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arXiv:2606.22849

Extended Thermodynamics and Renyi Entropy Beyond Fixed Central Charge

Chatchai Promsiri, Phuwadon Chunaksorn, Ratchaphat Nakarachinda, Ekapong Hirunsirisawat

Published Jun 22, 2026• 0 citations
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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 Rényi 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 $(\tilde{Q},\mathcal{V},μ_C)$ grand canonical ensemble. We demonstrate that the resulting entropy satisfies all four fundamental Rényi entropy inequalities throughout the admissible range of $μ_C$, establishing its consistency as a genuine Rényi measure. Physically, this novel measure extends conventional Rényi 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 Rényi 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.

DOI
10.48550/arxiv.2606.22849

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In this paper, we study the holographic R\'enyi entropy of a large interval on a circle at high temperature for the two-dimensional conformal field theory (CFT) dual to pure AdS3{\mathrm{AdS}}_{3}AdS3​ gravity. In the field theory, the R\'enyi entropy is encoded in the CFT partition function on nnn-sheeted torus connected with each other by a large branch cut. As proposed by Chen and Wu [Large interval limit of R\'enyi entropy at high temperature, arXiv:1412.0763], the effective way to read the entropy in the large interval limit is to insert a complete set of state bases of the twist sector at the branch cut. Then the calculation transforms into an expansion of four-point functions in the twist sector with respect to ${e}^{\ensuremath{-}\frac{2\ensuremath{\pi}TR}{n}}$. By using the operator product expansion of the twist operators at the branch points, we read the first few terms of the R\'enyi entropy, including the leading and next-to-leading contributions in the large central charge limit. Moreover, we show that the leading contribution is actually captured by the twist vacuum module. In this case by the Ward identity the four-point functions can be derived from the correlation function of four twist operators, which is related to double interval entanglement entropy. Holographically, we apply the recipe in [T. Faulkner, The entanglement R\'enyi entropies of disjoint intervals in AdS/CFT, arXiv:1303.7221] and [T. Barrella et al., Holographic entanglement beyond classical gravity, J. High Energy Phys. 09 (2013) 109] to compute the classical R\'enyi entropy and its one-loop quantum correction, after imposing a new set of monodromy conditions. The holographic classical result matches exactly with the leading contribution in the field theory up to ${e}^{\ensuremath{-}4\ensuremath{\pi}TR}$ and l6{l}^{6}l6, while the holographical one-loop contribution is in exact agreement with next-to-leading results in field theory up to ${e}^{\ensuremath{-}\frac{6\ensuremath{\pi}TR}{n}}$ and l4{l}^{4}l4 as well.
We describe a holographic approach to explicitly computing the universal logarithmic contributions to entanglement and Rényi entropies for free conformal scalar and spinor fields on even-dimensional spheres. This holographic derivation proceeds in two steps: first, following Casini and Huerta, a conformal mapping to thermal entropy in a hyperbolic geometry; then identification of the hyperbolic geometry with the conformal boundary of a bulk hyperbolic space and use of an AdS/CFT holographic formula to compute the resultant functional determinant. We explicitly verify the connection with the type-A trace anomaly for the entanglement entropy, whereas the Rényi entropy is computed with the aid of the Sommerfeld formula in order to deal with a conical defect. We show that as a by-product, the log coefficient of the Rényi entropy for round spheres can be efficiently obtained as the q-analog of a procedure similar to the one found by Cappelli and D’Appollonio that rendered the type-A trace anomaly.
Two-dimensional conformal field theories with a large central charge and a small number of low-dimension operators are studied using the conformal block expansion. A universal formula is derived for the Renyi entropies of N disjoint intervals in the ground state, valid to all orders in a series expansion. This is possible because the full perturbative answer in this regime comes from the exchange of the stress tensor and other descendants of the vacuum state. Therefore, the Renyi entropy is related to the Virasoro vacuum block at large central charge. The entanglement entropy, computed from the Renyi entropy by an analytic continuation, decouples into a sum of single-interval entanglements. This field theory result agrees with the Ryu-Takayanagi formula for the holographic entanglement entropy of a 2d CFT, applied to any number of intervals, and thus can be interpreted as a microscopic calculation of the area of minimal surfaces in 3d gravity.
In this paper, we calculate the Rényi entropy of one single interval on a circle at finite temperature in 2D CFT. In the low temperature limit, we expand the thermal density matrix level by level in the vacuum Verma module, and calculate the first few leading terms in e −π/T L explicitly. On the other hand, we compute the same Rényi entropy holographically. After considering the dependence of the Rényi entropy on the temperature, we manage to fix the interval-independent constant terms in the classical part of holographic Rényi entropy. We furthermore extend the analysis in [9] to higher orders and find exact agreement between the results from field theory and bulk computations in the large central charge limit. Our work provides another piece of evidence to support holographic computation of Rényi entropy in AdS3/CFT2 correspondence, even with thermal effect.