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REVIEW 3 major objections 5 minor 83 references

Phase transition in a doubly holographic model of closed $\mathbf{dS_{2} }$ spacetime

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper claims a third extremal surface, beyond the Hartman-Maldacena and island surfaces, controls the low-temperature phase of a doubly holographic dS2 model.

desk verdict The claimed new saddle S2 is a regulator-dependent boundary configuration rather than a genuine extremal surface, so the three-phase structure is unsupported despite some solid standard analysis. read the letter →

arxiv 2502.08380 v1 pith:ZZL2SDBX submitted 2025-02-12 hep-th

Wen-Hao Jiang , Cheng Peng , Yun-Song Piao This is my paper · ORCID
classification hep-th
keywords deSitterspacetimedoubleholographyislandformulaentanglemententropyphasetransitiongeneralizedmutualinformationHartman-Maldacenasurfacenegativegeodesic
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

The paper studies a doubly holographic model in which a closed $\mathrm{dS}_2$ brane with JT gravity and positive cosmological constant is glued to flat thermal baths and embedded in an $\mathrm{AdS}_3$ bulk. It claims that, besides the Hartman-Maldacena surface that connects the two bath intervals and the island surface, there is a third extremal surface, $S_2$, whose entropy can be the smallest at low temperature, so the radiation entropy is $S=\min\{S_1,S_2,S_3\}$. The paper proposes a generalized mutual information $I_{\mathrm{gen}}=2S_0-S_i$ that vanishes along the phase-transition curves, giving an interpretation of the three-way transition in terms of entanglement information. The new saddle is subtle: when it dominates, part of its bulk geodesic has negative length because its endpoints sit inside the bulk horizon. If correct, the result would make the low-temperature Page curve of this de Sitter model richer than the usual two-saddle story, with a definite triple point where the three phases meet.

What carries the argument

The machinery is the doubly holographic rewriting of the island formula as $S_R=\min\{\mathrm{ext}[\Gamma/(4G_N^{(3)})+A(\partial I)/(4G_N^{(2)})]\}$, with $\Gamma$ an extremal surface in the $\mathrm{AdS}_3$ bulk. In this embedding the brane location is fixed by the gluing condition $z_{\mathrm{dS}_2}=\epsilon |1-\omega_+\omega_-|/2$, and the three saddles are classified by which endpoints they connect. The new $S_2$ saddle is defined by taking the brane endpoints to the horizon, $p\to\infty$, with a regulator $\varepsilon$; its defining feature is the geodesic length $L_{34}=2\log(2\varepsilon^2\cosh(Ht)/\epsilon)$, which becomes negative when the saddle dominates. The generalized mutual information $I_i=2S_0-S_i$ is the diagnostic quantity whose zeros line up with the phase-transition curves $X_{ij}$.

What would settle it

Compute the fully regularized length of the $S_2$ configuration without splitting it into the two geodesic pieces used in Eq. (27): if the total length is positive while $S_2$ still beats $S_1$ and $S_3$ at low temperature, the negative-length effect is an artifact of how the surface was decomposed, whereas if the total length is negative and the entropy functional has no lower bound, the $S_2$ saddle is not a valid competitor and the predicted three-phase diagram would not appear in a consistent quantization.

Watch

Extended reading notes

Core claim

Working in the doubly holographic model in which a closed $\mathrm{dS}_2$ brane is glued to flat thermal baths and embedded in a BTZ bulk, the paper finds three competing extremal surfaces that contribute to the radiation entropy: the surface connecting the two bath intervals ($S_1$), a Hartman-Maldacena-type surface that connects the two baths and also joins the two $\mathrm{dS}_2$ brane endpoints at the horizon ($S_2$), and the island surface ($S_3$). The claim is that the true entropy is $S=\min\{S_1,S_2,S_3\}$, so the new $S_2$ saddle dominates at low temperature and creates a three-phase diagram with two critical temperatures. The paper further proposes that the transitions are diagnosed by a generalized mutual information $I_{\mathrm{gen}}=2S_0-S_i$, which vanishes along the transition curves; the same structure is found for the $\mathrm{AdS}_2$ eternal black hole with a thermal bath. The price is that, when $S_2$ dominates, the geodesic segment on the $\mathrm{dS}_2$ brane has negative length, because its endpoints lie inside the bulk horizon, and the paper proposes to count coincident bulk geodesics only once.

Load-bearing premise

The load-bearing assumption is that taking the endpoints to the horizon with a small cutoff is a legitimate way to define a surface's entropy, even though part of the surface comes out with a negative length when this surface wins; if a piece of a surface cannot have negative length, the new saddle and the three-phase transition disappear.

Editorial extensions

If this is right

  • If the claim is right, the Page-curve transition in this $\mathrm{dS}_2$ model is not two-phase but three-phase: at low temperature the new saddle mediates between the connected phase and the island phase.
  • The vanishing of the generalized mutual information $I_{\mathrm{gen}}=2S_0-S_i$ provides a diagnostic for when the dominant saddle changes, extending the known role of $I(A;B)=0$ in Page transitions.
  • The same three-way phase structure appears in the $\mathrm{AdS}_2$ eternal black hole with a thermal bath, so the new saddle is not an artifact of de Sitter signature.
  • The negative geodesic length inside the bulk horizon implies that bulk geodesic contributions inside horizons must be counted without multiplicity for the entropy to be bounded below.
  • The phase structure predicts a critical temperature where all three saddles meet; below it the connected $S_1$ phase never dominates.

