Pith. sign in

REVIEW 4 major objections 7 minor 2 cited by

The entanglement membrane in 2d CFT: reflected entropy, RG flow, and information velocity

T0 review · 4 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For two-dimensional CFTs, the entanglement membrane must carry an extra worldline degree of freedom to reproduce reflected entropy, which the ordinary degenerate-tension membrane gets wrong by a factor of two at late times.

desk verdict A genuinely new generalized membrane for 2d CFT with a sharp reflected-entropy prediction; the central claim inherits the standard EWCS conjecture, but the exact-checked core deserves refereeing. read the letter →

arxiv 2411.16542 v1 pith:NX4H4NYS submitted 2024-11-25 hep-th cond-mat.stat-mechcond-mat.str-el

Hanzhi Jiang , Márk Mezei , Julio Virrueta This is my paper · ORCID
classification hep-thcond-mat.stat-mechcond-mat.str-el MSC 81T4083C5781P40 PACS 04.70.-s11.25.Tq03.67.Mn
keywords entanglementmembranereflectedentropytwo-dimensionalCFTwedgecrosssectionRGflowBTZblackholeKasneruniverseinformationvelocity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the standard entanglement membrane—a minimal-curve description of how entanglement spreads in chaotic systems—cannot, on its own, account for reflected entropy in two-dimensional conformal field theories. The authors show holographically that a correct membrane description of a 2d CFT must carry an extra scalar degree of freedom on its worldline, the bulk depth $\xi=\log z$, in addition to the membrane shape. With this extra degree of freedom, the membrane reproduces the entanglement-wedge-cross-section result for reflected entropy after a quench, including a late-time plateau $S_R/2=\ell/2$ that the ordinary degenerate-tension membrane would instead predict as $\ell$. They further show that a relevant deformation of the CFT, dual to a three-dimensional black hole with a Kasner interior, gaps out the extra degree of freedom and recovers the ordinary non-degenerate membrane tension $E(v)$. If this is right, the distance between the generalized and ordinary membrane shows up not in entanglement entropy itself but in finer correlation measures, and 2d CFTs sit at a special point in the space of chaotic systems.

What carries the argument

The load-bearing object is the generalized membrane Lagrangian $L_{\rm eff} = \frac12 e^{-2\xi}(\dot x^2 - 1 - \dot\xi^2)$ obtained from the extremal-surface area functional in a three-dimensional black hole at large depth $z=e^\xi$, with degenerate tension $E(v)=1$ to leading order. The extra field $\xi(u)=\log z(u)$ is a worldline degree of freedom that encodes the bulk radial position of the extremal surface; it must be kept because in $d=2$ no finite plateau in $z$ exists and the extremal surface reaches exponentially deep into the black hole. The paper's main computations map the entanglement-wedge cross section to piecewise generalized-membrane configurations—non-equilibrium vertical segments and equilibrium plateau-plus-null segments—and evaluate their lengths. For the interpolation, a relevant deformation is modeled by an Einstein-scalar action whose interior solution is a Kasner universe; small-$\phi$ perturbation theory yields analytic expansions for $v_B$, $v_E$, and $E(v)$, and adding a $\phi^4$ interaction provides a UV-to-IR RG flow with degenerate tensions at both ends.

What would settle it

Compute the reflected entropy after a global quench in a 2d CFT directly, without invoking the entanglement-wedge-cross-section conjecture, for a bipartition made of intervals of length $\ell$ separated by $D\ll\ell$, and compare the late-time plateau of $S_R/2$ with the predicted $\ell/2$; a plateau at $\ell$ would falsify the generalized-membrane prediction. On the holographic side, one can also check the small-$\phi$ expansion of the membrane tension, e.g. $v_B = 1 - \frac{3\pi(3\pi-8)\Gamma(3/4)^2}{2\Gamma(1/4)^2}\bar\kappa^2$, against a direct conformal-perturbation-theory computation.

