthe area of the co dimension two static minimal surface anchored on a subsystem with longrectangular strip geometry in a CF T dual to an AdS -RN black hole. The subsystem A in thislong rectangular strip geometry is described by x ∈ (cid:20) − l , l (cid:21) , y ∈ (cid:20) − L , L (cid:21) , (19)where l ≪ L .The area A A of the surface anchored on this subsystem as described above may then beexpressed as A A = 2 L Z ∞ r c dr s f ( r ) (cid:18) − r c r (cid:19) . (20)where r c is the turning point of the minimal surface in the bulk in the x direction. The length l of the rectangular strip in the x direction is given by l = 2 Z ∞ r c r c drr s f ( r ) (cid:18) − r c r (cid:19) . (21)To evaluate the integrals described in eqs. (20) and (21) it is required to perform a coordinatetransformation from r to u = r c r . Eqs. (17), (20) and (21) may then be re-expressed in terms of The long rectangular geometry implies that the length L in the transverse direction ( y direction) is finite butlarge enough so that for any finite subsystem with length l in the longitudinal direction ( x direction), we have l ≪ L . – 9 –he variable u as f ( u ) = 1 − r h u r c − Q u r c r h + Q u r c , (22) A A = 2 Lr c Z f ( u ) − / u √ − u du , (23) l = 2 r c Z u f ( u ) − / √ − u du . (24)Having reviewed the techniques required for the perturbative computation of the area of thebulk minimal surfaces in AdS -RN geometries, homologous to subsystems with long rectangularstrip geometry in the dual CF T s, in subsection 3.2 we proceed to to describe the geometricalconfiguration involving the introduction of auxiliary subsystems required for the calculation ofthe entanglement negativity for this case. Having reviewed the computation of area for the co-dimension two bulk minimal surface anchoredon a subsystem in subsection 3.1, we now describe the geometrical construction required tocompute the holographic entanglement negativity for the bipartite configuration involving asingle subsystem in a
CF T dual to bulk AdS geometries. To this end it is necessary toconsider two auxiliary subsystems B and B , each of length b , on either side of the subsystem A along the direction of the x axis (henceforth referred to as the partitioning direction), specifiedby x ∈ (cid:20) − (cid:18) b + l (cid:19) , − l (cid:21) , y ∈ (cid:20) − L , L (cid:21) ; (25)and x ∈ (cid:20) l , (cid:18) b + l (cid:19)(cid:21) , y ∈ (cid:20) − L , L (cid:21) ; (26)respectively. Note that the subsystems B and B have been chosen to be symmetric alongthe partitioning direction. This leads to the equality of the minimal areas A B = A B and A A ∪ B = A A ∪ B . With this identification, the holographic entanglement negativity for theabove configuration may be computed from eq. (12), where the bipartite limit B → A c isimplemented by taking the limit b → ∞ .Having described the geometrical setup required to evaluate the holographic entanglementnegativity for different configurations involving a single subsystem, we now turn our attentionto the detailed computation of the holographic entanglement negativity for zero and finitetemperature bipartite mixed state configurations of a single subsystem in holographic CF T swith a conserved charge in the following subsections. In subsection 3.3, we consider differentscenarios for CF T s dual to non extremal AdS -RN configurations, while subsection 3.4 portraysthe cases involving bulk extremal AdS -RN geometries. AdS -RN black holes Having described the perturbative area computation techniques in subsection 3.1 and thenecessary geometrical construction in subsection 3.2, we are now in a position to describe in– 10 –etail the perturbative computation of the holographic entanglement negativity for the finitetemperature bipartite mixed state configuration of a single subsystem with a long rectangularstrip geometry in a
CF T dual to a bulk non extremal AdS -RN black hole. This necessitates theperturbative evaluation of the areas of the bulk static minimal surfaces homologous to relevantcombinations of the subsystem and auxiliary subsystems in the boundary, for different limitsinvolving the bulk parameters described in subsection 3.1. We now proceed to characterize theholographic entanglement negativity for the above mentioned configuration in these varied limitsin depth. We start with the regime of small charge and low temperature in subsection 3.3.1,moving on to the case of small charge and high temperature in subsection 3.3.2 before turningour attention to the limit of large charge and high temperature in subsection 3.3.3. In this subsection we describe the limit of small charge and low temperature. Utilizing eq. (18),the non extremality condition
T >
AdS -RN black hole may be re-expressed interms of the horizon radius r h and the charge Q as r h > √ Q . (27)In the regime of small charge and low temperature, it may be shown that Q/r h ∼ r c ≫ r h (refer to [27] for details). In this limit, employing eq. (22), the quantity f ( u ) − appearing ineqs. (23) and (24) may be expanded about u = 0 as follows [27, 35] f ( u ) − = 1 + 1 + α (cid:18) r h r c (cid:19) u + O "(cid:18) r h ur c (cid:19) , (28)where α ≡ Q r h . The expression for f ( u ) − obtained in eq. (28) may now be substituted back intoeqs. (23) and (24) to finally arrive at the following area A Σ of the co dimension two static bulkminimal surface anchored on a generic subsystem Σ with long rectangular strip geometry [27] A Σ = A div Σ + A finite Σ . (29)The divergent part A div Σ and the finite part A finite Σ of A Σ in eq. (29) are given by A div Σ = 2 (cid:18) La (cid:19) , (30) A finite Σ = k Ll + k r h (1 + α ) Ll + O (cid:2) r h l (cid:3) . (31)Here a is the UV cut off, l is the length of subsystem Σ in the partitioning direction, L is itslength along the transverse direction, and the constants k and k appearing in eq. (31) arelisted in appendix A.1 in eqs. (103) and (104) respectively.As discussed in [31], the subsystems B and A ∪ B in the CF T in the boundary withlengths b and ( l + b ) respectively become very large (in fact, infinite) along the partitioningdirection in the limit B → A c (that is, b → ∞ ). Thus even at low temperature, the minimalsurfaces corresponding to these subsystems will extend deep into the bulk, approaching theblack hole horizon, i.e., r c ≈ r h and u ≡ r c r h ≈
1. For this case we follow the method describedin [24, 27]. The Taylor expansion for the function f ( u ) appearing in eq. (22) may be expanded– 11 –round u = u as [27] f ( u ) = (cid:18) − Q r h (cid:19) (cid:18) − r h r c u (cid:19) + O h ( u − u ) i . (32)For this case, the finite part of the area A finite Σ of the co dimension two bulk minimal surfaceanchored on the generic subsystem Σ, as detailed in [27], may be given utilizing eqs. (23) and(24) by A finite Σ = Llr h + Lr h √ δ (cid:18) K ′ + K ′ ǫ (cid:19) + O (cid:2) ǫ (cid:3) . (33)where l is the length of subsystem Σ in the partitioning direction, δ ≡ Q √ r h and the constants K ′ and K ′ are given in appendix A.1 in eqs. (105) and (106) respectively. The parameter ǫ appearing in eq. (33) is given by [35] ǫ = 13 exp h −√ (cid:0) lr h − c − c δ (cid:1)i , (34)where the constants c and c in eq. (34) are listed in appendix A.1 in eqs. (107) and (108).The holographic entanglement negativity in the limit of small charge and low temperature,for the mixed state configuration of subsystem A may now be obtained utilizing eqs. (30), (31)and (33) from the construction described in eq. (12) as follows E A = 38 G (4) N " La + k Ll + k r h (1 + α ) Ll − Llr h + O (cid:2) r h l (cid:3) . (35)In eq. (35), the second term on the right hand side represents the holographic entanglementnegativity for the zero temperature pure state configuration of a single subsystem in a CF T dual to bulk pure AdS geometry, while the third and fourth terms describe the correctionsarising from the charge and the temperature.The corresponding holographic entanglement entropy of subsystem A may be obtained usingeqs. (30) and (31) from eq. (13) as S A = 14 G (4) N " La + k Ll + k r h (1 + α ) Ll + O (cid:2) r h l (cid:3) . (36)Comparing eqs. (35) and (36) we arrive at the following relation E A = 32 (cid:16) S A − S th A (cid:17) , (37)where we have used the fact that the thermal entropy is given by S th A = Llr h G (4) N . The relation in eq. (37) describes that the holographic entanglement negativity ischaracterized by the elimination of the thermal entropy from the holographic entanglemententropy for the finite temperature mixed state of a single subsystem. Note that in quantuminformation theory this is a universal result for configurations involving a single subsystem,signifying the characterization of entanglement negativity as the upper bound on the distillable Eq. (31) is utilized to compute A A , while eq. (33) is employed to compute A B and A A ∪ B . Finally we takethe bipartite limit b → ∞ . With the volume of subsystem A being V ≡ Ll , the volume dependence of this term becomes evident. – 12 –ntanglement. Reproduction of this relation for the holographic components of entanglementnegativity and entropy is remarkable and serves as a consistency check for the higher dimensionalholographic construction involving a single subsystem as advanced in [28], and further exploredin [31]. We further note that the cancellation of the thermal contribution is similar to thecorresponding AdS /CF T case, pointing towards it being a universal feature for the holographicentanglement negativity in all holographic CF T s.Having computed the holographic entanglement negativity for the case of small charge andlow temperature, we now advance to investigate the limit of small charge and high temperaturein subsection 3.3.2.
