Geometric and Renormalized Entropy in Conformal Field Theory

C. Holzhey; F. Larsen; F. Wilczek

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Geometric and Renormalized Entropy in Conformal Field Theory

C. Holzhey, F. Larsen, F. Wilczek

High Energy Physics - Theory · Submitted 18 Mar 1994 · 16 references

Pith's verdict: This is the paper that pins down the exact (c+c̄)/6 coefficient for the logarithmic divergence of geometric entropy, and the moving-mirror renormalized entropy is a real, checkable result rather than a fit.

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arXiv IDhep-th/9403108 (version 1)

Pith ID52XHQOHI, Pith's permanent record of this paper and its review

CitationC. Holzhey, F. Larsen, F. Wilczek. "Geometric and Renormalized Entropy in Conformal Field Theory." arXiv:hep-th/9403108 (1994). https://pith.science/paper/52XHQOHI

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Editor's note

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Short version: this is where the exact coefficient comes from, and it holds up. If you work on entanglement entropy or the information problem, you already know the formula; what's worth reading is how cleanly it's derived and how far the moving-mirror part carries it. Srednicki had numerical evidence for a log divergence with an approximate coefficient; this gives (c+c̄)/6 for a general CFT, exactly, with corrections vanishing as ε→0. Two independent routes — replica plus modular invariance, and the Cardy trace-anomaly argument — agree. The modular trick is the nice bit: the whole many-particle entropy falls out of the vacuum piece after τ→−1/τ, which is a genuinely economical way to see it. The anomaly derivation also makes the physical origin explicit: the divergence traces straight to the short-distance singularity in TT, which is the right story and is stated as such. Section 3's S_ren for the mirror is exact, and the match to thermal entropy at T=1/(8πM) in 4.3 is an independent check, not a fit. That's the part I'd point students to. Soft spots, in proportion: the state-independence of the UV divergence is argued, not proven in general, but the paper scopes itself to the mirror model where the cancellation is manifest, so I don't think this is load-bearing. Zero modes are ignored, a finite ε-independent term isn't excluded, fermions are deferred, and the mirror doesn't conserve energy — all acknowledged in the text. Section 4's information-problem discussion is the weakest stretch, and the authors say so themselves. One thing to flag for a referee: (4.23) has the log argument inverted relative to (4.22), so it carries the wrong sign, while (4.24) is correct; and (4.25) with the U'^2 ∝ u^{-2} condition below it is garbled — the extremal trajectory is U ∝ 1/u, giving U'^2 ∝ u^{-4}. Minor printed slips, but anyone checking the arithmetic will trip on them. Audience: entanglement entropy, Ryu-Takayanagi, black hole information. It deserves a serious referee, mostly to catch the Section 4.3 slips and to press on how far the renormalized-entropy definition generalizes beyond the conformal mirror.

Referee report

Major objections (2)

[Abstract and Section 3] The abstract states generally that 'the difference in the entropy between two such states is better behaved' (abstract:s4), but the paper proves this only for the conformal mirror states, where the map z → f(z) is exact. The paper does scope the claim in abstract:s6 ('in a simple model situation'), and Section 3 is explicit that state-independence of the UV-divergent piece is a heuristic expectation before the exact mirror computation. Still, the abstract's general phrasing is stronger than the body supports. I recommend either softening the abstract or adding one explicit sentence in Section 3 noting that the cancellation is verified here only within the conformal mirror model. This is a framing issue, not a derivation error.
[Eq. (4.23)] The argument of the logarithm is inverted relative to what follows from (4.22). Substituting U = c1 + c2 e^{-u/(4M)} into (4.22) gives S_ren = (1/12) ln[(4M)^2 (e^{(u2-u1)/8M} - e^{-(u2-u1)/8M})^2/(u2-u1)^2], whereas (4.23) prints the reciprocal. As printed, the large-(u2-u1) limit gives -(u2-u1)/(48M), the negative of (4.24). The final result (4.24) is correct, so this is a typographical inversion in an intermediate displayed equation, but it should be corrected because a reader reproducing (4.24) from (4.23) will get the wrong sign.