Watch

Extended reading notes

Core claim

On the paper's own terms: in a 2d CFT the entanglement membrane must be promoted to a generalized membrane carrying an extra worldline field $\xi(u)$ that records how deep the bulk extremal surface goes behind the horizon. When reflected entropy is computed holographically as half the entanglement wedge cross section, the generalized membrane gives $S_R/2 = \max\{0,\, T-D/2\}$ during linear growth and saturates at $S_R/2 = \ell/2$. The ordinary membrane with degenerate tension $E(v)=1$ incorrectly predicts the same linear-growth slope but a plateau $\ell$; the plateau mismatch survives even in the limit where the two theories agree on entanglement entropy. The paper also claims that adding a relevant scalar deformation, dual to a planar three-dimensional black hole with scalar hair and an interior Kasner universe, makes the $\xi$ degree of freedom gapped out, and the generalized membrane continuously reduces to the ordinary membrane with a non-degenerate $E(v)$. In higher dimensions the ordinary membrane does capture reflected entropy, so the extra degree of freedom is special to $d=2$.

Load-bearing premise

The argument rests on the holographic conjecture that reflected entropy equals the entanglement wedge cross section (from [27]), plus the assumption that the bulk-to-boundary membrane map is well-defined even though the paper concedes in footnote 17 that the map depends on which entanglement wedge cross section is computed.

Editorial extensions

If this is right

  • Reflected entropy, not entanglement entropy, is the observable that distinguishes the generalized 2d membrane from the ordinary one: the plateau $S_R/2$ is $\ell/2$ rather than $\ell$.
  • Adding a relevant deformation to a 2d CFT interpolates continuously from generalized to ordinary membrane theory, with the extra degree of freedom gapped out at a scale set by $e^{-2\xi_*} \sim \lambda^2/(2\pi^2)$.
  • In $d>2$ chaotic systems, reflected entropy after a quench is correctly captured by the ordinary membrane, with a plateau $\ell$ and a jump at the transition that vanishes only when $v_E=v_B$.
  • The information velocity $v_I(f)$ follows from the Legendre transform of the membrane tension; for spheres with chemical potential it saturates to $v_B$ above a critical $f_c$.
  • Saturated extremal surfaces project to butterfly-velocity cones of slope $v_B$ in the membrane picture, fixing earlier imprecise treatments of near-horizon membrane sections.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Our inference: if the generalized membrane's extra field really encodes the infinite-dimensional symmetry charges of the 2d CFT—an interpretation the paper speculates on but does not prove—then any chaotic 1+1d system with degenerate tension should show the same $\ell/2$ plateau, a prediction testable in dual-unitary circuits.
  • Our inference: the paper leaves the random-circuit counterpart of the generalized membrane open; a concrete extension would be to compute reflected entropy in a dual-unitary circuit and look for the worldline degree of freedom as a delocalized domain wall.
  • Our inference: the RG interpolation suggests a universal crossover in which a weakly deformed 2d CFT behaves like a 2d CFT at early times and like a higher-dimensional chaotic system at late times, with the crossover controlled by $1/\xi_*$.
  • Our inference: since the bulk-to-boundary projection depends on which entanglement wedge cross section is being computed, a fully autonomous effective theory would need a selection rule for the projection; until then the generalized membrane is best read as a reorganization of the holographic computation.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 7 minor

Summary. This paper studies the entanglement membrane description of the time evolution of entanglement in 2d CFTs and their relevant deformations. In 2d CFT the holographic derivation of the membrane yields a degenerate tension function E(v) = 1, and the authors propose a 'generalized membrane' with an extra worldline scalar ξ(u), the logarithm of the bulk depth in BTZ. They show that the generalized membrane reproduces the exact BTZ geodesic results for displaced half-spaces (App. B) and that, while entanglement entropy is insensitive to the extra degree of freedom, the reflected entropy (computed holographically via the entanglement wedge cross section, Eq. (3.3)) is not: the generalized membrane yields a plateau SR/2 = ℓ/2 (Eq. (3.12)), whereas the ordinary membrane gives ℓ (Sec. 4.3). The equilibrium EWCS underlying the plateau is constructed in Sec. 3.4 and checked against the exact solution in App. C.2. The paper further shows that a relevant deformation (hairy BTZ with Kasner interior) gaps out ξ and produces an ordinary non-degenerate tension, computed numerically and in conformal perturbation theory (Sec. 5), proposes a membrane description of reflected entropy in d > 2 (Eq. (4.12)) compared with numerical EWCS, and provides a membrane derivation of the information velocity of [1] (Sec. 6).