Having described the small charge and low temperature case in detail in subsection 3.3.1, in thissubsection we set out to study the small charge and high temperature regime, described by thecondition [27] r h l ≫
1, which implies δ ≡ Q √ r h ≪
1. As described in [31], at high temperatures,the turning point of the minimal surfaces approaches close to the black hole horizon, which isgiven by the condition r c ≈ r h .As described in [27, 35], the finite part of the area A finite Σ of the co dimension two bulkminimal surface anchored on a subsystem Σ in this case may be given by A finite Σ = Llr h + Lr h (cid:0) k + δ k (cid:1) + Lr h ǫ (cid:2) k + δ ( k + k log ǫ ) (cid:3) + O (cid:2) ǫ (cid:3) , (38)where l is the length of subsystem Σ in the partitioning direction, the parameter ǫ is given byeq. (34) (page 12), and the constants k , k , k , k and k are listed in appendix A.2 in eqs.(109), (110), (111), (112) and (113) respectively.Utilizing eqs. (30) and (38), the holographic entanglement negativity in the limit of smallcharge and high temperature, for the mixed state configuration of the single subsystem inquestion may now be obtained from the holographic construction by employing eq. (12) asfollows E A = 38 G (4) N " La + Lr h (cid:0) k + δ k (cid:1) + Lr h ǫ (cid:8) k + δ ( k + k log ǫ ) (cid:9) + O (cid:2) ǫ (cid:3) , (39)The parameter ǫ in eq. (39) is given by ǫ = 13 exp h −√ (cid:0) lr h − c − c δ (cid:1)i , (40)with the constants c and c given in appendix A.1 in eqs. (107) and (108).Substituting the expressions for area described in eqs. (30) and (38) into eq. (13), we maycompute the holographic entanglement entropy as follows S A = 14 G (4) N " La + Llr h + Lr h (cid:0) k + δ k (cid:1) + Lr h ǫ (cid:8) k + δ ( k + k log ǫ ) (cid:9) + O (cid:2) ǫ (cid:3) . (41)Comparing eq. (39) with eq. (41) we obtain E A = 32 (cid:16) S A − S th A (cid:17) , (42)– 13 –here as earlier we have identified S th A = Llr h G (4) N .Remarkably the holographic entanglement negativity in eq. (39) depends only on length L ,which is shared between the subsystem with the rest of the system. Note that this is equivalentto the area of the entangling surface between the subsystem and the rest of the system, whichfor the AdS /CF T scenario reduces to the length mentioned here. This is in contrast withthe holographic entanglement entropy reported in [27], and given in eq. (41), which scales withthe volume (which is the area in the AdS /CF T scenario) in this limit. However for theholographic entanglement negativity this volume dependent thermal term vanishes, resulting ina purely area dependent expression (length in the AdS /CF T scenario), which is in accordancewith quantum information theory results. We further note that this elimination of the thermalcontribution is similar to that of the AdS /CF T case reported in [28], which indicates that thiscancellation is possibly a universal feature of the holographic entanglement negativity for CF T sin any dimensions, besides providing a consistency check for the conjecture.Having examined the small charge and high temperature regime, we now move ahead toconsider the case of large charge and high temperature in subsection 3.3.3.
Having described the limit of small charge and high temperature in subsection 3.3.2, we nowturn our attention to the case of large charge and high temperature. In this limit the turningpoint of the co dimension two static minimal surface approaches close to the horizon, hence wehave r c ≈ r h and u ≡ r c r h ≈
1. In this limit the Taylor expansion of the lapse function f ( u )about u = u is given by eq. (32). Consequently in this case the finite part of the area A finite Σ ofthe co dimension two bulk minimal surface anchored on a subsystem Σ is given by eq. (33).Utilizing eqs. (30) and (33), the holographic entanglement negativity in the limit of largecharge and high temperature, for the mixed state configuration of the single subsystem A maynow be determined from eq. (12) as follows E A = 38 G (4) N " La + Lr h √ δ (cid:18) K ′ + K ′ ǫ (cid:19) + O (cid:2) ǫ (cid:3) , (43)with ǫ as given in eq. (40).The holographic entanglement entropy in this case may be determined from eq. (13) throughthe use of eqs. (30) and (33) as S A = 14 G (4) N " La + Llr h + Lr h √ δ (cid:18) K ′ + K ′ ǫ (cid:19) + O (cid:2) ǫ (cid:3) . (44)Comparing eqs. (43) and (44) we once again arrive at the following relation E A = 32 (cid:16) S A − S th A (cid:17) , (45)where as before we have S th A = Llr h G (4) N .Once again we observe from eq. (45) that the volume (area for AdS /CF T ) dependentthermal contribution drops out, leaving an expression for the holographic entanglementnegativity, given in eq. (43), that scales as the area (length for AdS /CF T ) of the entanglingsurface as earlier, in conformity with the quantum information theory. We further note that the– 14 –ancellation of the thermal term is similar to the corresponding AdS /CF T case, suggesting itto be a possible universal feature of the holographic entanglement negativity for all CF T s. Italso serves as a further consistency check for the holographic construction in [28].Having considered the three different limits pertaining to the charge and temperature of thenon extremal
AdS -RN black holes, we now proceed to consider the corresponding limits for theextremal AdS -RN black holes in subsection 3.4. AdS -RN black holes Having described in detail the holographic entanglement negativity for different limits of thefinite temperature mixed state configuration of a single subsystem with a long rectangular stripgeometry in a
CF T dual to a bulk non extremal AdS -RN black hole in subsection 3.3, in thissubsection we now proceed to investigate the cases involving an extremal black hole. Thisinvolves the zero temperature mixed state configuration of a single subsystem with a longrectangular strip geometry in a CF T dual to a bulk extremal AdS -RN black hole, in thelimits of small and large charge. We describe the small charge scenario in subsection 3.4.1, whilesubsection 3.4.2 portrays the regime of large charge. In the small charge regime, the suitable Taylor expansion of the function f ( u ) − about u = 0may be obtained as follows [27, 35] f ( u ) − = 1 + 2 r h r c u + O "(cid:18) r h r c u (cid:19) . (46)Utilizing eq. (46), the finite part of the area A finite Σ of the co dimension two bulk minimal surfaceanchored on a subsystem Σ may be obtained from eqs. (23) and (24) as [27] A finite Σ = k Ll + 4 k r h Ll + O (cid:2) r h l (cid:3) , (47)where as before l denotes the length of subsystem Σ in the direction of partition, and theconstants k and k are given in appendix A.1 in eqs. (103) and (104) respectively.As argued earlier (following [31]) in subsection 3.3.1, the subsystems B and A ∪ B becomeinfinite along the partitioning direction in the limit B → A c , and the corresponding minimalsurfaces approach the black hole horizon. Thus for these surfaces, we have r c ≈ r h and u ≡ r c r h ≈