Minor objections (7)

[Eq. (4.25) and the sentence below (4.24)] The displayed trajectory 'U = c1 + c2 u - c3' is garbled; the extremal Reissner-Nordström trajectory should be of the form U = c1 - c3/u (or U ∝ 1/u). Correspondingly, the condition below (4.24) should read U' ∝ u^{-2} (equivalently U'^2 ∝ u^{-4}), not U'^2 ∝ u^{-2}; the latter would give U ∝ ln u, which is not the extremal trajectory and would not yield a finite S_ren.
[Section 2.1, first paragraph] 'Roughly speaking, it the logarithm of the number of states' is missing a verb; it should read 'it is the logarithm'.
[Section 2.2, after (2.13)] 'it looses physical meaning' should be 'it loses physical meaning'.
[Section 2.2, after (2.5)] 'it must not appear appear in the final result' contains a duplicated word.
[Section 4.2] 'restricted either to either side of A' contains a duplicated 'either'.
[Section 2.2] The paper uses both ε and ǫ for the smearing parameters; a single symbol throughout would help the reader.
[Section 2.2, 0-mode caveat] The paper appropriately flags that 0-modes are ignored and can introduce infrared-cutoff dependence. It would help to state explicitly whether this affects the coefficient of the logarithmic divergence or only the additive constant.

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The paper

The paper claims

Geometric entropy in a 1+1 conformal field theory diverges logarithmically with the ultraviolet cutoff, but the difference between two states stays finite, and for a moving mirror that difference reproduces the thermal entropy of Hawking辐射.

The reading

The paper asks what happens to entropy when the microscopic model is a relativistic quantum field theory, where an infinite number of degrees of freedom per unit volume makes the entropy of a sharply localized region diverge. The authors show, for a general conformal field theory in 1+1 dimensions, that the geometric entropy of an interval of length $Sigma$ regulated by a short-distance cutoff $epsilon$ is $S = (c+bar c)/6 ln(Sigma/epsilon)$, with $c$ and $bar c$ the central charges of the left- and right-moving sectors; the coefficient is exact up to terms that vanish as $epsilon to 0$. They obtain this three ways: from the thermodynamics of a free photon gas on the torus, from modular invariance of the torus partition function, and from the conformal anomaly, which ties the divergence directly to the short-distance singularity in the product of two energy-momentum tensors. The point of the exercise is that the divergent piece is universal and state-independent, so subtracting the vacuum value leaves a finite, physically meaningful renormalized entropy. For states produced by a moving mirror, that renormalized entropy is a closed expression in the mirror trajectory, and for the trajectory that models a Schwarzschild black hole it coincides with the entropy of a thermal bath at the Hawking temperature.

What carries the argument

The machinery is the conformal mapping of the traced-over region onto a strip, followed by a replica trick: the $n$-th power of the reduced density matrix is a partition function on a torus of circumference proportional to $n$, and $S = (1 - n, d/dn)_{n=1} ln Z(n)$. A modular transformation $tau to -1/tau$ swaps the cycles of that torus, converting a sum over all highly excited multi-particle states into a single vacuum contribution proportional to $c+bar c$, which is what makes the general-${cal C}$ result exact rather than merely leading. The same coefficient reappears from the trace anomaly: rescaling the cutoff and using $langle T^mu_{ mu} rangle propto c$ times the curvature of the cone with opening $2pi n$ gives $partial S / partial ln epsilon = -(c+bar c)/6$. For the mirror, the map $z to f(z)$ that straightens the trajectory does all the work, and the anomalous transformation law of $T(z)$ supplies both the radiated flux and the entropy.

Extended reading notes

Core claim

The central result is a pair of formulas. First, for any conformal field theory in 1+1 dimensions the geometric entropy of an interval relative to the vacuum is $S = frac{c+bar c}{6} ln(Sigma/epsilon)$, with the coefficient fixed by the central charge and corrections vanishing as the cutoff is removed. Second, for a state created by a perfectly reflecting mirror moving along $bar z = f(z)$, the entropy relative to the stationary-mirror vacuum is $S_{rm ren} = frac{c}{12} ln!left[ frac{(f(z_2)-f(z_1))^2}{(z_2-z_1)^2, f'(z_1) f'(z_2)} right]$, which is independent of the smearing and finite as the cutoff goes to zero. Applied to the trajectory $U = c_1 + c_2 e^{-u/4M}$ that models collapse to a Schwarzschild black hole, this gives, for intervals long compared with $8M$, $S_{rm ren} approx frac{pi}{6}(u_2-u_1) cdot frac{1}{8pi M}$ — exactly the entropy a thermal gas at temperature $T = 1/(8pi M)$ would carry. The authors also rewrite the result as $S_{rm ren} = -frac{1}{24} ln(C/C_0)$, where $C$ is the two-point correlation of the observed energy-momentum tensor and $C_0$ its vacuum value, so the entropy measures excess correlations in the radiation rather than a count of particles.