Significance. The central claim — that the ordinary degenerate membrane fails for reflected entropy in 2d CFT and that an extra worldline degree of freedom is required — is sharp and, if upheld, is a significant step in understanding the limits of the membrane effective theory. The paper is technically strong: the scaling-limit membrane results are checked against exact geodesic solutions in Apps. B and C; the small-deformation perturbation theory (Sec. 5.2) is analytic and internally consistent (notably, the λ² log λ terms cancel in the constraint E(vB) = vB); the construction has no free parameters; and the CFT twist computation in App. B.6.2 independently verifies the entropy result. The ℓ/2 vs ℓ plateau distinction (Sec. 4.3) is a concrete, falsifiable prediction. The main caveats are that the reflected-entropy interpretation inherits the EWCS conjecture (3.3), the new equilibrium plateau lacks a direct boundary computation, the generalized membrane is acknowledged (footnote 17, Sec. 7) not to be an autonomous effective theory, and the d > 2 numerical support (Fig. 14, App. D) is unquantified.

major comments (4)
  1. [Sec. 3.4, Eq. (3.12)] Eq. (3.12) is internally inconsistent as printed. The two EWCS candidates have lengths T − D/2 (Eq. (3.5); exact version (C.8)) and ℓ/2 (Eq. (3.11); exact version (C.22)), so the minimal surface switches at T = (ℓ + D)/2, where the two lengths are equal. As printed, (3.12) keeps the linear branch until T = ℓ + D/2, where that branch has the value ℓ, and then asserts a plateau ℓ/2 for T > ℓ + D/2, producing a downward discontinuity of size ℓ/2. The correct plateau onset is (ℓ + D)/2, which is also what follows from the condition µB ≥ 0 in (C.23) and from taking the vE = vB = 1 limit of Eq. (4.12) with the plateau value ℓ/2. Please correct (3.12), the Fig. 11 caption, and any downstream statements that use this transition time.
  2. [Sec. 3.1, Sec. 3.4, App. B.6.2] The headline claim — that the ordinary membrane fails for reflected entropy in 2d CFT and that the generalized membrane is required — is a statement about reflected entropy only if the EWCS/reflected-entropy duality (3.3) holds. That duality is cited to [27] and is a conjecture. The only boundary CFT computation in the paper (App. B.6.2) verifies the displaced-half-space von Neumann entropy, not the EWCS; the non-equilibrium EWCS is covered by the CFT computation of [30], but the new equilibrium plateau of Sec. 3.4 is not. I recommend either providing (or citing) a CFT replica computation for the equilibrium regime — feasible along the lines of App. B.6.2 — or, at a minimum, stating explicitly in the abstract and in Sec. 4.3 that the ℓ/2 plateau is a prediction of the holographic EWCS under conjecture (3.3).
  3. [Sec. 3.4 footnote 17, Sec. 7] The paper's own caveats are in tension with the unqualified abstract claim that 'in order to correctly capture the reflected entropy in 2d CFT, one needs to add an additional degree of freedom.' Footnote 17 concedes that reproducing the heuristic picture of [30] for the reflected setup requires a bulk-to-boundary mapping that depends on which EWCS is being computed, and Sec. 7 concedes that no direct map of ξ(u) to CFT quantities is known and that no random-circuit interpretation exists. Moreover, the generalized membrane is derived from, and tested against, the same BTZ geodesic action (compare Eq. (2.9) with Apps. B and C), so within this paper it is a scaling-limit reorganization of the holographic computation rather than an independent effective theory. I recommend rescoping the abstract and introduction accordingly, and identifying a concrete test of autonomy (e.g., the footnote-17 setup or a dual-unitary circuit analog).
  4. [Sec. 4.2, App. D, Figs. 14 and 39] The numerical support for the d > 2 membrane prediction (4.12) is not quantified. No error bars are given; the data in Fig. 14 are for O(1) values of D (D ≤ 3.3), far from the D ≫ 1 scaling limit; and App. D reports that 'precise numerical agreement remains elusive' while Sec. 4.2 states the membrane description is 'surprisingly accurate.' The apparent systematic discrepancies in the plateau and in the onset time (Fig. 39) should be reconciled with the claimed agreement: either provide error bars, a convergence study in 1/D, or an explicit statement of which features of (4.12) are validated. Because the comparison in Sec. 4.3 (Fig. 15) uses (4.12) for the d > 2 curve, the strength of the ℓ vs ℓ/2 distinction partly rests on this unquantified numerical check.
minor comments (7)
  1. [Eq. (4.13)] The jump size is written as ΔSR/(2D) = (1/2)(1 − vE/vB)D, which is dimensionally inconsistent; it should be ΔSR/(2D) = (1/2)(1 − vE/vB) (equivalently, ΔSR/2 = (D/2)(1 − vE/vB)).
  2. [Sec. 5.3] The text gives the minimum of the potential as ϕ* = 3√2/µ, but the Fig. 20 caption quotes ϕ* = 2.12 for µ = 1, which corresponds to ϕ* = 3/√(2µ); the formula in the text appears to have a factor of 2 error.
  3. [App. B.4.2 (citation)] The statement that Im t = −iπ in the thermal double is attributed to reference [38] (Klebanov and Witten); the appropriate reference for the Schwarzschild-time continuation is [21] (Hartman and Maldacena) or a related paper on BTZ geodesics.
  4. [Eq. (2.11)] The rescaling 'introduce p ≡ e^{−2ξ}p' reuses the symbol p for the rescaled momentum; a distinct symbol (e.g., p̃) would avoid confusion, and the subscript on ξp should be typeset explicitly.
  5. [App. D, Fig. 39] The Fig. 39 caption contains the typo 'Comparisson' (should be 'Comparison'); several figure captions also have a stray space before the colon in 'Figure X :'.
  6. [Sec. 6.2] The branch selection for the parametric curves in Eq. (6.12) ('multiple solutions for λ') is discussed only in one sentence; a brief explanation of how the physical branch is identified would improve reproducibility.
  7. [Sec. 4.1, App. D] A brief statement of numerical method and code/data availability would strengthen reproducibility of the EWCS numerics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central results are derived by explicit holographic computation and cross-checked against exact geodesic solutions, not by fitting or self-referential reduction.