1. For this case the Taylor expansion for the function f ( u ) appearing in eq. (22) may beexpanded around u = u as [27, 35] f ( u ) = 6 (cid:18) − r h r c u (cid:19) + O h ( u − u ) i . (48)For this case, the finite part of the area A finite Σ of the co dimension two bulk minimal surfaceanchored on a subsystem Σ is given by A finite Σ = Llr h + Lr h (cid:0) K + K √ ǫ + K ǫ (cid:1) + O h ǫ / i , (49)where l again denotes the length of subsystem Σ in the direction of partition. The constants K , K and K appearing in eq. (49) are detailed in appendix A.3 in eqs. (114), (115) and (116).– 15 –he parameter ǫ appearing in eq. (49) is given by [27] ǫ = π lr h − k l ) , (50)where the constant k l has been listed in appendix A.3 in eq. (117).Utilizing eqs. (30), (47) and (49), it is now possible to express the holographic entanglementnegativity in the limit of small charge, for the zero temperature pure state configuration of thesingle subsystem in question from the holographic construction in [28] by employing eq. (12) asfollows E A = 38 G (4) N " La + k Ll + 4 k r h Ll − Llr h + O (cid:2) r h l (cid:3) . (51)Using eqs. (30) and (47), we may derive the holographic entanglement entropy from eq. (13) asfollows S A = 14 G (4) N " La + k Ll + 4 k r h Ll + O (cid:2) r h l (cid:3) . (52)Comparison of eq. (51) with eq. (52) leads us to E A = 32 (cid:16) S A − S th A (cid:17) , (53)where S th A = Llr h G (4) N .We note that the first two terms on the right hand side of eq. (51) characterize theholographic entanglement negativity for the zero temperature pure state configuration of asingle subsystem in a CF T dual to the bulk pure AdS geometry, while the other terms denotethe correction stemming from the charge of the extremal AdS -RN black hole. We furtherobserve the recurrence of the elimination of the “thermal” contribution from the holographicentanglement negativity as described in eq. (53). Emergence of this universal feature againserves to corroborate the conjecture advanced in [28].Having considered the case of small charge in detail, in subsection 3.4.2 we proceed todescribe the corresponding large charge scenario. Having detailed the computation of the holographic entanglement negativity in the limit ofsmall charge in subsection 3.4.1, in this subsection we compute the same for its large chargecounterpart. In the large charge scenario, as reasoned in the non extremal case in subsection3.3.3, we have r c ≈ r h and u ≡ r c r h ≈
1. In this limit the Taylor expansion of the lapse functionaround u = u is given by eq. (48). Hence the finite part of the area A finite Σ of the co dimensiontwo bulk minimal surface anchored on a subsystem Σ for this configuration is given by eq. (49).As before, substituting the expressions for area described in eqs. (30) and (49) into eq. (12),we may obtain the holographic entanglement negativity in this limit of large charge, for the zero See footnote 7 on page 12 for details. Here thermal contribution stands for the entropy due to the ground state degeneracy of the black holemicrostates, which corresponds to the degeneracy of the vacuum of the dual
CF T . – 16 –emperature pure state configuration of the single subsystem under consideration as follows E A = 38 G (4) N " La + Lr h (cid:0) K + K √ ǫ + K ǫ (cid:1) + O h ǫ / i , (54)where the parameter ǫ is given by ǫ = π lr h − k l ) , (55)with the constant k l given in appendix A.3 in eq. (117).The corresponding holographic entanglement entropy may be obtained from eq. (13),utilizing eqs. (30) and (49) as S A = 38 G (4) N " La + Llr h + Lr h (cid:0) K + K √ ǫ + K ǫ (cid:1) + O h ǫ / i , (56)Comparing eqs. (54) and (56) we derive the following relation E A = 32 (cid:16) S A − S th A (cid:17) , (57)where as earlier we have identified S th A = Llr h G (4) N .Remarkably even for the extremal case in the regime of large charge, it is once again seen thatthe expression for the holographic entanglement negativity in eq. (54) scales as the area (lengthfor AdS /CF T ) of the entangling surface, as observed earlier for the various non extremal casesdescribed in subsection 3.3. Note that as before we have been able to derive the universal resultin eq. (57), providing a further validation of the proposal in [28].The results derived in subsections 3.3 and 3.4 for the holographic entanglement negativity ofa single subsystem in CF T s dual to various non extremal and extremal AdS -RN configurationsfor various limits of the relevant parameters are consistent with quantum information theory.We notice that in the limit of small charge and low temperature the leading contribution to theholographic entanglement negativity comes from the zero temperature configuration of singlesubsystem in the CF T dual to the bulk pure AdS geometry, while the correction term dependson the charge and the temperature. This is owing to the fact that the bulk static minimal surfacehomologous to the subsystem is located near the boundary far away from the black hole horizon,and thus the leading contribution arises from the near boundary pure AdS geometry. On theother hand for both the regimes of small charge and high temperature, and large charge andhigh temperature, the leading part is purely dependent on the area (length in AdS /CF T ) ofthe entangling surface between the subsystem and the rest of the system, while the volume (areain AdS /CF T ) dependent thermal term vanishes. In this case the dominant contribution stemsfrom the entanglement between the degrees of freedom at the entangling surface (entangling linefor AdS /CF T ). For the case of small charge with extremal AdS -RN black hole, the leadingcontribution involves the bulk pure AdS geometry term, and a correction term dependent onthe charge. However for the large charge regime, the leading contribution comes only from theterms dependent purely on the area (length in AdS /CF T ) of the relevant entangling surface,while once again the volume (area in AdS /CF T ) dependent thermal term drops out.We further observe that the holographic entanglement negativity is purely a function ofthe area of the entangling surface in the CF T even in the regimes of large charge and/orhigh temperature, whereas the corresponding holographic entanglement entropy is dominated– 17 –y volume dependent thermal contribution. This ensues from the precise cancellation of thevolume dependent thermal terms in the expressions for the holographic entanglement negativity.As discussed in the previous subsections, this elimination is reproduces the correspondingphenomena observed also for the
AdS /CF T scenario, and thus hints towards a universalcharacteristic of the entanglement negativity for holographic CF T s. Naturally these resultsprovide strong consistency checks, and serve to further substantiate the higher dimensionalholographic construction advanced in [28]. Having computed the holographic entanglementnegativity for different mixed state configurations of a single subsystem with long rectangularstrip geometry in a
CF T dual to extremal and non extremal AdS -RN black holes in differentlimits of charge and temperature, we now turn our attention to similar configurations for generichigher dimensional AdS d +1 /CF T d scenarios in section 4. CF T d dual to AdS d +1 -RN Having described in detail the computation of the holographic entanglement negativity for thevarious mixed state configurations of a single subsystem in the
