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arXiv:2505.02897 · Communications in Mathematical Physics

Universality of the Microcanonical Entropy at Large Spin

Sridip Pal, Jiaxin Qiao, Balt C. van Rees

Published Oct 30, 2025• 1 citation
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Abstract

Abstract We consider rigorous consequences of modular invariance for two-dimensional unitary non-rational CFTs with $$c > 1$$ c > 1 . Simple estimates for the torus partition function can lead to remarkably strong results. We show in particular that the spectral density of spin- J operators must grow like $$\exp \left( \pi \sqrt{\frac{2}{3}(c-1) J} \right) /\sqrt{2J}$$ exp π 2 3 ( c - 1 ) J / 2 J in any twist interval at or above $$(c-1)/12$$ ( c - 1 ) / 12 , with a known twist-dependent prefactor. This proves that the large J spectrum becomes dense even without averaging over spins. For twists below $$(c-1)/12$$ ( c - 1 ) / 12 we establish that the growth must be strictly slower. Finally, we estimate how fast the maximal gap between two spin- J operators must go to zero as J becomes large.

DOI
10.1007/s00220-025-05442-y

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A bstract We prove the conjecture proposed by Hartman, Keller and Stoica (HKS) [1]: the grand-canonical free energy of a unitary 2D CFT with a sparse spectrum below the scaling dimension c12 \frac{c}{12} 12c​ c 12 + ϵ and below the twist c12 \frac{c}{12} 12c​ c 12 is universal in the large c limit for all β L β R ≠ 4 π 2 . The technique of the proof allows us to derive a one-parameter (with parameter α ∈ (0, 1]) family of universal inequalities on the unitary 2D CFT partition function with general central charge c ⩾ 0, using analytical modular bootstrap. We derive an iterative equation for the domain of validity of the inequality on the ( β L , β R ) plane. The infinite iteration of this equation gives the boundary of maximal-validity domain, which depends on the parameter α in the inequality. In the c → ∞ limit, with the additional assumption of a sparse spectrum below the scaling dimension c12 \frac{c}{12} 12c​ c 12 + ϵ and the twist αc12 \frac{\alpha c}{12} 12αc​ αc 12 (with α ∈ (0, 1] fixed), our inequality shows that the grand-canonical free energy exhibits a universal large c behavior in the maximal-validity domain. This domain, however, does not cover the entire ( β L , β R ) plane, except in the case of α = 1. For α = 1, this proves the conjecture proposed by HKS [1], and for α < 1, it quantifies how sparseness in twist affects the regime of universality. Furthermore, this implies a precise lower bound on the temperature of near-extremal BTZ black holes, above which we can trust the black hole thermodynamics.
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