full rationale

The paper's derivation chain is self-contained in the sense required for circularity analysis. The generalized membrane Lagrangian (2.9)-(2.10) is obtained directly from the BTZ geodesic action, and the reflected-entropy plateaus are genuine outputs: the generalized membrane gives S_R/2 = l/2 (Sec. 3.4, Eq. (3.11)) and this is verified against the exact EWCS geodesic solution in Appendix C.2 (Eq. (C.22)), while the higher-dimensional ordinary-membrane result (4.12) is compared against direct numerical holographic computation in Fig. 14. No parameter is fitted to the quantity it is later said to predict; v_E and v_B are computed from the black hole metric. The identification S_R = EWCS, Eq. (3.3), is an external conjecture cited to Dutta-Faulkner [27] and is explicitly acknowledged in Sec. 7 as still holographic; an unproven external assumption is a correctness/risk issue, not circularity. Self-citations such as [4] and [25] are to standard membrane results that are re-derived or used as background, and no load-bearing claim reduces to a self-citation chain. Footnote 17 concedes that the bulk-to-boundary mapping is setup-dependent, which weakens the autonomy of the effective theory but does not make the derivation equivalent to its inputs by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The central claims rely on the holographic dictionary, the EWCS/reflected entropy conjecture, the end-of-the-world brane model of quenches, the scaling limit that defines the generalized membrane, and the standard quantization of the hairy scalar. No parameters are fitted to data; the deformation coupling is a physical input.