AdS /CF T scenario in section3, we now proceed to analyze the corresponding cases for the AdS d +1 /CF T d scenario. Althoughwe employ a similar perturbative computation of the areas of the corresponding bulk staticminimal surfaces anchored on the respective subsystems as earlier, the relevant parameters ofthe AdS d +1 -RN black hole, characterizing the various limits for this case are the temperature T , and the chemical potential µ conjugate to the charge Q . It is convenient to express theholographic entanglement negativity in terms of the total energy ε of the dual CF T d , and an effective temperature T eff , both of which are functions of the temperature and the chemicalpotential. Note that the parameter ε is related to the expectation value of the component T of the energy momentum tensor [50].To this end we first review the perturbative calculation of the area of extremal surfacesrequired for the computation of the negativity in subsection 4.1 before moving on to the nonextremal AdS d +1 -RN configurations in subsection 4.2 and finally concluding with its extremalcounterparts in subsection 4.3. AdS d +1 -RN The metric for the
AdS d +1 -RN black hole (with d ≥
3) may be expressed as ds = 1 z (cid:18) − f ( z ) dt + dz f ( z ) + d~x (cid:19) ,f ( z ) = 1 − M z d + ( d − Q ( d − z d − ,A t ( z ) = Q (cid:16) z d − H − z d − (cid:17) , (58)where the AdS length scale has been set to unity, and Q and M are the mass and charge ofthe black hole respectively, and f is the lapse function. The horizon z H is determined by thesmallest real root of the equation f ( z ) = 0. The chemical potential µ , conjugate to the charge Q , is then defined as follows µ ≡ lim z → A t ( z ) = Qz d − H , (59)– 18 –hile the Hawking temperature is T = − π ddz f ( z ) (cid:12)(cid:12)(cid:12)(cid:12) z = z H = d πz H − ( d − Q z d − H d ( d − ! . (60)The total energy ε of the system, which is a dimensionless quantity with limits d − d − ≤ ε ≤ ε ( T, µ ) = b − n r d π b b (cid:16) µ T (cid:17) , (61)where the numerical constants b and b are listed in appendix B in eqs. (118) and (119)respectively.The lapse function, chemical potential and temperature may then be expressed in terms of ε as follows f ( z ) = 1 − ε (cid:18) zz H (cid:19) d + ( ε − (cid:18) zz H (cid:19) d − , (62) µ = 1 z H s ( d − d −
2) ( ε − , (63) T = 2 ( d − − ( d − ε πz H . (64)The effective temperature T eff , which describes the number of microstates for a given temperatureand chemical potential, may be defined as [50] T eff ( T, µ ) ≡ d πz H = T " s d π b b (cid:18) µ T (cid:19) . (65)We now proceed to review the computation the area of a co dimension two bulk staticminimal surface anchored on a subsystem A with long rectangular strip geometry in a CF T d dual to an AdS d +1 -RN black hole. The subsystem A is specified by x ≡ x ∈ (cid:20) − l , l (cid:21) , x i ∈ (cid:20) − L , L (cid:21) , i = 2 , . . . , d −
2; (66)with l ≪ L . The area A A of the co dimension two bulk extremal surface anchored on thissubsystem may then be expressed as A A = 2 L d − z d − ∗ Z l/ dxz ( x ) d − = 2 L d − z d − ∗ Z z ∗ a dzz d − r f ( z ) h z d − ∗ − z d − i , (67)where z ∗ is the turning point of the bulk extremal surface in the x direction, and a is the UV– 19 –ut off of the CF T d . The length l of the rectangular strip in the x direction is related to z ∗ as l = 2 Z z ∗ dz r f ( z ) h(cid:0) z ∗ z (cid:1) d − − i . (68)The integral in eq. (68) may be expressed as the following double sum [50] l = z ∗ d − ∞ X n =0 n X k =0 Γ (cid:2) + n (cid:3) Γ h d ( n + k +1) − k d − i ε n − k (1 − ε ) k Γ [1 + n − k ] Γ [ k + 1] Γ h d ( n + k +2) − k − d − i (cid:18) z ∗ z H (cid:19) nd + k ( d − . (69)The area of the static extremal surface in eq. (67) may likewise be described as a double sumas follows [50] A A = 2 d − (cid:18) La (cid:19) d − + 2 L d − z d − ∗ √ π Γ h − d − d − i d −
1) Γ h d − i + L d − ( d − z d − ∗ ∞ X n =1 n X k =0 Γ (cid:2) + n (cid:3) Γ h d ( n + k − − k +22( d − i ε n − k (1 − ε ) k Γ [1 + n − k ] Γ [ k + 1] Γ h d ( n + k ) − k +12( d − i (cid:18) z ∗ z H (cid:19) nd + k ( d − . (70)In what follows, we express the area and the turning point of the corresponding bulk staticextremal surface as perturbative expansions in terms of the specified parameters ε and T eff , forvarious limits of the temperature T and the chemical potential µ of the dual CF T d . The resultsthus obtained are then utilized to compute the holographic entanglement negativity for variouspure and mixed configurations of a single subsystem in the dual CF T d dual to AdS d +1 -RNnon extremal and extremal black holes, from the conjecture as described in eq. (11). Thesetup required for such computation is a higher dimensional generalization of that described insubsection 3.2.Here again we consider two auxiliary subsystems B and B on either side of the subsystem A along the direction of the x axis (henceforth referred to as the partitioning direction), specifiedby x ∈ (cid:20) − (cid:18) b + l (cid:19) , − l (cid:21) , x i ∈ (cid:20) − L , L (cid:21) ; (71)and x ∈ (cid:20) l , (cid:18) b + l (cid:19)(cid:21) , x i ∈ (cid:20) − L , L (cid:21) ; (72)respectively, where i = 2 , . . . , d −
2. Here L denotes the length of the strip in the remaining ( d − B and B to be symmetricalong the partitioning direction. This leads to the equality of the minimal areas A B = A B and A A ∪ B = A A ∪ B . With this identification, the holographic entanglement negativity for theabove configuration may be computed from eq. (12).Having outlined the procedure and the subsystem geometry to compute the holographicentanglement negativity for a single subsystem, we now proceed to describe these evaluations indetail in the following subsections. The cases involving CF T d dual to non extremal AdS d +1 -RNblack holes have been considered in subsection 4.2, while subsection 4.3 describes the extremalscenarios. – 20 – .2 Non extremal AdS d +1 -RN black holes Having considered the perturbative are computation techniques in subsection 4.1, we are nowin a position to take up the detailed computation of the holographic entanglement negativityfor different finite temperature mixed state configurations of a single subsystem with a longrectangular strip geometry in
CF T d s dual to non extremal AdS d +1 -RN black holes for variouslimits of the temperature T and the chemical potential µ . This involves the perturbativeexpansion of the areas of the bulk static extremal surfaces homologous to relevant combinationsof the subsystem and auxiliary subsystems, as mentioned in subsection 4.1. We set off with theregime of small chemical potential and low temperature in subsection 4.2.1, proceeding next tothe case of small chemical potential and low temperature in subsection 4.2.2 before finishingwith the large chemical potential and low temperature limit in subsection 4.2.3. The regime of small chemical potential and low temperature is described by the conditions µl ≪ T l ≪
1. Depending on whether T ≪ µ or T ≫ µ [50], this gives rise to two differentsituations as described below. (i) T l ≪ µl ≪ We first consider the case where T ≪ µ , which, together with µl ≪ T l ≪
1, may be recastas
T l ≪ µl ≪
1. In this limit, the parameters ε ( T, µ ) and T eff ( T, µ ), given by eqs. (61) and eq.(65), may be expanded about Tµ = 0 to the leading order as [50] ε ≈ b − nπ √ b b d (cid:18) Tµ (cid:19) , (73) T eff ≈ (cid:18) µdπ √ b b + T (cid:19) . (74)In this regime the turning point of the bulk static extremal surface remains far away from theblack hole horizon, i.e., z ∗ ≪ z H . Thus eq. (69) may be utilized to expand z ∗ in terms of (cid:16) lz H (cid:17) d as z ∗ = l Γ h d − i √ π Γ h d d − i −
