A covariant holographic entanglement entropy proposal

“, GKP-W relation [38, 39] for the AdS/CFT correspondence.”

“, GKP-W relation [38, 39] for the AdS/CFT correspondence.”

“In string theory a natural realization of the holographic principle is manifested by the AdS/CFT correspondence [18, 19] which gives us a precise map between a quantum gravity theory on an asymptotically AdS spacetime M and an ordinary QFT on the conformal boundary ∂M of M.”

“to the bulk-boundary relation within the AdS/CFT context in § 4. In § 5 we illustrate the calculations of the entanglement entropy using our covariant proposal and demonstrate the consistent agreement with the minimal surface prescri pt ion of [24, 10].”

“Roughly speaking, the holographic principle states that the number of degrees of freedom in a quantum theory of gravity scales with the area of the system, in contrast to standard QFTs where the entropy is extensive and scales with the volume [15, 16, 17].”

“Recently, entanglement entropies of various 1+1 and 2+1 dimensional condensed matter systems have been actively investigated in order to understand zero temperature quantum phase transitions [11, 4, 12, 13, 14].”

“Using the relation between the central charge of the dual CFT and the bulk Newton’s constant c = 3 2G (3) N [50], we obtain the expression in the dual CFT language”

“The same general result (8.6) can be simply derived holographically using the HRT [34] prescription in Banados geometries [44].”

“J H E P 0 1 ( 2 0 2 3 ) 1 0 5 It should be pointed out that volume-complexity1 was also introduced for some subregions in the bulk spacetime such as in a region [11, 12] enclosed by a (H)RT hypersurface [13, 14] or inside a Wheeler-DeWitt (WDW) patch [15].”

“The Vaidya solution is the simplest example of a time dependent gravitational background which is explored in the context of the RT conjecture [6,68,69].”

“The original derivation of the membrane theory relies on the HRT prescription [3] for the computation of the holographic entanglement entropy in dynamical settings.”

“Perhaps the most prominent of all these examples is the holographic computations of entanglement entropy of the dual boundary theory of interest which is particularly useful in determining the entanglement entropy of pure states [4, 5, 6, 7, 8].”

“When γA∪B = γwormhole, the computation of S(A ∪ B) follows from the Hubeny-Rangamani-Takayanagi (HRT) prescription [29] where the determinants of induced metrics corresponding to the extremal surfaces are,”

“More precisely, the generalized entropy of (II) is well-approximated, to leading order in 1/GN , in (III) by a classical entanglement surface computed via the Ryu-Takayanagi (RT) prescription [91] (or its covariant extension [92]) — the surface is extremal and thus must satisfy appropriate boundary condition on the br…”

“A cornerstone of this relation is the Ryu-Takayanagi (RT) formula [ 2, 3 ] and its generalizations [ 4–7 ] , which relate extremal areas in spacetime to entanglement entropies of the corresponding quantum state in the holo-graphically dual theory.”

“Intriguingly, within the AdS/CFT framework [7, 8, 9], entanglement entropy has been discovered to correlate with minimal surfaces in the corresponding spacetime [10, 11], adhering to a principle akin to the area law observed in black holes.”

“Subsequently the proposal of [18, 19] has been generalized to time-dependent situations [20] and, more relevant for this paper, to confining backgrounds [21] (see also [22]).”

“In holographic duality, key insights concerning the microscopic fabric of spacetime—the Ryu-Takayanagi proposal [1, 2] with its various reformulations and generalizations [3–5], holographic entropy inequalities [6, 7], error correction [8] or the analogy to tensor networks [9–11]—all stipulate the existence of some pr…”

“For covariant cases, RT formula is reformulated into the Hubeny-Rangamani-Takayanagi (HRT) formula [13, 14].”

“Later, RT formula is extended to the covariant case, which is dubbed as the Hubeny-Rangamani-Takayanagi (HRT) formula [8, 9].”

“[6] V.”

“Via the HRT formula [38, 39], this translates into finding extremal surfaces anchored at boundary curves γ(y) in the null surfaces in asymptotically AdS space.”

“of HRT surfaces in standard charged and rotating black holes6 3 Discussion6 1 Introduction As is by now well established [1,2], in AdS/CFT the Ryu-Takayangi [3,4] and HubenyRangamani-Takayanagi (HRT) [5] prescriptions generally describe the von Neumann entropy of CFT regions Ain terms of the area of an appropriate bul…”

“The covariant version of the RT proposal for the bulk metric, which is either non-static or even non-stationary, was presented in [6] and the HEE was derived for rotating BTZ black holes as a stationary but non-static spacetime.”

“Furthermore, a covariant generalization of this formula was proposed in [11].”

“The lengths of the extremal curves may now be obtained in the un-primed coordinates through the standard Poincar´e AdS 3 result as follows [10,11] (5.16) In the above expression, (cid:15) is the UV cut-off for the original BCFT 2 in the primed coordinates and the second logarithmic term arises due…”

“A prime example is the holographic entanglement entropy proposed by Ryu-Takayanagi[4, 5], manifesting the idea that geometry is emergent.”

“To find area of γ wormhole we use the prescription given by Hubeny–Rangamani–Takayanagi (HRT) [20].”

“The island formula may then be obtained through the RT prescription [9–11] in the corresponding higher-dimensional bulk perspective.”

