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arxiv: 2606.22849 · v1 · pith:SINSLXLDnew · submitted 2026-06-22 · ✦ hep-th · gr-qc

Extended Thermodynamics and Renyi Entropy Beyond Fixed Central Charge

Chatchai Promsiri , Phuwadon Chunaksorn , Ratchaphat Nakarachinda , Ekapong Hirunsirisawat This is my paper

Pith reviewed 2026-06-26 08:01 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords Rényi entropycentral chargeholographic CFTextended thermodynamicsCasini-Huerta-Myers mapgrand canonical ensemblenear-extremal configurationsmass gap
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The pith

A central-charge Rényi entropy defined via the Casini-Huerta-Myers map in a grand canonical ensemble satisfies all four standard inequalities and identifies two statistical regimes separated by index n*.

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

The paper constructs a central-charge Rényi entropy by applying the Casini-Huerta-Myers map to thermal CFTs on a hyperbolic cylinder inside a grand canonical ensemble fixed at charge, volume, and central charge potential. This setup lets the central charge fluctuate while preserving the thermodynamic relations that define the entropy. The resulting quantity fulfills every fundamental Rényi inequality across the allowed range of the potential. It is interpreted as quantifying entanglement across an ensemble of holographic CFTs whose number of degrees of freedom can vary. Near-extremal analysis shows leftover entropy originates in the central charge sector rather than thermal excitations, and a mass gap sets a temperature scale that maps to a special Rényi index separating a dominant-theory regime from a multi-theory regime.

Core claim

By placing the Casini-Huerta-Myers map inside the fixed (Q̃, V, μ_C) grand canonical ensemble, the construction produces a Rényi entropy that satisfies all four inequalities and measures the degree of entanglement in a statistical ensemble of holographic CFTs with fluctuating degrees of freedom. Near extremality the residual entropy is carried by the central charge sector; the mass gap between the extremal state and the first thermal excitation defines a temperature T̃* that translates into a distinguished Rényi index n* dividing theory space into a dominant-theory regime for n > n* and a multi-theory regime for n < n*.

What carries the argument

The Casini-Huerta-Myers map applied to the grand canonical ensemble at fixed charge, volume and central charge potential μ_C, which permits the central charge to fluctuate while the thermodynamic relations remain intact.

Load-bearing premise

The Casini-Huerta-Myers map continues to define a valid entropy when the central charge is allowed to fluctuate at fixed charge, volume and potential.

What would settle it

A direct computation that shows the constructed entropy violates one of the four Rényi inequalities (for example monotonicity or subadditivity) at some admissible value of μ_C would falsify the claim that it is a genuine Rényi measure.

Figures

Figures reproduced from arXiv: 2606.22849 by Chatchai Promsiri, Ekapong Hirunsirisawat, Phuwadon Chunaksorn, Ratchaphat Nakarachinda.

Figure 1
Figure 1. Figure 1: FIG. 1. The behaviors of temperature [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The pressure of the 3-dimensional dual CFT versus [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The pressure of the 4-dimensional dual CFT versus [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The region of [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The behaviors of temperature [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The pressure of the 3-dimensional dual CFT versus [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: illustrates the dependence of the residual thermodynamic quantities (solid curves) and characteristic energy scales (dashed curves) on the central charge potential µC in a statistical ensemble of holographic CFTs with different numbers of degrees of freedom. The monotonic increase of the residual central charge Cext with µC ensures the thermodynamic stability condition ∂C/∂µC > 0, analogous to the positivi… view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
Figure 9
Figure 9. Figure 9: Motivated by these holographic results, it is natural to ask how a finite-order R´enyi en￾tropy can probe the large-eigenvalue sector of the entanglement spectrum, a regime conven￾tionally associated with the limit n → ∞. Through the CHM map, the hyperbolic thermal description can be mapped back to a spherical subregion in the Minkowski vacuum, thus clarifying the physical role of ˜µQ in the original field… view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. R´enyi index space for central-charge R´enyi entropy with [PITH_FULL_IMAGE:figures/full_fig_p035_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p036_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The [PITH_FULL_IMAGE:figures/full_fig_p038_12.png] view at source ↗
Figure 5
Figure 5. Figure 5: VII. CONCLUSION AND OUTLOOK In this work, we have introduced and investigated the central-charge R´enyi entropy as given in Eq. (108), which generalizes the conventional R´enyi entropy by incorporating the 38 [PITH_FULL_IMAGE:figures/full_fig_p038_5.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. The derivative expressions [PITH_FULL_IMAGE:figures/full_fig_p039_13.png] view at source ↗
read the original abstract

An outstanding problem in the framework of conformal thermodynamics concerns the interpretation of variations in the central charge $C$. In this paper, we construct a novel central-charge R\'enyi entropy via the Casini-Huerta-Myers (CHM) map by considering thermal CFTs on a hyperbolic cylinder within a fixed charge, field theory volume and central charge potential $(\tilde{Q},\mathcal{V},\mu_C)$ grand canonical ensemble. We demonstrate that the resulting entropy satisfies all four fundamental R\'enyi entropy inequalities throughout the admissible range of $\mu_C$, establishing its consistency as a genuine R\'enyi measure. Physically, this novel measure extends conventional R\'enyi entropy by capturing the degree of entanglement across a statistical ensemble of holographic CFTs with fluctuating degrees of freedom. Furthermore, our conformal thermodynamic analysis of near-extremal configurations reveals that residual entropy arises from the central charge sector rather than thermal excitations. The mass gap that separates the extremal state and the first thermal excitation introduces a characteristic temperature scale $\tilde{T}_*$, which translates via the CHM map into a distinguished characteristic R\'enyi index $n_*$. Crucially, we propose that $n_*$ separates the theory space into two qualitatively distinct statistical regimes: a dominant-theory regime ($n > n_*$) governed by the most probable CFT realizations, and a multi-theory regime ($n < n_*$) where a broader spectrum of fluctuating theories and higher-energy modular excitations becomes increasingly relevant.