assumptions (6)
  • domain assumption Holographic duality: entanglement entropy is computed by HRT/RT surfaces and reflected entropy by the entanglement wedge cross section, Eq. (3.3).
    The entire membrane derivation and the reflected entropy results rely on AdS/CFT and the Dutta-Faulkner conjecture.
  • domain assumption The holographic quench is modelled by a one-sided black hole with an end-of-the-world brane, Eq. (3.4).
    Used in Sec. 3 to set up the reflected entropy dynamics, following [21].
  • domain assumption The scaling limit, with u, x of order Λ and z = e^ξ of order e^Λ, is valid, and the leading-order effective action (2.9) captures the entropy and EWCS.
    The generalized membrane is defined by this large-depth asymptotic expansion; exact solutions in Appendices B and C support it.
  • domain assumption The hairy BTZ solution with m^2 = -3/4 and Kasner interior is the correct holographic dual of a relevant deformation of the CFT.
    Sec. 5 uses the Einstein-scalar system as a holographic RG flow and assumes standard quantization and the source falloff (5.6).
  • standard math Standard CFT twist operator methods and the Brown-Henneaux central charge are valid.
    Used in Appendix B.6.2 to cross-check the entanglement entropy calculation.
  • standard math The null energy condition implies v_E ≤ v_B and the membrane tension constraints in (2.8).
    Used to justify properties of E(v), cited from [25] and [26].
invented entities (1)
  • Generalized membrane with extra worldline scalar ξ(u), the log of bulk depth independent evidence
    purpose: To capture entanglement entropy and reflected entropy in 2d CFT where the ordinary membrane has degenerate tension E(v)=1 and fails for reflected entropy.
    The extra degree of freedom is defined as log of bulk depth from holography; it yields the sharp plateau value SR/2 = ℓ/2 for reflected entropy, checked against exact EWCS computations and the CFT results of [30]. However, the authors state in Sec. 7 that a direct map to field-theoretic quantities is still missing.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The entanglement membrane in 2d CFT: reflected entropy, RG flow, and information velocity." pith.science (2026). https://pith.science/paper/NX4H4NYS

@misc{pith2026241116542,
  author       = {Pith},
  title        = {Pith review of: The entanglement membrane in 2d CFT: reflected entropy, RG flow, and information velocity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NX4H4NYS}},
  note         = {Machine review of arXiv:2411.16542}
}
read the original abstract

The time evolution of entanglement entropy in generic chaotic many-body systems has an effective description in terms of a minimal membrane, characterised by a tension function. For 2d CFTs, a degenerate tension function reproduces several results regarding the dynamics of the entropy; this stands in contrast to higher dimensions where the tension is non-degenerate. In this paper we use holography to show that, in order to correctly capture the reflected entropy in 2d CFT, one needs to add an additional degree of freedom to the membrane description. Furthermore, we show that the conventional non-degenerate membrane tension function emerges upon introducing a relevant deformation of the CFT, dual to a planar BTZ black hole with scalar hair and with an interior Kasner universe. Finally, we also study the membrane description for reflected entropy and information velocity arXiv:1908.06993 in higher dimensions.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quantum chaos and pole skipping in two-dimensional conformal perturbation theory

    hep-th 2025-09 conditional novelty 6.0 of 10

    A deformed 2D CFT's stress-tensor pole-skipping point shifts at O(lambda^2); at h=1/2 the shift matches the holographic butterfly velocity.

  2. Islands, Double Holography, and the Entanglement Membrane

    hep-th 2024-12 conditional novelty 6.0 of 10

    A double-holographic Page curve is realized as an entanglement membrane with a vertical growing segment before the Page time and two saturated butterfly-velocity lines exiting through a boundary after it.

Reference graph

Works this paper leans on

63 extracted references · 6 canonical work pages · cited by 2 Pith papers

  1. [4]

    Mezei, Membrane theory of entanglement dynamics from holography , Phys

    M. Mezei, Membrane theory of entanglement dynamics from holography , Phys. Rev. D 98 (2018) 106025 [ 1803.10244]

    arXiv 2018

  2. [30]

    Kudler-Flam, Y

    J. Kudler-Flam, Y. Kusuki and S. Ryu, Correlation measures and the entanglement wedge cross-section after quantum quenches in two-dimensional conformal field theories , JHEP 04 (2020) 074 [ 2001.05501]

    arXiv 2020

  3. [27]