Load-bearing premise

The subtraction that defines the renormalized entropy assumes the ultraviolet-divergent part of the geometric entropy is the same for every state, so that subtracting the vacuum value leaves something finite and physically meaningful; the paper demonstrates this for the conformal mirror states, where the map from the moving to the stationary mirror is exact, but for a general excited state it is asserted rather than derived.

Editorial extensions

If this is right

The coefficient of the logarithmic divergence in geometric entropy is fixed by the central charge of the theory, so any 1+1 conformal field theory has the same universal short-distance entropy law regardless of its detailed field content.
For a mirror trajectory that asymptotes to the Schwarzschild form, the renormalized geometric entropy of a long interval equals the thermodynamic entropy of Hawking radiation at $T = 1/(8pi M)$, giving a microscopic derivation of that entropy rather than an assumption of thermality.
The renormalized entropy is not additive and not locally positive: it can shrink as the interval is enlarged, and it can go negative when the observed state is more ordered than the vacuum.
Trajectories that produce no Hawking radiation, such as the extremal Reissner-Nordstrom form $U = c_1 + c_2 u - c_3$, give vanishing renormalized entropy, and a mirror that returns to rest drives the entropy rapidly back to zero even after a long thermal period.
Because the divergent part is state-independent, the difference between the geometric entropies of any two low-energy states of a conformal theory is a finite, cutoff-independent quantity.

Reading between the lines

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

If the renormalized entropy really is the right bookkeeping quantity, then the standard argument that a pure state evolving into apparently thermal radiation violates quantum mechanics loses much of its force: the thermal appearance is a reflection of vacuum correlations across the horizon, not a genuine loss of information.
The identification $S_{rm ren} = -frac{1}{24} ln(C/C_0)$ suggests a general diagnostic: for any state of a conformal theory, compute the ratio of the observed two-point stress-tensor correlation to its vacuum value and read off an entropy. Whether this survives away from the mirror setting is an open test.
The result hints that entropy in field theory should be treated as a relational quantity tied to a choice of region and a choice of reference state, rather than as an extensive substance carried by the state alone — a view with consequences for how horizon entropy is counted in more realistic settings.
The paper's own caveat about prescribed mirror motion means the energy budget of evaporation is not accounted for; a version of the calculation in which the mirror recoils would be the natural next check of whether the entropy really returns to zero at the end of evaporation.

Original abstract

In statistical physics, useful notions of entropy are defined with respect to some coarse graining procedure over a microscopic model. Here we consider some special problems that arise when the microscopic model is taken to be relativistic quantum field theory. These problems are associated with the existence of an infinite number of degrees of freedom per unit volume. Because of these the microscopic entropy can, and typically does, diverge for sharply localized states. However the difference in the entropy between two such states is better behaved, and for most purposes it is the useful quantity to consider. In particular, a renormalized entropy can be defined as the entropy relative to the ground state. We make these remarks quantitative and precise in a simple model situation: the states of a conformal quantum field theory excited by a moving mirror. From this work, we attempt to draw some lessons concerning the ``information problem'' in black hole physics

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.42 · human participants 0.01 · controlled comparison 0.52 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.13 · mentions code or data availability 0.02.

