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arxiv: 2606.01025 · v1 · pith:L2HPPGYFnew · submitted 2026-05-31 · ✦ hep-th

Black holes with quantum corrections in 3d: The case of Page curve in Lindblad, greybody factor, and Lyapunov exponent

Mahdis Ghodrati This is my paper

Pith reviewed 2026-06-28 17:00 UTC · model grok-4.3

classification ✦ hep-th
keywords black holesquantum correctionsPage curveLindblad formalismgreybody factorLyapunov exponentCotler-Jensen theoryexceptional points
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The pith

Black holes modeled as open quantum systems via Lindblad dynamics exhibit zig-zag behavior in the Page curve due to exceptional points.

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

The paper treats black holes with quantum corrections as open quantum systems and applies the Lindblad formalism to explain the zig-zag pattern seen in the Hawking-Page curve during radiation. Exceptional points in the Lindblad dynamics are presented as the mechanism responsible for this non-monotonic behavior. Quantum corrections are computed for parameters of the three-dimensional Cotler-Jensen theory, an AdS3 model with reparametrized modes, and compared throughout to the Jackiw-Teitelboim case. The work further derives quantum-corrected greybody factors for several black hole solutions and analyzes the resulting changes to the Lyapunov exponent, effective potential, and quantum information properties.

Core claim

Treating black holes that include quantum corrections as open quantum systems, the Lindblad equation accounts for the zig-zag shape of the Hawking-Page curve through the appearance of exceptional points; the same framework supplies explicit quantum corrections to parameters in the 3d Cotler-Jensen theory, produces modified greybody factors, and shifts the Lyapunov exponent relative to the uncorrected or JT cases.

What carries the argument

The Lindblad master equation applied to an open-system description of the black hole, with exceptional points as the feature that produces the zig-zag in the Page curve.

If this is right

  • The zig-zag in the Hawking-Page curve arises directly from exceptional points in the Lindblad spectrum.
  • Quantum corrections to the parameters of 3d Cotler-Jensen theory differ in detail from those obtained in JT gravity.
  • Greybody factors for black hole solutions acquire explicit quantum corrections that can be evaluated for each geometry.
  • The Lyapunov exponent and the associated potential receive shifts once quantum corrections are included.
  • The quantum information structure, including entanglement features, is modified by the same corrections.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The Lindblad-plus-exceptional-point picture could be tested in other AdS models or in higher-dimensional gravity to see whether similar non-monotonic Page curves appear.
  • Exceptional points may correspond to thermodynamic critical points, offering a dynamical-systems route to black hole phase transitions.
  • Small-system numerical simulations of the Lindblad equation could provide a concrete check of the predicted zig-zag without requiring full quantum gravity.
  • Altered Lyapunov exponents might translate into changed scrambling times inside the quantum-corrected geometry.

Load-bearing premise

Black hole radiation including quantum corrections can be modeled as an open quantum system whose dynamics are captured by the Lindblad equation, and that exceptional points are the main driver of the zig-zag Page curve.

What would settle it

A numerical diagonalization of the Lindblad operator for a quantum-corrected 3d black hole that either produces or fails to produce exceptional points whose eigenvalues trace the observed zig-zag in the Page curve, or a direct computation of the greybody factor that matches or deviates from the predicted quantum correction.

Figures

Figures reproduced from arXiv: 2606.01025 by Mahdis Ghodrati.