12 ( d + 1) 2 d − − d Γ h d − i Γ h d − i d +1 π d +12 Γ h + d − i Γ h d d − i d ε (cid:18) lz H (cid:19) d + O "(cid:18) lz H (cid:19) d − . (75)The area of the minimal surface anchored on a generic subsystem Σ may likewise be computedfrom eq. (70) and re-expressed in terms of the parameters ε and T eff as follows [50] A Σ = 2 d − (cid:18) La (cid:19) d − + S (cid:18) Ll (cid:19) d − + ε S S (cid:18) πT eff d (cid:19) d L d − l + O h ( T eff l ) d − i . (76)where l is the length of subsystem Σ in the partitioning direction, and the numerical constants S and S are listed in appendix B in eqs. (120) and (121) respectively.– 21 –s discussed in subsection 3.3.1 following the arguments in [31], the subsystems B and A ∪ B in the CF T d become infinite along the direction of partition in the bipartite limit (i.e., B → A c ). Thus the minimal surfaces homologous to these subsystems penetrate into the bulkenough to approach the black hole horizon, i.e., z ∗ ≈ z H . For these surfaces the appropriateTaylor expansion gives rise to the following expression for the area [50] A Σ = 2 d − (cid:18) La (cid:19) d − + (cid:18) πT eff d (cid:19) d − L d − l + (cid:18) πT eff d (cid:19) d − L d − (cid:16) N + N ( b − ε ) (cid:17) + O (cid:20) Tµ (cid:21) . (77)The numerical constants N and N in eq. (77) are listed in appendix B in eqs. (122) and (123)respectively.Utilizing eqs. (76) and (77), the holographic entanglement negativity in the limit of smallchemical potential and low temperature (with T ≪ µ ), for the mixed state configuration ofsubsystem A may now be obtained from the construction in [28] by employing eq. (12) asfollows E A = 38 G ( d +1) N " d − (cid:18) La (cid:19) d − + S (cid:18) Ll (cid:19) d − + ε S S (cid:18) πT eff d (cid:19) d L d − l − (cid:18) πT eff d (cid:19) d − L d − l + O h ( T eff l ) d − i . (78)The corresponding holographic entanglement entropy is then computed from eq. (13) throughthe use of eq. (76) as follows S A = 14 G ( d +1) N " d − (cid:18) La (cid:19) d − + S (cid:18) Ll (cid:19) d − + ε S S (cid:18) πT eff d (cid:19) d L d − l + O h ( T eff l ) d − i . (79)Comparing eq. (78) with eq. (79) we arrive at E A = 32 (cid:16) S A − S th A (cid:17) , (80)where S th A = G ( d +1) N (cid:16) πT eff d (cid:17) d − V with V ≡ L d − l being the volume of the subsystem underconsideration.The first two terms on the right hand side of eq. (78) for the holographic entanglementnegativity correspond to the holographic entanglement negativity for the zero temperature purestate configuration of a single subsystem in a CF T d dual to bulk pure AdS d +1 geometry, while therest of the terms characterize the perturbative corrections due to the temperature and chemicalpotential of the black hole. As has been seen with the AdS /CF T scenario, we observe fromeq. (80) the universal relation between the holographic entanglement negativity and entropy,signifying the elimination of the volume dependent thermal contribution, which serves as astrong consistency check of our computation, and the holographic conjecture in [28]. As earlier, eq. (76) is employed to calculate A A , while A B and A A ∪ B are obtained utilizing eq. (77), followedby implementation of the bipartite limit b → ∞ . – 22 – ii) µl ≪ T l ≪ Next we consider the limit T ≫ µ . It may be consolidated with µl ≪ T l ≪ µl ≪ T l ≪
1. For this case the parameters ε ( T, µ ) and T eff ( T, µ ) described by eqs. (61) and(65) may have the following Taylor expansions around µT = 0 [50] ε = 1 + d ( d − π ( d − (cid:16) µT (cid:17) + O (cid:20)(cid:16) µT (cid:17) (cid:21) , (81) T eff = T " d ( d − π ( d − (cid:16) µT (cid:17) + O (cid:20)(cid:16) µT (cid:17) (cid:21) . (82)Once again the conditions T l ≪ µl ≪ z ∗ ≪ z H . The expression for z ∗ maybe obtained by expanding eq. (69) in terms of (cid:16) lz H (cid:17) d , and is same as the one given in eq. (75).The area of the extremal surface anchored on a generic subsystem Σ is once again determinedby expansion of eq. (70) in terms of (cid:16) lz H (cid:17) d as [50] A Σ = 2 d − (cid:18) La (cid:19) d − + S (cid:18) Ll (cid:19) d − + ε S S (cid:18) πT eff d (cid:19) d L d − l + O h ( T eff l ) d − i , (83)where the numerical constants S and S are the same as in eq. (76), and are given in appendixB in eqs. (120) and (121) respectively.We note here that in spite of the areas of the bulk static minimal surface as given in eqs.(76) and (83) having identical forms for both T l ≪ µl ≪ µl ≪ T l ≪
1, the underlyingparameters ε and T eff in these equations have different expressions for the two cases as may beobserved by comparing eqs. (73) and (74), with eqs. (81) and (82) respectively.As outlined in the discussion just preceding eq. (77) (page 22), the subsystems B and A ∪ B become infinite along the partitioning direction in the limit B → A c , and consequentlythe corresponding extremal surfaces extend deep into the bulk so that the turning points ofthese surfaces are close to the horizon (i.e., z ∗ ≈ z H ). The relevant expression for the area forthese surfaces is given as follows [50] A Σ = 2 d − (cid:18) La (cid:19) d − + (cid:18) πT eff d (cid:19) d − L d − l + (cid:18) πT eff d (cid:19) d − L d − γ d (cid:16) µT (cid:17) . (84)The function γ d (cid:0) µT (cid:1) in eq. (84) admits a perturbative expansion in µT , which has been describedin appendix B in eq. (124).The holographic entanglement negativity for the required finite temperature mixed stateconfiguration of the single subsystem in question with long rectangular strip geometry, in thelimit of small chemical potential and low temperature (with T ≫ µ ) may then be computed from eqs. (83) and (84), using the holographic conjecture described by eq. (12), and is given inform by eq. (78). Relations described in eqs. (79) and (80), and the discussions just after eq.(80) also follow. Once again A A is computed by applying eq. (83), whereas eq. (84) is utilized to obtain A B and A A ∪ B , andafterwards we take the bipartite limit b → ∞ . – 23 – .2.2 Small chemical potential - high temperature Having described the computation of the holographic entanglement negativity for the mixed statein question for the two cases in the limit of small chemical and low temperature in subsection4.2.1, we now proceed to determine the same in the limit of small chemical potential and hightemperature. As detailed in [50], this regime is given by the conditions µ ≪ T and T l ≫