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In statistical physics, useful notions of entropy are defined with respect to some coarse graining procedure over a microscopic model. Here we consider some special problems that arise when the microscopic model is taken to be relativistic quantum field theory. These problems are associated with the existence of an infinite number of degrees of freedom per unit volume. Because of these the microscopic entropy can, and typically does, diverge for sharply localized states. However the difference in the entropy between two such states is better behaved, and for most purposes it is the useful quantity to consider. In particular, a renormalized entropy can be defined as the entropy relative to the ground state. We make these remarks quantitative and precise in a simple model situation: the states of a conformal quantum field theory excited by a moving mirror. From this work, we attempt to draw some lessons concerning the ``information problem'' in black hole physics

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	In statistical physics, useful notions of entropy are defined with respect to some coarse graining procedure over a microscopic model.	background 1.00	0.04	0.03	asserted 1.00
s1	Here we consider some special problems that arise when the microscopic model is taken to be relativistic quantum field theory.	background 0.66	0.48	0.02	asserted 0.89
s2	These problems are associated with the existence of an infinite number of degrees of freedom per unit volume.	background 1.00	0.07	0.03	asserted 1.00
s3	Because of these the microscopic entropy can, and typically does, diverge for sharply localized states.	background 0.97	0.12	0.06	asserted 0.61
s4	However the difference in the entropy between two such states is better behaved, and for most purposes it is the useful quantity to consider.	background 0.54	0.16	0.12	asserted 0.83
s5	In particular, a renormalized entropy can be defined as the entropy relative to the ground state.	background 0.46	0.34	0.13	asserted 0.95
s6	We make these remarks quantitative and precise in a simple model situation: the states of a conformal quantum field theory excited by a moving mirror.	method 0.86	0.85	0.07	asserted 0.93
s7	From this work, we attempt to draw some lessons concerning the ``information problem'' in black hole physics	interpretation 0.83	0.39	0.22	unclear 0.47

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.96.

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arXiv listing abstract v1

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References

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C. Holzhey, Princeton University thesis , 1992, (unpublished)

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M. Srednicki, Phys. Rev. Lett. 71, (1993) 666

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L. Susskind, Some Speculations About Black Hole Entropy in String Theory , RU-93-44,hep-th/9309145

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M. Srednicki, private communication

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J. Preskill, seminar at Princeton University

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E. Keski-Vakkuri and S. Mathur, Evaporating Black Holes and Entropy , MIT-CTP-2272,hep-th/9312194

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P. Ginsparg, Applied Conformal Field Theory , in Fields, Strings, and Critical Phenomena Les Houches, Session XLIX, 1988, eds. E.Br´ ezin and J. Zinn- Justin

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J. L. Cardy, Conformal Invariance and Statistical Mechanics , in Fields, Strings, and Critical Phenomena Les Houches, Session XLIX, 1988, eds. E.Br´ ezin and J. Zinn-Justin

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[16]

T-D. Chung and Herman Verlinde, Dynamical Moving Mirrors and Black Holes, PUPT-1430,hep-th/9311007 35 This figure "fig1-1.png" is available in "png" format from: http://arxiv.org/ps/hep-th/9403108v1 This figure "fig1-2.png" is available in "png" format from: http://arxiv.org/ps/hep-th/9403108v1 This figure "fig1-3.png" is available in "png" format from: h...

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hep-th 2012-06 conditional novelty 7.0 of 10

Sharp corners, cones, and creases in the boundary of an entangled region produce new universal logarithmic terms in entanglement entropy, with coefficients fixed by central charges in even dimensions.
A refinement of entanglement entropy and the number of degrees of freedom
hep-th 2012-02 conditional novelty 7.0 of 10

A differential operator in the entangling size R turns entanglement entropy into a UV-finite quantity that flows monotonically in three dimensions but not in four.
Towards a derivation of holographic entanglement entropy
hep-th 2011-02 accept novelty 7.0 of 10

For spherical entangling surfaces, holographic entanglement entropy is derived rather than conjectured, by mapping the CFT vacuum to a thermal state on a hyperbolic space and reading off a topological black hole's hor...
On Holographic Entanglement Entropy and Higher Curvature Gravity
hep-th 2011-01 conditional novelty 7.0 of 10

For Lovelock gravity, holographic entanglement entropy is computed by extremizing the Jacobson-Myers functional of intrinsic surface curvature, not Wald's horizon entropy formula.
Holographic c-theorems in arbitrary dimensions
hep-th 2010-11 conditional novelty 7.0 of 10

A holographic flow function built from tuned higher-curvature gravity decreases monotonically and equals the A-type trace anomaly coefficient in even dimensions and a universal entanglement-entropy coefficient in all ...
Entanglement entropy for the n-sphere
hep-th 2010-07 accept novelty 7.0 of 10