    Dutta and T

    S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section , 1905.00577

    arXiv 1905

  4. [1]

    Couch, S

    J. Couch, S. Eccles, P. Nguyen, B. Swingle and S. Xu, Speed of quantum information spreading in chaotic systems , Phys. Rev. B 102 (2020) 045114 [ 1908.06993]

    arXiv 2020

  5. [2]

    Nahum, J

    A. Nahum, J. Ruhman, S. Vijay and J. Haah, Quantum Entanglement Growth Under Random Unitary Dynamics , Phys. Rev. X 7 (2017) 031016 [ 1608.06950]

    arXiv 2017

  6. [3]

    Jonay, D.A

    C. Jonay, D.A. Huse and A. Nahum, Coarse-grained dynamics of operator and state entanglement, 1803.00089

    arXiv

  7. [5]

    Zhou and A

    T. Zhou and A. Nahum, Emergent statistical mechanics of entanglement in random unitary circuits, Phys. Rev. B 99 (2019) 174205 [ 1804.09737]

    arXiv 2019

  8. [6]

    Zhou and A

    T. Zhou and A. Nahum, Entanglement Membrane in Chaotic Many-Body Systems , Phys. Rev. X 10 (2020) 031066 [ 1912.12311]

    arXiv 2020

Show all 63 references
  1. [7]

    Rampp, S.A

    M.A. Rampp, S.A. Rather and P.W. Claeys, Entanglement membrane in exactly solvable lattice models, Phys. Rev. Res. 6 (2024) 033271 [ 2312.12509]

    2024 arXiv

  2. [8]

    Vardhan and S

    S. Vardhan and S. Moudgalya, Entanglement dynamics from universal low-lying modes , 2407.16763

  3. [9]

    Mezei and J

    M. Mezei and J. Virrueta, Exploring the Membrane Theory of Entanglement Dynamics , JHEP 02 (2020) 013 [ 1912.11024]

    2020 arXiv

  4. [10]

    Calabrese and J.L

    P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems , J. Stat. Mech. 0504 (2005) P04010 [ cond-mat/0503393]

    2005 arXiv

  5. [11]

    Calabrese and J

    P. Calabrese and J. Cardy, Quantum Quenches in Extended Systems , J. Stat. Mech. 0706 (2007) P06008 [ 0704.1880]

    2007 arXiv

  6. [12]

    Asplund, A

    C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Entanglement Scrambling in 2d Conformal Field Theory , JHEP 09 (2015) 110 [ 1506.03772]

    2015 arXiv

  7. [13]

    Davison and L.V

    R.A. Davison and L.V. Delacretaz, Universal thermalization dynamics in (1+1)d QFTs , 2409.09112

    arXiv

  8. [14]

    Rakovszky, F

    T. Rakovszky, F. Pollmann and C.W. von Keyserlingk, Sub-ballistic Growth of R´ enyi Entropies due to Diffusion , Phys. Rev. Lett. 122 (2019) 250602 [ 1901.10502]

    2019 arXiv

  9. [15]

    Huang, Dynamics of R´ enyi entanglement entropy in diffusive qudit systems, 2008.00944

    Y. Huang, Dynamics of R´ enyi entanglement entropy in diffusive qudit systems, 2008.00944

    2008 arXiv

  10. [16]

    ˇZnidariˇ c,Entanglement growth in diffusive systems , Commun

    M. ˇZnidariˇ c,Entanglement growth in diffusive systems , Commun. Phys. 3 (2020) 100. – 64 –

    2020

  11. [17]

    Rakovszky, F

    T. Rakovszky, F. Pollmann and C.W. von Keyserlingk, Comment on ”Entanglement growth in diffusive systems” , 2010.07969

    2010 arXiv

  12. [18]

    Ryu and T

    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT , Phys. Rev. Lett. 96 (2006) 181602 [ hep-th/0603001]

    2006 arXiv

  13. [19]

    Ryu and T

    S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy , JHEP 08 (2006) 045 [hep-th/0605073]

    2006 arXiv

  14. [20]

    Hubeny, M

    V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [ 0705.0016]