1. Theparameters ε and T eff may be given in the limit µ ≪ T (refer to case (ii), subsection 4.2.1 fordetails) by eqs. (81) and (82). The other condition T l ≫ z ∗ ≈ z H . Thus the area of thebulk extremal surface may be obtained by perturbative expansion of eq. (70) around z ∗ z H = 1,and is given by eq. (84).Substituting eq. (84) into eq. (12), the holographic entanglement negativity in the limit ofsmall chemical potential and high temperature, for the mixed state configuration of the singlesubsystem under consideration may be determined as follows E A = 38 G ( d +1) N " d − (cid:18) La (cid:19) d − + (cid:18) πT eff d (cid:19) d − L d − γ d (cid:16) µT (cid:17) . (85)Next we employ eq. (84) to compute the corresponding holographic entanglement entropy fromeq. (13) as follows S A = 14 G ( d +1) N " d − (cid:18) La (cid:19) d − + (cid:18) πT eff d (cid:19) d − L d − l + (cid:18) πT eff d (cid:19) d − L d − γ d (cid:16) µT (cid:17) . (86)Comparing eqs. (85) and (86) we again obtain E A = 32 (cid:16) S A − S th A (cid:17) , (87)where S th A = G ( d +1) N (cid:16) πT eff d (cid:17) d − L d − l as before.Once again we observe that the expression for the holographic entanglement negativity ineq. (85) for the mixed state in question is purely dependent on the area L d − of the entanglingsurface between the single subsystem with the rest of the system. As earlier this indicates thevanishing of the volume dependent thermal contribution, as may be seen from eq. (87). This isagain in agreement with the usual quantum information theory expectations, and constitutes afairly strong consistency check for the holographic construction proposed in [28].Having completed the computation of the holographic entanglement negativity in the smallchemical potential and high temperature regime, we now turn our attention to the same for thecase of large chemical potential and low temperature in subsection 4.2.3. With the calculation of the holographic entanglement negativity for the small chemical potentialand high temperature limit in subsection 4.2.2, we continue in this subsection with the largechemical potential and low temperature regime, is described by the conditions µl ≫ T ≪ µ .As described earlier (case (i), subsection 4.2.1 ), the condition T ≪ µ implies that theparameters ε ( T, µ ) and T eff ( T, µ ) admit of the expansions described by eqs. (73) and (74)respectively. Similar to subsection 4.2.2, the condition µl ≫ z ∗ ≈ z H . Hence, the areaof the bulk extremal surface may once again be obtained through the perturbative expansion of– 24 –q. (70) around z ∗ z H = 1 [50], and is given by eq. (77).Utilizing eq. (77), we may now obtain the holographic entanglement negativity in the limitof large chemical potential and low temperature, for the mixed state configuration of subsystem A , from the construction advanced in [28] by employing eq. (12) as follows E A = 38 G ( d +1) N " d − (cid:18) La (cid:19) d − + (cid:18) πT eff d (cid:19) d − L d − (cid:16) N + N ( b − ε ) (cid:17) + O (cid:20) Tµ (cid:21) . (88)Eq. (77) may then be put to use to determine the holographic entanglement entropy for thiscase from eq. (13) as follows S A = 14 G ( d +1) N " d − (cid:18) La (cid:19) d − + (cid:18) πT eff d (cid:19) d − L d − l + (cid:18) πT eff d (cid:19) d − L d − (cid:16) N + N ( b − ε ) (cid:17) + O (cid:20) Tµ (cid:21) . (89)Once again comparison of eqs. (88) and (89) leads us to E A = 32 (cid:16) S A − S th A (cid:17) , (90)where as earlier S th A = G ( d +1) N (cid:16) πT eff d (cid:17) d − L d − l .Again the holographic entanglement negativity in eq. (88), obtained in the limit of largechemical potential and high temperature, is purely dependent on the area of the relevantentangling surface. As before this happens due to the precise cancellation of the volumedependent thermal term described by eq. (90), in conformity with the standard quantuminformation theory expectations, and serves to further substantiate the holographic conjectureproposed in [28].Having described in detail the computation of the holographic entanglement negativity forthe required single subsystem with long rectangular strip geometry for various cases involving CF T d s dual to bulk non extremal AdS d +1 -RN black holes, in subsection 4.3 we turn our attentionto the same for two different cases involving extremal AdS d +1 -RN black holes. AdS d +1 -RN black holes Having obtained the holographic entanglement negativity for the required finite temperaturemixed state configurations in the
CF T d s dual to the bulk non extremal AdS d +1 -RN black holesin subsection 4.2, we now set out to outline the procedure for zero temperature mixed stateconfigurations dual to bulk extremal AdS d +1 -RN black holes. The relevant parameters for these– 25 –xtremal configurations are given as [50] Q = d ( d − L ( d − z d − H , (91) ε = b , (92) µ = 1 z H r b b z H s d ( d − d − , (93) T eff = µd π √ b b . (94)In the above equations the parameter Q describes the charge of the extremal AdS d +1 -RN blackhole, while T eff as earlier denotes the effective temperature.In the next two subsections we proceed to obtain perturbative expansions of the area ofthe co dimension two bulk extremal surface homologous to the single subsystem with longrectangular strip geometry, for two different limits of the charge Q . The expression so obtainedwill then be employed to compute the holographic entanglement negativity for the configurationunder consideration from the conjecture proposed in [28]. In subsection 4.3.1 we portray thecase involving a small chemical potential, while subsection 4.3.2 describes the large chemicalpotential configuration. The limit of small chemical potential is defined by the condition µl ≪
1, which entails z ∗ ≪ z H . The solution of eq. (68) in this limit once again leads to eq. (75) [50]. The area of theextremal surface anchored on a generic subsystem Σ may be similarly obtained from eq. (70)and re-expressed in terms µ as follows [50] A Σ = 2 d − (cid:18) La (cid:19) d − + S (cid:18) Ll (cid:19) d − + 2 ( d − d − S S µ ( d − p d ( d − ! d L d − l + O h ( µl ) d − i . (95)where the constants S and S are identical to those in the earlier cases described in subsection4.2.1, and are given in appendix B in eqs. (120) and (121) respectively.As discussed in earlier subsections, the turning points of the extremal surfaces correspondingto the infinite auxiliary subsystems approach the black hole horizon (i.e., z ∗ ≈ z H ). Theappropriate area expansion for these surfaces are given as [50] A Σ = 2 d − (cid:18) La (cid:19) d − + µ ( d − p d ( d − ! d − L d − l + µ ( d − p d ( d − ! d − L d − N ( b ) , (96)where N ( b ) is the value of N ( ε ) at ε = b . It is listed in appendix B in eq. (125).Utilizing eqs. (95) and (96), it is now possible to compute the holographic entanglementnegativity in the limit of small chemical potential, for the pure state configuration of the single– 26 –ubsystem in question by employing eq. (12) as follows E A = 38 G ( d +1) N d − (cid:18) La (cid:19) d − + S (cid:18) Ll (cid:19) d − + 2 ( d − d − S S µ ( d − p d ( d − ! d L d − l − µ ( d − p d ( d − ! d − L d − l + O h ( µl ) d − i . (97)We then employ eq. (95) to obtain the following expression for the holographic entanglemententropy from eq. (13) S A = 14 G ( d +1) N d − (cid:18) La (cid:19) d − + S (cid:18) Ll (cid:19) d − + 2 ( d − d − S S µ ( d − p d ( d − ! d L d − l + O h ( µl ) d − i . (98)Through the comparison of eqs. (97) and (98) we arrive at the following relation E A = 32 (cid:16) S A − S th A (cid:17) , (99)where we have S th A = G ( d +1) N (cid:18) µ ( d − √ d ( d − (cid:19) d − L d − l. In eq. (97) we note that the first two terms describe the holographic entanglement negativityfor the zero temperature pure state configuration of a single subsystem with long rectangularstrip geometry in a
CF T d dual to bulk pure AdS d +1 geometry. The rest of the terms representthe correction due to the chemical potential of the black hole. We further observe the cancellationof the volume dependent thermal contribution in eq. (99) indicating the possible universalityof this feature for holographic entanglement negativity, while serving to validate the conjecturein [28].With the computation of the negativity for the case of small charge having been described,we now conclude this section with the large charge scenario in subsection 4.3.2. Having considered the small charge regime in subsection 4.3.1, in this subsection we take up thelarge chemical potential regime, specified by the condition µl ≫