Figure 1
Figure 1. Figure 1: The behavior of V(z) versus hH, which is non-monotonic and affects the non-monotonicity of energy gaps versus coupling. where for the case of thermofield double state (TFD), one can write |TFD⟩ = 1 √ Z X n e −βEn/2 |n⟩L ⊗ |n⟩R . (2.27) With the Hamiltonian Htot = HL + HR, the time evolution would be U(t) = e −iHtott , and the fermionic bilinear coupling would be e iµV , V = 1 qN Hint, Hint = i X N i=1 ψ i … view at source ↗
Figure 2
Figure 2. Figure 2: The schematic structure of graviton exchanges between heavy and light operators discussed [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The propagations between a heavy and light operators and the diagrammatic rules discussed [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The behavior of the product of propagation [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The difference between classic and quantum corrected greybody factor for [PITH_FULL_IMAGE:figures/full_fig_p035_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The classical 5d absorption factor with 1-loop correction, and a separate plot of δ(ω). One could consider the Schwarzian soft mode which are being coupled to the boundary insertion that creates or absorbs the bulk scalar, and then integrating out the soft mode dresses the boundary two-point, or the scattering kernel, producing loop corrections which are suppressed by inverse powers of C i.e. powers of 1/S… view at source ↗
Figure 7
Figure 7. Figure 7: The difference between classic and quantum corrected greybody factor for [PITH_FULL_IMAGE:figures/full_fig_p039_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The grey body factor with 1-loop correction for WBTZ geometry. [PITH_FULL_IMAGE:figures/full_fig_p040_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The quantum corrected versus classical greybody factor for [PITH_FULL_IMAGE:figures/full_fig_p044_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The quantum corrected versus classical greybody factor for [PITH_FULL_IMAGE:figures/full_fig_p045_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The quantum corrected versus classical greybody factor for [PITH_FULL_IMAGE:figures/full_fig_p046_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The quantum corrected versus classical greybody factor for [PITH_FULL_IMAGE:figures/full_fig_p046_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The quantum corrected versus classical greybody factor for [PITH_FULL_IMAGE:figures/full_fig_p047_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The quantum corrected versus classical greybody factor for [PITH_FULL_IMAGE:figures/full_fig_p047_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The relationship between α and hH. The one-loop O(1/C) correction to the block can be found by the reparameterization field ϕ = α0θ + √ϵ C as B(h; θ1, θ2) = α0 2 sin(α0θ12 2 ) !2h  1 + h √ C J ′(1) 12 . ϵ + 1 C  h 2 2 (J ′(1) 12 . ϵ) 2 + hJ ′(2) 12 . ϵ + O  1 C3/2  . (5.9) Finally, the bilocal operator can be written as ⟨B(hL;w, 0)⟩α =  α 2 sin(αw 2 ) 2hL  1 + hL c Vh/c(w) + h 2 L c Vh2/c(w) + O … view at source ↗
Figure 16
Figure 16. Figure 16: The behavior of each contribution to bilocal operator [PITH_FULL_IMAGE:figures/full_fig_p050_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The behavior of bilocal operator B and the Lyapunov exponent for various light operator weight hL while α = 0.1 and c = 3. 5 10 15 20 25 30 t -2×10-16 0 4×10-16 Β c=100 c=50 c=30 c=20 c=15 20 40 60 80 100 t 0.00 0.02 0.04 0.06 0.08 0.10 λL c=100 c=50 c=30 c=20 c=15 [PITH_FULL_IMAGE:figures/full_fig_p051_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The behavior of bilocal operator B and the Lyapunov exponent for various big central charge c, while hL = 0.5 and α = 0.1. From Figures 18 and 19, one could see that at large times and for any central charge c, the Lyapunov exponent reach to a specific λL, so at large time the value of λL would be independent of c, but is related to hL, α, or hH. Generally, one could see that by increasing c, λL would inc… view at source ↗
Figure 19
Figure 19. Figure 19: The behavior of bilocal operator B and the Lyapunov exponent for various low central charge c, while hL = 0.5 and α = 0.1. 6 Quantum corrections to other thermodynamical vari￾ables of black holes We could also check the effects of quantum correction on other variables of black holes such as potential, off-shell geometries, quantum width, quantum information structures, etc, which we discuss some of them b… view at source ↗
Figure 20
Figure 20. Figure 20: The change in effective potential with reparametrization modes of Cotler-Jensen theory for [PITH_FULL_IMAGE:figures/full_fig_p054_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: The classic and 1-loop quantum corrected greybody factor in Cotler-Jensen theory for various [PITH_FULL_IMAGE:figures/full_fig_p055_21.png] view at source ↗
read the original abstract

In this work, first we discuss black holes with quantum corrections as an open quantum system and apply Lindblad formalism to explain the ``zig-zag" behavior in the Hawking- Page curve and radiation process, specially by considering the effects of exceptional points. Then, we calculate quantum corrections for various parameters of $3d$ Cotler-Jensen theory which is $\text{AdS}_3$ with reparametrized modes. At each step of the calculations, we compare the results with the case of JT. We then calculate quantum-corrected greybody factor for various black hole solutions, and specially for the case of Cotler-Jensen theory. Finally, we study the effects of quantum corrections on Lyapunov exponent, potential, and quantum information structure of black holes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper models black holes with quantum corrections in 3d Cotler-Jensen theory as open quantum systems using the Lindblad master equation, attributing the zig-zag behavior of the Hawking-Page curve to exceptional points in the non-Hermitian spectrum. It computes quantum corrections to parameters in the theory (comparing throughout to JT gravity), derives quantum-corrected greybody factors for various solutions, and examines the impact of these corrections on Lyapunov exponents, potentials, and quantum information measures.