    2007 arXiv

  15. [21]

    Hartman and J

    T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [ 1303.1080]

    2013 arXiv

  16. [22]

    Liu and S.J

    H. Liu and S.J. Suh, Entanglement Tsunami: Universal Scaling in Holographic Thermalization , Phys. Rev. Lett. 112 (2014) 011601 [ 1305.7244]

    2014 arXiv

  17. [23]

    Liu and S.J

    H. Liu and S.J. Suh, Entanglement growth during thermalization in holographic systems , Phys. Rev. D 89 (2014) 066012 [ 1311.1200]

    2014 arXiv

  18. [24]

    Roberts, D

    D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [1409.8180]

    2015 arXiv

  19. [25]

    Mezei, On entanglement spreading from holography , JHEP 05 (2017) 064 [ 1612.00082]

    M. Mezei, On entanglement spreading from holography , JHEP 05 (2017) 064 [ 1612.00082]

    2017 arXiv

  20. [26]

    Mezei and D

    M. Mezei and D. Stanford, On entanglement spreading in chaotic systems , JHEP 05 (2017) 065 [1608.05101]

    2017 arXiv

  21. [28]

    Takayanagi and K

    T. Takayanagi and K. Umemoto, Entanglement of purification through holographic duality , Nature Phys. 14 (2018) 573 [ 1708.09393]

    2018 arXiv

  22. [29]

    Akers, T

    C. Akers, T. Faulkner, S. Lin and P. Rath, The Page curve for reflected entropy , JHEP 06 (2022) 089 [ 2201.11730]

    2022 arXiv

  23. [31]

    Mezei and J

    M. Mezei and J. Virrueta, The Quantum Null Energy Condition and Entanglement Entropy in Quenches, 1909.00919

    1909 arXiv

  24. [32]

    Mezei and W

    M. Mezei and W. van der Schee, Black holes often saturate entanglement entropy the fastest , Phys. Rev. Lett. 124 (2020) 201601 [ 2001.03172]

    2020 arXiv

  25. [33]

    Breitenlohner and D.Z

    P. Breitenlohner and D.Z. Freedman, Stability in Gauged Extended Supergravity , Annals Phys. 144 (1982) 249

    1982

  26. [34]

    Caceres, A

    E. Caceres, A. Kundu, A.K. Patra and S. Shashi, Page curves and bath deformations , SciPost Phys. Core 5 (2022) 033 [ 2107.00022]

    2022 arXiv

  27. [35]

    Witten, Anti-de Sitter space and holography , Adv

    E. Witten, Anti-de Sitter space and holography , Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150]

    1998 arXiv

  28. [36]

    Gubser, I.R

    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [ hep-th/9802109]. – 65 –

    1998 arXiv

  29. [37]

    de Haro, S.N

    S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS / CFT correspondence , Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230]

    2001 arXiv

  30. [38]

    Klebanov and E

    I.R. Klebanov and E. Witten, AdS / CFT correspondence and symmetry breaking , Nucl. Phys. B 556 (1999) 89 [ hep-th/9905104]

    1999 arXiv

  31. [39]

    Frenkel, S.A

    A. Frenkel, S.A. Hartnoll, J. Kruthoff and Z.D. Shi, Holographic flows from CFT to the Kasner universe , JHEP 08 (2020) 003 [ 2004.01192]

    2020 arXiv

  32. [40]

    Kasner, Geometrical theorems on Einstein ’s cosmological equations, Am

    E. Kasner, Geometrical theorems on Einstein ’s cosmological equations, Am. J. Math. 43 (1921) 217

    1921

  33. [41]

    Belinski and I.M

    V.A. Belinski and I.M. Khalatnikov, Effect of Scalar and Vector Fields on the Nature of the Cosmological Singularity, Sov. Phys. JETP 36 (1973) 591

    1973

  34. [42]

    Hartnoll, C.P

    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101 (2008) 031601 [ 0803.3295]

    2008 arXiv

  35. [43]

    Hartnoll, C.P

    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic Superconductors, JHEP 12 (2008) 015 [0810.1563]

    2008 arXiv

  36. [44]