The logarithmic coefficient of the entanglement entropy for a sphere and a conformal scalar is computed exactly in any dimension: -1/3 in 2D, 1/90 in 4D, and zero in odd spacetime dimensions.
Entanglement Renyi entropies in holographic theories
hep-th 2010-06 accept novelty 7.0 of 10

Fursaev's derivation of the Ryu-Takayanagi formula is shown to use non-saddle-point geometries; corrected Renyi entropy calculations in two-dimensional holographic theories reproduce the Ryu-Takayanagi predictions, in...
Entanglement entropy of two disjoint intervals in conformal field theory
hep-th 2009-05 conditional novelty 7.0 of 10

The paper derives an exact formula for the Renyi entanglement entropy of two disjoint intervals in a free compactified boson, expressed via Riemann-Siegel theta functions, valid for all integer replica indices n.
On String Theory Duals of Lifshitz-like Fixed Points
hep-th 2009-05 accept novelty 7.0 of 10

A type IIB supergravity solution with anisotropic scaling exponent 3/2 is constructed from a D3-D7 brane system, with an interpolating RG flow to AdS5 and direction-dependent entanglement entropy.
Near Extremal Black Hole Entropy as Entanglement Entropy via AdS2/CFT1
hep-th 2007-10 conditional novelty 7.0 of 10

Near-extremal black hole entropy equals the entanglement entropy between two dual conformal quantum mechanics on the two AdS2 boundaries, supported by a BTZ microscopic counting and an exact free-fermion computation.
Entanglement and correlation functions following a local quench: a conformal field theory approach
cond-mat.stat-mech 2007-08 accept novelty 7.0 of 10

Conformal field theory gives closed-form time dependence of entanglement entropy and correlation functions after joining two decoupled halves of a one-dimensional quantum system.
Ground state entanglement in quantum spin chains
quant-ph 2003-04 accept novelty 7.0 of 10

The entanglement entropy of a spin block in a critical chain scales as (c/3) log L, with c the conformal central charge, and saturates off criticality at a value set by the correlation length.
Eternal Black Holes in AdS
hep-th 2001-06 conditional novelty 7.0 of 10

The maximally extended AdS black hole is dual to two copies of the boundary CFT in the thermofield double entangled state, and summing over geometries restores unitarity.
Entanglement entropy of black holes from the first law of thermodynamics
hep-th 2026-10 conditional novelty 6.0 of 10

Entanglement entropy of quantum matter in Killing horizons can be computed on-shell by integrating the first law of thermodynamics along a one-parameter family of equilibrium backgrounds, with the result agreeing with...
Anomaly quenching and dynamical cooling of Hawking evaporation in Horndeski gravity
gr-qc 2026-08 conditional novelty 6.0 of 10

In a 2D CGHS black hole with a linear Horndeski G3 coupling, the derived first-order correction lowers the Hawking temperature as mass decreases, and an assumed resummation leads to a cold remnant with logarithmically...
Toward an Observable Algebra for de Sitter Space: Gap Protection and Modular Dressings
hep-th 2026-07 conditional novelty 6.0 of 10

In global dS2 the fundamental complement of a finite union of arcs is empty unless some complementary gap has length at least π; a commutant-based algebra model reproduces this and exhibits a discontinuous 'activation...
CFT Dual for Timelike Geodesic in Lorentzian dS
hep-th 2026-06 unverdicted novelty 6.0 of 10

Timelike geodesic-integrated Wightman functions in dS are dual to bulk-operator correlators with a linear combination of an OPE block and its Casimir partner.
The Entanglement Wedge Polygon
hep-th 2026-06 unverdicted novelty 6.0 of 10

The paper defines the entanglement wedge polygon as the intersection of entanglement wedges external to individual homology regions and studies its topological and geometric properties in AdS examples.
Apparent Fermionic Spectra for Bosonic Radiation: Accelerated Charge Kinematics
quant-ph 2026-06 unverdicted novelty 6.0 of 10

Accelerated point charges produce apparent Fermi-Dirac photon spectra via special kinematics without thermal equilibrium or horizons.
Nonlocal Nonstabilizerness from Holographic Schwinger Pair Production
hep-th 2026-05 unverdicted novelty 6.0 of 10

In holographic Schwinger pair production, the excess capacity of entanglement is √λ(d−2)/(d−1)³ — positive for d>2, zero for d=2 — so the produced pair carries nonlocal magic for d>2.
The Maximal Entanglement Limit in Statistical and High Energy Physics
quant-ph 2026-01 unverdicted novelty 6.0 of 10