1. With this condition it maybe observed from eq. (93) that the horizon radius is large, and the turning point of the extremalsurface is thus close to the horizon (i.e., z ∗ ≈ z H ). In this regime, the area of the bulk staticextremal surface anchored on a generic subsystem Σ with long rectangular strip geometry maybe determined by perturbative computation of the integral in eq. (70) around z ∗ z H = 1, and isgiven by eq. (96).The holographic entanglement negativity in the limit of large chemical potential, for the purestate configuration of the single subsystem under consideration may now be derived through the Similar to the extremal
AdS -RN configurations, the thermal contribution characterizes the entropy due tothe ground state degeneracy of the black hole, corresponding to the vacuum degeneracy of the dual CF T . – 27 –se of eq. (96) by employing eq. (12) as follows E A = 38 G ( d +1) N d − (cid:18) La (cid:19) d − + µ ( d − p d ( d − ! d − L d − N ( b ) . (100)The corresponding expression for the holographic entanglement entropy is then computed fromeq. (96) utilizing eq. (13) as S A = 14 G ( d +1) N d − (cid:18) La (cid:19) d − + µ ( d − p d ( d − ! d − L d − l + µ ( d − p d ( d − ! d − L d − N ( b ) . (101)Once again comparing eqs. (100) and (101) we arrive at E A = 32 (cid:16) S A − S th A (cid:17) , (102)where as before we have identified S th A = G ( d +1) N (cid:18) µ ( d − √ d ( d − (cid:19) d − L d − l. We once again take note that the holographic entanglement negativity computed in eq.(100) for the large chemical potential regime is purely dependent on the area of the appropriateentangling surface, due to the elimination of the volume dependent thermal contributiondescribed in eq. (102), which agrees with quantum information theory expectations. Furthermorethe universality of the relation in eq. (102) substantiates the construction in [28] as earlier.We note that all the holographic entanglement negativity results obtained in subsections4.2 and 4.3 for the various pure and mixed state configuration of a single subsystem with longrectangular strip geometry in
CF T d s dual to various non extremal and extremal AdS d +1 -RNconfigurations in the appropriate limits of the relevant parameters are in conformity withthe quantum information theory expectations. Furthermore the leading contribution to theholographic entanglement negativity for the AdS d +1 /CF T d framework exhibits similar behaviorto that of the corresponding cases in the AdS /CF T scenario in the corresponding limits of therelevant parameters.In the limit of small chemical potential and low temperature, the leading contribution arisesfrom the zero temperature configuration in the CF T d dual to bulk pure AdS d +1 geometry, plusthe correction terms due to the chemical potential and the temperature of the black hole. Inthis case the leading contribution comes from the near boundary pure AdS d +1 geometry, asthe bulk static minimal surface anchored on the subsystem is located near the boundary faraway from the black hole horizon. In contrast, for both the cases of small chemical potentialand high temperature, and large chemical potential and low temperature, the leading termsare purely dependent on the area of the entangling surface between the subsystem with therest of the system, and the volume dependent thermal term cancels. For these regimes thearea dependent entanglement terms dominate the holographic entanglement negativity. For theextremal AdS d +1 -RN configurations, the dominant contribution for the small chemical potentialregime consists of the bulk pure AdS d +1 geometry term, and correction terms dependent on the– 28 –hemical potential. Furthermore for the large chemical potential limit, the leading contributionarises only from the terms dependent purely on the area of the entangling surface, with thevolume dependent thermal terms canceling. The reason for these behaviors are identical tothose for the non extremal cases described earlier.Interestingly, similar to the AdS /CF T scenario for large chemical potential and/or hightemperature, the leading terms for the holographic entanglement negativity depends purelyon the area of the relevant entangling surface. In contrast the corresponding holographicentanglement entropy is dominated by volume dependent thermal terms. Furthermore justas for the AdS /CF T scenario the cancellation of the volume dependent thermal term for the AdS d +1 /CF T d framework is similar to that observed for the lower dimensional AdS /CF T scenario indicating once again the possible universality of this feature for the holographicentanglement negativity in CF T d s. These similarities serve as strong consistency checks, besidesfurther validation of the higher dimensional holographic construction proposed in [28]. To summarize we have investigated the application of a higher dimensional holographicentanglement negativity proposal for bipartite states described by a single connected subsystemin
CF T d s dual to bulk AdS d +1 geometries advanced in [28] to specific examples. The proposalinvolved a specific algebraic sum of the areas of co dimension two bulk static minimal surfaceshomologous to appropriate combinations of the subsystem in question and certain auxiliarysubsystems in the dual CF T d . In this context we have established a construction to computethe holographic entanglement negativity for such bipartite states characterized by subsystemswith long rectangular strip geometries in CF T d s with a conserved charge dual to bulk nonextremal and extremal AdS d +1 -RN black holes.In this connection we have reviewed in details the perturbative techniques utilized by us tocompute the areas of the relevant bulk co dimension two surfaces homologous to appropriatecombinations of subsystems in the dual CF T d with a conserved charge as mentioned above.To establish the perturbative techniques in this context, we have first considered specific casesinvolving various limits of the charge and the temperature in the AdS /CF T scenario andobtained the holographic entanglement negativity for the bipartite states described by a singlesubsystem with long rectangular strip geometries in CF T s with a conserved charge dual tobulk non extremal and extremal AdS -RN black holes. Interestingly for all the cases consideredby us we have demonstrated the elimination of the thermal contributions for the holographicentanglement negativity similar to the case of CF T d s dual to bulk AdS d +1 -Schwarzschild blackholes described in the literature. The above feature for the holographic entanglement negativityis also observed for the corresponding AdS /CF T scenario. This justifies its characterizationas the upper bound on the distillable entanglement in quantum information theory and alsoindicates a possible universality for holographic CF T d s. Furthermore, this serves as a strong nontrivial consistency check for the higher dimensional holographic for the entanglement negativityas proposed and exemplified in the literature.Subsequently we have computed the corresponding holographic entanglement negativityfor the above subsystem configuration in the general AdS d +1 /CF T d scenario. In this contextwe have