Reading between the lines

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

  • The paper's fix, counting coincident geodesics once, restores a lower bound but leaves open why a bulk geodesic inside a horizon should be assigned length at all; a more systematic prescription would follow from a replica calculation of the $S_2$ saddle, which this paper does not do.
  • The three-phase diagram has a triple point where $S_1$, $S_2$, and $S_3$ meet; if it is physical, similar triple points should appear in other doubly holographic de Sitter models with two boundaries, and the low-temperature connected phase would be governed by a horizon-anchored surface rather than by the usual Hartman-Maldacena surface.
  • The generalized mutual information $I_2=2S_0-S_2$ may plausibly be interpreted as a conditional or reflected entropy of the two radiation intervals; testing that interpretation in a boundary conformal field theory calculation would give a dual check of the phase transition beyond the bulk geodesic computation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper studies a doubly holographic model of two-dimensional closed de Sitter spacetime following the construction of [26]. It computes three candidate holographic entanglement entropies: the connected Hartman-Maldacena-type surface S1, an island surface S3, and a new configuration S2 whose endpoints lie on the dS2 brane near the horizon. The paper claims that S2 can dominate at low temperature, producing a three-way phase transition, and it introduces a generalized mutual information I_gen to interpret the transitions. It also notes that the S2 saddle contains a geodesic with negative length when dominant, attributes this to endpoints inside the bulk BTZ horizon, and proposes to count coincident geodesics only once. Section VI repeats the analysis for an AdS2 eternal black hole and finds a similar phase structure.

Significance. If the S2 saddle were legitimate, the paper would provide a concrete doubly holographic dS2 setup with a three-saddle phase structure and a possible mutual-information interpretation. The computations of S1 and S3 from standard geodesic lengths are explicit, and the paper is commendably honest about the negative-length subtlety, even proposing a resolution. No parameters are fitted to data; all phase boundaries are analytic. However, the legitimacy of S2 is the load-bearing element of the paper, and I do not find it established: S2 is a regulator-dependent boundary limit rather than an interior extremum of the generalized entropy, and the proposed fix does not remove the negative length. The central claim therefore rests on an ad hoc prescription.

major comments (3)
  1. [§II, Eqs. (14)–(15)] S2 is not an interior extremum of the generalized entropy. Differentiating Eq. (14) with respect to p gives dS2/dp = -(cH/3) tanh(Hp) + 2H^2 φ_r sech^2(Hp); no finite-p stationary point is a minimum for positive φ_r, and the p-dependent part is minimized only at p → ∞, i.e. on the horizon. The second cutoff ε introduced in Eq. (15) is therefore an ad hoc regulator for a boundary configuration, and the phase boundaries X_12 and X_23 in Eq. (19) depend on this regulator. Since the existence of S2 as a legitimate saddle is the basis for the claimed three-phase structure and for the generalized mutual information interpretation, this is a load-bearing issue and not a presentation detail.
  2. [§IV, Eq. (27) and §V] The negative geodesic length is not resolved by the proposed counting rule. The paper states that when S2 dominates, L34 < 0, and Eq. (29) shows the endpoints lie inside the bulk BTZ horizon. The rule in Section V, that coincident geodesics contribute only once, prevents the entropy from being made arbitrarily negative by adding many geodesics, but it does not make L34 positive and is not derived from the replica or quantum-extremal-surface prescriptions. Without a positive-length definition of this saddle, the entropy S = min(S1, S2, S3) in Eq. (18) and the phase diagram in Fig. 6 are not supported.
  3. [Abstract vs. §II and §VII] The abstract claims there exists "a new extremal surface besides the Hartman-Maldacena surface and the island surface," but §II (after Eq. (14)) says the S2 saddle "seems coincident with the HM surface of dS spacetime," and §VII says "we wonder whether it could be regarded as a kind of HM surface." These statements are inconsistent about the central novelty of the paper. The authors need to state unambiguously whether S2 is a genuinely new surface or a realization of the HM surface, and adjust the abstract and discussion accordingly.
minor comments (5)
  1. [Abstract and §VII] There are typos: "We purpose a simple solution" should be "We propose a simple solution," and in Section VII "ths issue" and "transtion" should be corrected.
  2. [Fig. 6 caption] The caption contains a stray "3" in "q= 0.2 3"; this should be cleaned up.
  3. [Notation throughout] The two cutoffs ϵ (embedding scale) and ε (horizon cutoff) are visually very similar; Eqs. (15), (27), and (29) would benefit from distinct symbols and clear definitions at first use.
  4. [§III, Eq. (21)] The generalized mutual information is defined only for i = 1, 2 in the expression I_gen = 2S0 - S_i; the text should clarify how it is meant to apply to the island phase i = 3, or why it is only needed for the connected-type saddles.
  5. [§VI, Eqs. (41)–(44)] The matching of X'_23 with I'_2 = 0 is explicitly qualitative since both diverge in the H → 0 limit; this should be stated as a consistency check rather than as an exact coincidence.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: phase comparison and mutual-information check are independent; only minor background self-citations.