Quantum systems reach a Maximal Entanglement Limit where entanglement geometry produces thermal reduced density matrices and probabilistic behavior in statistical and high-energy physics.
Generalizing quantum dimensions: Symmetry-based classification of local pseudo-Hermitian systems and the corresponding domain walls
hep-th 2025-11 unverdicted novelty 6.0 of 10

Generalized quantum dimensions from SymTFTs classify massless and massive RG flows in pseudo-Hermitian systems and relate coset constructions to domain walls.
Curious QNEIs from QNEC: New Bounds on Null Energy in Quantum Field Theory
hep-th 2025-10 unverdicted novelty 6.0 of 10

Derives new state-independent lower bounds on semi-local integrals of null energy flux in QFTs of two and higher dimensions using QNEC, strong subadditivity, and modular Hamiltonians.

See all 167 Pith citations

How later papers used it

627 later papers cite it

Evidence: substantial. 625 of the citing papers share no author with it, and 91 of those carry a confident label from Pith's citation classifier.

Pith read 37 of the 869 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

13 later papers by other authors are listed below: 9 used its method or tool and 4 compared their own results with it.

Uses its method or tool 9 of 56 papers with this label

Used a technique introduced by this paper involving the trace anomaly.

Sine-Gordon Theory : Entanglement entropy and holography (2016) · 1.00 · paraphrased: the licence does not allow quoting
Computed the Rényi entropy from the replica trick of this paper on the θ-direction.

Quantum Correction of the Wilson Line and Entanglement Entropy in the Pure AdS$_3$ Einstein Gravity Theory (2019) · 0.99 · paraphrased: the licence does not allow quoting
Calculated the Rényi entropy from the replica trick of this paper.

U(1) CS Theory vs SL(2) CS Formulation: Boundary Theory and Wilson Line (2020) · 0.99 · paraphrased: the licence does not allow quoting
Used the replica trick often given as this paper to compute the von Neumann entanglement entropy.

Entanglement in interacting quenched two-body coupled oscillator system (2022) · 0.98 · paraphrased: the licence does not allow quoting
Used the replica trick of this paper to compute, since a reduced density matrix is lost in QFT.

AdS$_3$ Einstein Gravity and Boundary Description: Pedagogical Review (2023) · 0.98 · paraphrased: the licence does not allow quoting
Introduced the renormalized (or relative) entanglement as a regularized entropy of this paper for the reduced density operator of A.

The entropic boundary law in BF theory (2008) · 0.94 · paraphrased: the licence does not allow quoting
Specifically, for $N$ free scalar fields, the renormalised entropy of the right-moving radiation collected by a distant observer over a retarded-time interval $[u_1,u_2]$ is governed by the Holzhey-Larsen-Wilczek (HLW) [ref] /Fiola-Preskill-Strominger-Trivedi (FPST) [ref] formula:

Anomaly quenching and dynamical cooling of Hawking evaporation in Horndeski gravity (2026) · Entanglement Entropy of the Hawking Radiation · 0.80
Introduced geometric entropy characterizing entanglement of the outgoing fields inside an interval with anything outside it, following this paper.

Dilaton gravity with a boundary: from unitarity to black hole evaporation (2020) · 0.86 · paraphrased: the licence does not allow quoting
Calculated the central charge for open boundary conditions following this paper from the underlying CFT.

Multicriticality in stochastic dynamics protected by self-duality (2025) · 0.86 · paraphrased: the licence does not allow quoting

Compares with it 4 of 32 papers with this label

Found its own result identical to the universal value of the mutual information for adjacent intervals in a conformal field theory given by this paper, identifying a quantity with the UV cutoff.

$T\bar{T}$, the entanglement wedge cross section, and the breakdown of the split property (2020) · 0.99 · paraphrased: the licence does not allow quoting
The two are related by a rotation of the polar axis, $θ_std=π2 - θ$, so that $sinθ_0$ appearing in those references is replaced by $cosθ_0$ throughout this work, with no change in the underlying physics.] [ref] [equation] The comparison tells us that the central charge of the CFT$_2$ dual to the wedge $W_4$ takes the imaginary value [equation] which agrees with the value [ref] inferred from the partition function.