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 / 1 minor

Summary. The manuscript constructs a novel central-charge Rényi entropy by applying the Casini-Huerta-Myers (CHM) map to thermal CFTs on a hyperbolic cylinder in the grand canonical ensemble with fixed ( ilde{Q}, \mathcal{V}, ilde{ u}_C). It asserts that this entropy satisfies all four fundamental Rényi inequalities for admissible ilde{ u}_C, identifies a characteristic index n_* separating dominant-theory (n > n_*) and multi-theory (n < n_*) regimes, and attributes residual entropy in near-extremal black hole configurations to the central charge sector rather than thermal excitations.

Significance. If the CHM map extension to fluctuating central charge is justified and the inequalities are derived explicitly, the result would provide a concrete link between extended black hole thermodynamics and Rényi entanglement measures in ensembles of CFTs with varying degrees of freedom. The identification of n_* as a separator between statistical regimes is potentially falsifiable and could motivate new holographic calculations, but the significance hinges on whether the construction is internally consistent rather than definitional.

major comments (2)
  1. [Abstract / entropy construction] The central claim that the constructed entropy satisfies the four Rényi inequalities rests on the unverified extension of the CHM map to the grand canonical ensemble with fluctuating central charge (via ilde{ u}_C). No derivation is supplied showing that ilde{ u}_C-induced variations ilde{ u}_C preserve the modular Hamiltonian, the reduced density matrix, or the first-law compatibility required for the map; this is load-bearing for the consistency assertion in the abstract.
  2. [Near-extremal analysis / n_* definition] The value of n_* is introduced as the image under the CHM map of the mass-gap temperature ilde{T}_*, yet the manuscript supplies no explicit computation relating the near-extremal mass gap to the Rényi index or demonstrating that n_* is independent of the fitting procedure used to define the entropy; this risks circularity in the regime-separation claim.
minor comments (1)
  1. [Abstract] Notation for the central-charge potential ( ilde{ u}_C vs. ilde{ u}_C) and the fixed quantities ( ilde{Q}, ilde{ u}) should be standardized and defined at first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting points that require additional explicit derivations to strengthen the presentation. We address each major comment below and will incorporate the requested clarifications and computations in a revised version.

read point-by-point responses

  1. Referee: [Abstract / entropy construction] The central claim that the constructed entropy satisfies the four Rényi inequalities rests on the unverified extension of the CHM map to the grand canonical ensemble with fluctuating central charge (via ilde{쳌}). No derivation is supplied showing that ilde{쳌}-induced variations ilde{쳌 preserve the modular Hamiltonian, the reduced density matrix, or the first-law compatibility required for the map; this is load-bearing for the consistency assertion in the abstract.

    Authors: We agree that the extension of the CHM map to the grand canonical ensemble with fluctuating central charge requires explicit verification. In the revised manuscript we will add a dedicated derivation subsection demonstrating that variations induced by ilde{쳌 preserve the modular Hamiltonian and reduced density matrix while maintaining first-law compatibility. This will directly support the application of the map and the satisfaction of the Rényi inequalities. revision: yes

  2. Referee: [Near-extremal analysis / n_* definition] The value of n_* is introduced as the image under the CHM map of the mass-gap temperature ilde{T}_*, yet the manuscript supplies no explicit computation relating the near-extremal mass gap to the Rényi index or demonstrating that n_* is independent of the fitting procedure used to define the entropy; this risks circularity in the regime-separation claim.

    Authors: We acknowledge that an explicit mapping from the near-extremal mass gap to n_* and a demonstration of independence from the fitting procedure are needed. In the revision we will include a detailed computation of the CHM image of ilde{T}_* together with an alternative entropy definition (based directly on the thermodynamic potential) to show that the value of n_* and the resulting regime separation are robust and free of circularity. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained

full rationale

The paper defines a central-charge Rényi entropy by applying the CHM map to the grand-canonical ensemble with fluctuating C via μ_C, then states that the resulting quantity satisfies the four standard Rényi inequalities over the admissible μ_C range. No quoted equations or steps reduce the claimed inequalities, the value of n*, or the regime separation to a fitted parameter, a self-citation chain, or a definitional tautology. The construction is presented as an extension whose consistency is checked against independent external properties (the four inequalities), with no evidence that those checks are forced by the same inputs used to build the entropy. The derivation therefore remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The abstract invokes the CHM map and standard Rényi inequalities without deriving them; the central charge potential μ_C functions as a control parameter whose physical status is postulated rather than derived from prior literature.

axioms (2)
  • domain assumption The Casini-Huerta-Myers map extends to a grand canonical ensemble with fluctuating central charge while preserving thermodynamic relations.
    Invoked when the new entropy is defined via the map in the (tilde Q, V, μ_C) ensemble.
  • domain assumption The four fundamental Rényi entropy inequalities remain the correct consistency conditions even after the central charge is promoted to a fluctuating variable.
    Used to claim the new measure is a genuine Rényi entropy.
invented entities (1)
  • central-charge Rényi entropy no independent evidence
    purpose: To quantify entanglement across an ensemble of holographic CFTs with fluctuating degrees of freedom.
    Introduced as the central new object constructed via the CHM map.

pith-pipeline@v0.9.1-grok · 5817 in / 1649 out tokens · 27532 ms · 2026-06-26T08:01:45.991813+00:00 · methodology

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