“…boundary CFT is proportional to the area of a minimal spacelike surface (referred to as the RT surface) that ends on the boundary of 𝐴 in the bulk AdS space 11 , we may See [113] for the covariant generalization. gain new insights into how the gauge structure influences entanglement properties…”

“the entanglement entropy of progressively larger boundary intervals is understood to probe deeper in the bulk via the Ryu-Takayanagi prescription [17, 18]), in agreement with the expectation that points deep in the bulk are sensitive to dual infrared energy scales.”

“In recent years a coherent picture of a particular section of the dictionary, under the heading ‘subregion duality’, has emerged [3–14].”

“A further direction worth exploring is a possible generalization of entwinement to find a dual for geodesics not affixed to a constant time slice, in the spirit of the HRT proposal [65] (see also [66] for the related concept of residual entropy).”

“This idea is supported by the Ryu Takayanagi formula [1, 2] and its covariant generalization [3], which relate the area of particular extremal surfaces in the bulk, to the entanglement entropies in the dual conformal field theory (CFT).”

“puting relevant Ryu-Takayanagi [28, 29] or the covariantized HRT surfaces [30] [8–10].”

“Advances in this direction rely on a deeper understanding of the calculation of the entropy of Hawking radiation, which has been possible thanks to the identification of a new contribution to the entanglement entropy, following from a quantum generalization [7] of the holographic entanglement entropy prescription [8,…”

“The Ryu-Takayanagi proposal has been generalized to covariant formulae to suit generic backgrounds in [7], and in this paper we will study the behavior of extremal codimension-two surfaces in backgrounds which are small deformations of pure AdS spacetime.”

“Extremizing A then gives SEE [17, 19], SEE = A[Wext] 4GN .”

“Their proposal, originally conjectural and referring only to static situations, was extended to the covariant setting in [5] by taking to be an extremal surface, and later proved in”

“We also left out the case F ≥ 0 and μ+ 6= μ− which requires using the Hubeny, Rangamani, and Takayanagi prescription [35].”

“The RT proposal [2], in its generalized form due to [3], asserts that, for theories with an Einstein gravity dual, the entanglement entropy is obtained by a simple geometric calculation,”

“Left: starting from a TFD state with inverse temperature β0, fixing the area of the HRT surface γR, corresponding to the time slice τ = 0, results in a Euclidean saddle with a conical singularity (red) with opening angle 2πm, where m = β0β .”

“This will also provide a new renormalization scheme for the area Ryu-Takayanagi surface of a spherical ball subregion of boundary CFTd in d ≥ 2.”

“Through the holographic entanglement entropy formula [65–67], the entanglement entropy measures the distance between AdS boundary and the EOW brane.”

“This investigation may be useful in the AdS/CFT correspondence through the Hubeny-Rangamani-Takayanagi conjecture, and as part of the entanglement tsunami picture [34–37].”

“In particular, AdS/CFT provides a geometrical method to compute entanglement entropy [3, 4] (see [5, 6] for its derivations).”

“e region we rst recall the covariant version of the Ryu-Takayangi proposal (usually called the HRT or Hubeny-Rangamani-Takayanagi proposal) for holographic entanglement entropy of a boundary region R [38,39]. According to the HRT proposal, the entanglement entropy of R is equal to the area of the extremal surface R, w…”

“In the context of AdS / CFT correspondence [ 1 ] , the Ryu-Takayanagi (RT) formula [ 2–4 ] un-covers a relation between bulk geometry and boundary entanglement, by claiming that the entanglement entropy (EE) S A for a region A in the boundary CFT is given by the area of a minimal surface E A…”

“The holographic calculation of the EE, which was proposed by Ryu and Takayanagi (RT) [1, 2] and its covariant generalization [3], drew much attention due to its elegance and implications in condensed matter physics and gravity theories.”

“Although the prescription of the extremal quantum surface was proposed in the framework of holography [23], [24], the island rule is applicable to black holes in more general theories.”

“One such example is the holographic calculation of entanglement entropy [4].”

“inates and Penrose diagrams 17 B SdS3 and extremal surfaces 18 C Analytic expressions for the area integrals 20 1 Introductionand summary Over the last several years, holographic entanglement entropy [1, 2, 3, 4] has been under substantial investigation, both from the point of view of gaining new insights on strongly…”

“Recently, a prescription (similar to the Ryu-Takayanagi’s proposal for the CFT entanglement entropy [16, 17]) has been proposed for the BMSFT entanglememnt entropy [12] that relates it to the area of some particular”

“J H E P 0 3 ( 2 0 2 0 ) 1 8 6 two independent approaches; the first one based on some symmetry considerations and the second on the HRRT holographic prescription [33, 34].”

“Note that even though the location of the particle is time-dependent, the background spacetime is static6 and at any given time we may use a Ryu-Takayanagi (RT) surface [26, 27] (instead of its dynamical generalization — the HRT surface [28]).”

“This formula comes from the holographic entanglement entropy(HEE) formula (or RT formula) [10, 11] and its quantum version, i.e. the quantum extremal surface (QES) [12, 13].”

“The entanglement entropy, the unique measure of bi-partite entanglement for pure states, can capture the spatial structure of spacetime through the spacelike extremal surfaces in the bulk [3, 6].”