Significance. If the central mapping from Lindblad dynamics and exceptional points to the Page-curve zig-zag can be made explicit and derived from the underlying action, the work would provide a concrete open-system mechanism for non-unitary effects in quantum-corrected 3d gravity and a useful benchmark against JT gravity. The multi-faceted comparisons (greybody factors, Lyapunov exponents) add value, but the current qualitative treatment of the Page curve limits immediate impact.

major comments (2)
  1. [Lindblad formalism and Page curve discussion (abstract and main text sections on open-system modeling)] The central claim that Lindblad dynamics with exceptional points explains the zig-zag in the Hawking-Page curve is load-bearing yet unsupported by derivation: the manuscript posits the Lindblad equation and attributes the behavior to EPs without deriving the jump operators from the 3d Cotler-Jensen action or the quantum-corrected metric, and without an explicit solution of the master equation that maps EP coalescence to the entanglement-entropy time series.
  2. [Quantum corrections and greybody factor calculations] In the sections comparing quantum corrections and greybody factors to JT gravity, the calculations appear to proceed by direct substitution into the corrected metric or action, but no error analysis, convergence checks, or parameter ranges are provided to establish that the reported modifications are robust rather than artifacts of the chosen regularization or truncation.
minor comments (2)
  1. [Introduction and theory setup] Notation for the reparametrized modes in the Cotler-Jensen theory is introduced without a dedicated table or appendix summarizing the dictionary between 3d quantities and their JT counterparts.
  2. [Figures and results sections] Figure captions for the Page curve and Lyapunov plots should explicitly state the numerical method used to extract the zig-zag or exponent values.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below, agreeing where additional clarification or checks are warranted.

read point-by-point responses

  1. Referee: [Lindblad formalism and Page curve discussion] The central claim that Lindblad dynamics with exceptional points explains the zig-zag in the Hawking-Page curve is load-bearing yet unsupported by derivation: the manuscript posits the Lindblad equation and attributes the behavior to EPs without deriving the jump operators from the 3d Cotler-Jensen action or the quantum-corrected metric, and without an explicit solution of the master equation that maps EP coalescence to the entanglement-entropy time series.

    Authors: We acknowledge that the Lindblad equation is introduced as an effective model for the open-system dynamics without an explicit derivation of the jump operators from the 3d Cotler-Jensen action or metric. The zig-zag is attributed to exceptional points identified in the non-Hermitian spectrum, but no full solution of the master equation connecting EP coalescence directly to the entanglement-entropy time series is provided. The manuscript focuses on the phenomenological application rather than a first-principles derivation. We will revise the relevant sections to emphasize the effective nature of the description, clarify the assumptions, and note the limitations, including the need for future work on deriving the jump operators from the underlying theory. This constitutes a partial revision. revision: partial

  2. Referee: [Quantum corrections and greybody factor calculations] In the sections comparing quantum corrections and greybody factors to JT gravity, the calculations appear to proceed by direct substitution into the corrected metric or action, but no error analysis, convergence checks, or parameter ranges are provided to establish that the reported modifications are robust rather than artifacts of the chosen regularization or truncation.

    Authors: We agree that robustness checks are needed. The quantum corrections and greybody factors are computed via direct substitution of the corrected parameters into the relevant expressions, with comparisons to JT gravity performed at the level of effective quantities. In the revised manuscript we will add error analysis, specify the parameter ranges employed, and include convergence checks with respect to the regularization and truncation order. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper posits the Lindblad master equation as a model for black holes with quantum corrections treated as open systems, then attributes zig-zag Page-curve behavior to exceptional points while computing corrections to greybody factors and Lyapunov exponents in the 3d Cotler-Jensen theory and comparing to JT gravity. No quoted equations or self-citations in the provided description reduce any claimed prediction or result to an input by construction; the Lindblad application and comparisons are presented as external formalisms applied to the target theory rather than internally fitted or self-defined quantities. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no specific free parameters, axioms, or invented entities can be extracted; the work appears to rest on standard assumptions from quantum gravity and open quantum systems without introducing new ones visible here.

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