discussed the various regimes for the perturbative expansions of the holographicentanglement negativity characterized by the total energy and an effective temperature of the CF T d which are functions of the temperature and the chemical potential conjugate to the charge.It is observed that similar to the AdS /CF T context, the holographic entanglement negativity– 29 –nvolves the elimination of the thermal contributions conforming to its characterization as anupper bound on the distillable entanglement and indicating the universality of this feature alsoobserved in the AdS /CF T scenario. Once again this constitutes a significant consistency checkfor the higher dimensional construction for the holographic entanglement negativity in this case.As mentioned earlier in the introduction there is an alternate holographic proposal for theentanglement negativity which involves the area of the back reacted bulk minimal entanglementwedge cross section due to the cosmic brane for the bulk conical defect in the replica limit.For subsystems with spherical entangling surfaces this back reaction may be described by anoverall dimension dependent constant. While this was explicitly verified in the AdS /CF T context it still requires substantiation for higher dimensional examples. For the AdS /CF T context it has been demonstrated that the algebraic combination of the lengths of bulk geodesiccombinations relevant to the computation of the holographic entanglement negativity from theproposal described earlier was shown to be proportional to the area of the minimal entanglementwedge cross section where the proportionality constant depends on the back reaction factor. Thisgeometrical conclusion is expected to also hold for higher dimensions and in this connection ourresults presented here may also be modified by such an overall constant factor arising from theback reaction of the bulk cosmic brane. However for the subsystems with long rectangular stripgeometries considered here, the explicit computation of this constant is a non trivial open issuefor future investigations. We emphasize here however that all the conclusions following from ourresults will not be affected by such an overall constant factor.We mention here that the explicit computation of the holographic entanglement negativityfor other subsystem geometries from the proposal utilized here remains a non trivial openissue for future investigations. Our results serves to further substantiate the holographicentanglement negativity proposal for bipartite states in CF T d s in the context of a generic higherdimensional AdS d +1 /CF T d scenario which may possibly find interesting applications to diverseissues involving mixed state entanglement for disparate areas ranging from many body systemsin condensed matter physics to quantum gravity and black holes. Furthermore they are alsoexpected to provide insight into a possible bulk proof for the holographic proposal consideredhere which remains an outstanding issue. We hope to address these interesting problems in thenear future. Acknowledgment
We would like to thank Vinay Malvimat for crucial suggestions and discussions. The workof Sayid Mondal is supported in part by the Taiwan’s Ministry of Science and Technology(grant No. 106-2112-M-033-007-MY3 and 109-2112-M-033-005-MY3) and the National Centerfor Theoretical Sciences (NCTS). – 30 – ppendix A Non extremal and extremal
AdS -RN A.1 Non extremal
AdS -RN (Small charge - low temperature) The constants k and k appearing in eq. (31) (page 11) in subsection 3.3.1 and in eq. (47) (page15) in subsection 3.4.1 are given as follows k = − π Γ (cid:2) (cid:3) Γ (cid:2) (cid:3) , (103) k = Γ (cid:2) (cid:3) (cid:2) (cid:3) . (104)The constants K ′ and K ′ appearing in eq. (33) (page 12) in subsection 3.3.1 are given by K ′ = − √ π Γ (cid:2) (cid:3) Γ (cid:2) (cid:3) + ln 4 −
108 + 12 ∞ X n =2 n − (cid:2) n + (cid:3) Γ [ n + 1] Γ (cid:2) n +34 (cid:3) Γ (cid:2) n +54 (cid:3) − n ! + π , (105) K ′ = π − . (106)The constants c and c in eq. (34) (page 12) in subsection 3.3.1 and in eq. (40) (page 13)in subsection 3.3.2 are given as follows c = √ π (cid:2) (cid:3) Γ (cid:2) (cid:3) + ∞ X n =1 Γ (cid:2) n + (cid:3)
2Γ [ n + 1] Γ (cid:2) n +34 (cid:3) Γ (cid:2) n +54 (cid:3) − √ n ! , (107) c = 1 √ − ∞ X n =0 Γ (cid:2) n + (cid:3) Γ [ n + 1] Γ (cid:2) n +64 (cid:3) Γ (cid:2) n +84 (cid:3) − √ ! + 32 ∞ X n =0 Γ (cid:2) n + (cid:3) Γ [ n + 1] Γ (cid:2) n +74 (cid:3) Γ (cid:2) n +94 (cid:3) − √ ! . (108) A.2 Non extremal
AdS -RN (Small charge - high temperature) The constants k , k , k , k and k in eq. (38) (page 13) in subsection 3.3.2 are given as follows k = ∞ X n =1 n − (cid:2) n + (cid:3) Γ [ n + 1] Γ (cid:2) n +34 (cid:3) Γ (cid:2) n +54 (cid:3) − √ n ! + π √ √ π Γ (cid:2) − (cid:3) Γ (cid:2) (cid:3) , (109)– 31 – = 3 π − (cid:2) (cid:3) Γ (cid:2) (cid:3) Γ (cid:2) (cid:3) + 3 ∞ X n =1 n + 2 Γ (cid:2) n + (cid:3) Γ [ n + 1] Γ (cid:2) n +64 (cid:3) Γ (cid:2) n +84 (cid:3) − √ n ! − ∞ X n =1 n + 3 Γ (cid:2) n + (cid:3) Γ [ n + 1] Γ (cid:2) n +74 (cid:3) Γ (cid:2) n +94 (cid:3) − √ n ! , (110) k = − √ π √ , (111) k = 2 √ − √ √ π Γ (cid:2) (cid:3) Γ (cid:2) (cid:3) Γ (cid:2) (cid:3) , (112) k = − √ . (113) A.3 Extremal
AdS -RN (Small charge) The constants K , K and K appearing in eq. (49) (page 15) in subsection 3.4.1 are listed asfollows K = 2 √ " − √ π Γ (cid:2) (cid:3) Γ (cid:2) (cid:3) + ln 44 − √ π √ πζ (cid:18) (cid:19) + √ π ∞ X n =2 n − (cid:2) n +34 (cid:3) Γ (cid:2) n +54 (cid:3) − n √ n ! , (114) K = − π √ , (115) K = 2 √ (cid:20) − √ π + √ πζ (cid:18) (cid:19)(cid:21) . (116)The constant k l appearing in eq. (50) (page 16) in subsection 3.4.1 and in eq. (55) (page17) in subsection 3.4.2 is given as follows k l = r π " Γ (cid:2) (cid:3) (cid:2) (cid:3) + ∞ X n =1 Γ (cid:2) n +34 (cid:3) (cid:2) n +54 (cid:3) − √ n ! + ζ (cid:18) (cid:19) . (117)– 32 – ppendix B Non extremal and extremal AdS d +1 -RN The numerical constants b and b appearing in eq. (61) (page 19) and in eq. (65) (page 19) insubsection 4.1 are given as follows b = 2 ( d − d − , (118) b = dd − . (119)The constants S and S appearing in eq. (76) (page 21) and in eq. (83) (page 23) insubsection 4.2.1, and in eq. (95) (page 26) in subsection 4.3.1 are given as follows S = 2 d − π d − Γ h − d − d − i ( d −
1) Γ h d − i Γ h d d − i Γ h d − i d − , (120) S = Γ h d − i d +1 − d − π − d Γ h d d − i d Γ h + d − i Γ h d − i Γ h − d − d − i + 2 d − ( d −
2) Γ h d − i √ π ( d + 1) . (121)The numerical constants N and N appearing in eq. (77) (page 22) in subsection 4.2.1 aregiven as follows N = 2 √ π Γ h − d − d − i d −
1) Γ h d − i + 2 Z dx p − x d − x d − p − b x d + b x d − − x d − p − x d − ! , (122) N = Z dx x p − x d − p − b x d + b x d − ! (cid:18) − x d − − b x d + b x d − (cid:19) . (123)The function γ d (cid:0) µT (cid:1) appearing in eq. (84) (page 23) in subsection 4.2.1 is given as follows γ d (cid:16) µT (cid:17) = N (1) + d ( d − π ( d − (cid:16) µT (cid:17) Z dx x p − x d − √ − x d ! (cid:18) − x d − − x d (cid:19) + O (cid:20)(cid:16) µT (cid:17) (cid:21) , (124)– 33 –here the expression for the numerical constant N ( ε ) is given in eq. (125).The numerical constant N ( ε ) in eq. (124), which also appears in eq. (96) (page 26) insubsection 4.3.1, is given as follows N ( ε ) = 2 √ π Γ h − d − d − i d −
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