    Freedman, S.S

    D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner, Renormalization group flows from holography supersymmetry and a c theorem , Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017]

    1999 arXiv

  37. [45]

    Girardello, M

    L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, Novel local CFT and exact results on perturbations of N=4 superYang Mills from AdS dynamics , JHEP 12 (1998) 022 [hep-th/9810126]

    1998 arXiv

  38. [46]

    Girardello, M

    L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, The Supergravity dual of N=1 superYang-Mills theory, Nucl. Phys. B 569 (2000) 451 [ hep-th/9909047]

    2000 arXiv

  39. [47]

    Hartnoll and N

    S.A. Hartnoll and N. Neogi, AdS black holes with a bouncing interior , SciPost Phys. 14 (2023) 074 [2209.12999]

    2023 arXiv

  40. [48]

    Eccles, W

    S. Eccles, W. Fischler, T. Guglielmo, J.F. Pedraza and S. Racz, Speeding up the spread of quantum information in chaotic systems , JHEP 12 (2021) 019 [ 2108.12688]

    2021 arXiv

  41. [49]

    Bertini, P

    B. Bertini, P. Kos and T. Prosen, Entanglement spreading in a minimal model of maximal many-body quantum chaos , Phys. Rev. X 9 (2019) 021033 [ 1812.05090]

    2019 arXiv

  42. [50]

    Piroli, B

    L. Piroli, B. Bertini, J.I. Cirac and T. Prosen, Exact dynamics in dual-unitary quantum circuits, Phys. Rev. B 101 (2020) 094304 [ 1911.11175]

    2020 arXiv

  43. [51]

    Czech, J.L

    B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The Gravity Dual of a Density Matrix , Class. Quant. Grav. 29 (2012) 155009 [ 1204.1330]

    2012 arXiv

  44. [52]

    Headrick, V.E

    M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [ 1408.6300]

    2014 arXiv

  45. [53]

    Wall, Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy, Class

    A.C. Wall, Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy, Class. Quant. Grav. 31 (2014) 225007 [ 1211.3494]

    2014 arXiv

  46. [54]

    Akers, T

    C. Akers, T. Faulkner, S. Lin and P. Rath, Reflected entropy in random tensor networks , JHEP 05 (2022) 162 [ 2112.09122]

    2022 arXiv

  47. [55]

    Akers, T

    C. Akers, T. Faulkner, S. Lin and P. Rath, Reflected entropy in random tensor networks. Part II. A topological index from canonical purification , JHEP 01 (2023) 067 [ 2210.15006]. – 66 –

    2023 arXiv

  48. [56]

    Akers, T

    C. Akers, T. Faulkner, S. Lin and P. Rath, Entanglement of purification in random tensor networks, Phys. Rev. D 109 (2024) L101902 [ 2306.06163]

    2024 arXiv

  49. [57]

    Akers, T

    C. Akers, T. Faulkner, S. Lin and P. Rath, Reflected entropy in random tensor networks III: triway cuts , 2409.17218

    arXiv

  50. [58]

    Fidkowski, V

    L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, The Black hole singularity in AdS / CFT, JHEP 02 (2004) 014 [ hep-th/0306170]

    2004 arXiv

  51. [59]

    Banados, C

    M. Banados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time , Phys. Rev. Lett. 69 (1992) 1849 [ hep-th/9204099]

    1992 arXiv

  52. [60]

    Banados, M

    M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2+1) black hole , Phys. Rev. D 48 (1993) 1506 [ gr-qc/9302012]

    1993 arXiv

  53. [61]

    Brown and M

    J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity , Commun. Math. Phys. 104 (1986) 207

    1986

  54. [62]

    Calabrese and J.L

    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory , J. Stat. Mech. 0406 (2004) P06002 [ hep-th/0405152]

    2004 arXiv

  55. [63]

    Hubeny, Extremal surfaces as bulk probes in AdS/CFT , JHEP 07 (2012) 093 [1203.1044]

    V.E. Hubeny, Extremal surfaces as bulk probes in AdS/CFT , JHEP 07 (2012) 093 [1203.1044]. – 67 –

    2012 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.