full rationale

The central derivation is self-contained. The entropies S1, S2 and S3 are computed as independent geodesic lengths (Eqs. 13, 15, 17) from the doubly holographic embedding taken from [26], and the phase rule S = min[S1, S2, S3] (Eq. 18) is a direct comparison rather than a fit. The equality conditions X_ij (Eq. 19) are algebraic consequences of equating pairs of these entropies. The proposed generalized mutual information I_gen = 2S0 - S_i (Eq. 21) is defined from the same entropies, but checking I_i = 0 against the transition curves is a genuine consistency test: the coincidence at low temperature is shown by explicit approximate algebra (Eqs. 25-26, 42-44), not imposed by construction. The regulator dependence and negative geodesic length of the S2 saddle (Eqs. 27-29) are substantive correctness concerns about whether S2 is a legitimate extremal surface, but they are not circularity: they do not reduce a claimed output to an input. The self-citations, e.g. [42] and [59], occur in background lists of dS island literature and carry no load in the derivation. Hence the paper is not circular; the score reflects only the minor, non-load-bearing self-citations.

Assumptions & free parameters 3 free parameters · 3 assumptions · 1 invented entities

The central claim rests on domain assumptions from earlier doubly holographic constructions, an ad hoc multiplicity rule to handle negative geodesic lengths, and several cutoff parameters chosen by hand.

free parameters (3)
  • cutoff ε for p→∞ = small, typically 0.01 in figures
    Introduced to regulate the divergence when dS brane endpoints are taken to the horizon (p→∞); S2 and phase boundaries depend on it.
  • cutoff ξ for outer bath endpoint = ξ→0
    Used in the AdS2 section to define the outer endpoint of the radiation region; appears in I'_1 and I'_2.
  • AdS3 embedding scale ϵ = set ϵbH=ϵg=ϵ
    The two embedding scales are equated for convenience; all entropy formulas are expressed in terms of this single scale.
assumptions (3)
  • domain assumption The island formula (Eq. 2) with the extremal surface prescription min{ext} is valid for doubly holographic constructions in dS spacetime.
    The paper assumes the doubly holographic dictionary of [26] applies to dS2 with a thermal bath, which is still under debate (noted in the introduction).
  • domain assumption The embedding of the dS2 brane and flat bath into AdS3 via z_dS2 = (ϵ_g/2)|1-ω+ω-| and z_bath = ϵ_b H/(sqrt(ω+ω-)) is the correct bulk description.
    This is taken from [26] and is the basis for all geodesic length calculations.
  • ad hoc to paper Contributions from bulk geodesics with the same limiting endpoints should be counted without multiplicity.
    Proposed in Sec. V to prevent the entropy from being unbounded below; it has no independent derivation and does not remove the negative length of individual geodesics.
invented entities (1)
  • Generalized mutual information I_gen
    purpose: Proposed to interpret phase transitions between saddles
    Defined as 2S0 - S_i; its zero is compared with transition curves, but it is constructed from the same entropies that define the phases.

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Cite this review

Pith. "Pith review of Phase transition in a doubly holographic model of closed $\mathbf{dS_{2} }$ spacetime." pith.science (2026). https://pith.science/paper/ZZL2SDBX

@misc{pith2026250208380,
  author       = {Pith},
  title        = {Pith review of: Phase transition in a doubly holographic model of closed $\mathbfdS_2 $ spacetime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZZL2SDBX}},
  note         = {Machine review of arXiv:2502.08380}
}
abstract

Double holography has been proved to be a powerful method in comprehending the spacetime entanglement. In this paper we investigate the doubly holographic construction in ${\mathrm{dS_{2}} }$ spacetime. We find that in this model there exists a new extremal surface besides the Hartman-Maldacena surface and the island surface, which could lead to a more complex phase structure. We then propose a generalized mutual entropy to interpret the phase transition. However, this extremal surface has a subtle property that the length of a part of the geodesic is negative when this saddle is dominant. This is because the negative part of the geodesic is within the horizon of the bulk geometry. We move to the ${\mathrm{AdS_{2}} }$ spacetime and find this subtlety still exists. We purpose a simple solution to this issue.

Figures

Figures reproduced from arXiv: 2502.08380 by the authors.

Figure 1
Figure 1. FIG. 1: A schematic of ( [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The Penrose diagram of the flat spacetime with thermal bath glued to the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The schematic of the surface that connects the two sides of the system (red dashed line). [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The schematic of the surface that connects the endpoints ’1’ with ’2’ and ’3’ with ’4’ (green [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The schematic of the island surface that connects the endpoints ’1’ with ’3’ and ’2’ with [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Phase diagram of the system if we remove [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The curve of [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: The Penrose diagram of the two sided eternal black hole in AdS [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: The phase transition diagram with the parameter set by [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

Discussion (0). Continue with ORCID to comment.

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