Effective dynamics and quantum information in de Sitter wedge holography (2026) · Holographic entanglement entropy · 0.98
Checked that its own final results reproduce those of this paper as a special case when the interface is absent.

Entanglement through conformal interfaces (2008) · 0.97 · paraphrased: the licence does not allow quoting
Obtained the entanglement entropy, reproducing the well-known result of this paper.

Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT (2017) · 0.91 · paraphrased: the licence does not allow quoting

How the literature uses it

391 arXiv papers cite this in 568 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 333 sentences without a confident label

Builds on it 3 papers

This generalizes the results of [ref] to a case with symmetries different from that of a conformal theory, and appropriate comparisons with a CFT$_2$ will be made along the way.

Entanglement Entropy in Warped Conformal Field Theories · Rindler method · 0.99
In order to carry out the analysis independently of the boundary conditions, it is desirable to generalize the analysis in [ref] based on the conformal invariance to our case with interfaces.

Entanglement through conformal interfaces · Introduction · 0.97
In this case Eq. ([ref]) is likely to result in a finite difference of entanglement entropies between the squeezed vacua $|J_1⟩$ and $|J_2⟩$, therefore providing a generalization of the results derived in [ref] .

Entanglement entropy of squeezed vacua on a lattice · Discussion · 0.94

Uses its method or tool 37 papers

The central charge is obtained from the prefactor of the logarithmic contribution [ref] to the bipartite entanglement entropy [ref] .

Precursor of Laughlin state of hard core bosons on a two leg ladder · Phases from central charge and vortex number · 0.96
We use the standard “replica” approach to compute the entropy [ref] .

Topological Entanglement Entropy in Chern-Simons Theories and Quantum Hall Fluids · Introduction · 0.99
We use a technique introduced by Holzhey, Larsen and Wilczek (HLW) [ref] involving the trace anomaly.

Sine-Gordon Theory : Entanglement entropy and holography · Introduction · 1.00

Compares with it 24 papers

We can compare equation [ref] with the standard result for a CFT$_2$ on $S^2$ with a central charge $c$, given by [ We use the angular convention $ds^2=dθ^2+cos^2θ dΩ_d-2^2$, rather than $ds^2=dθ^2+sin^2θ dΩ_d-2^2$ adopted in [ref] .

Effective dynamics and quantum information in de Sitter wedge holography · Holographic entanglement entropy · 0.97
This precisely matches $2 S_3-2α(ρ_A)$ which was computed in [ref] .

R\'enyi Mutual Information in Quantum Field Theory · Introduction · 0.98
This scaling form is almost identical to the celebrated EE result [ref] , upon substituting $c/3$ with $b/4$.

Shannon and R\'enyi mutual information in quantum critical spin chains · Introduction · 1.00

Cites it as support 5 papers

Entropy [ref] is well behaved and there is no information loss.

Schwarzschild Metric with Planck Length · Planck · 0.99
There is also a fair amount of evidence to support this conjecture [ref] : * As shown in [ref] , it reproduces precisely the entanglement entropy of a $d=2$ CFT for an interval of length $ℓ$ on a circle of circumference $2πR$ [ref]

On Holographic Entanglement Entropy and Higher Curvature Gravity · Introduction · 0.96
For the half line in a 2d CFT, it is a fact that $c_n$ is indeed independent of $n$ [ref] .

Analyticity of replica correlators and Modular ETH · 2d CFT · 0.98

Cites it as background 145 papers

In this regard, a major breakthrough came from Ryu and Takayanagi [ref] who discovered a precise relationship between geometry, in the form of minimal-area surfaces, to entanglement entropy, a central concept in quantum information [ref] .

Entanglement Entropy and Subregion Complexity in Thermal Perturbations around Pure-AdS Spacetime · Introduction · 0.96
The technology to compute entanglement entropy is best developed in the case of 1+1 dimensional CFTs [ref] (, [ref] for earlier work and [ref] for generalizations to integrable non-conformal systems).

Holographic entanglement entropy for disconnected regions · Introduction · 0.92
As predicted by conformal field theory (CFT) [ref] , $(1+1)$-dimensional critical systems host a sub-extensive entanglement structure with logarithmic growing EE [ref] .

A critical state under weak measurement is not critical · Introduction · 0.89

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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