# Universality of the Microcanonical Entropy at Large Spin

## TL;DR

This paper explores how entropy behaves in certain 2D quantum field theories at high spin values, showing a universal growth pattern in the number of operators.

## Contribution

The paper proves a universal exponential growth of the spectral density of spin-J operators in non-rational CFTs with $c > 1$.

## Key findings

- The spectral density of spin-J operators grows exponentially with a specific formula at or above a critical twist value.
- The growth is strictly slower for twists below the critical threshold of $(c-1)/12$.
- The maximal gap between spin-J operators decreases as spin increases.

## Abstract

We consider rigorous consequences of modular invariance for two-dimensional unitary non-rational CFTs with \documentclass[12pt]{minimal}
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\begin{document}$$c > 1$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$\exp \left( \pi \sqrt{\frac{2}{3}(c-1) J} \right) /\sqrt{2J}$$\end{document}expπ23(c-1)J/2J in any twist interval at or above \documentclass[12pt]{minimal}
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\begin{document}$$(c-1)/12$$\end{document}(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 \documentclass[12pt]{minimal}
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\begin{document}$$(c-1)/12$$\end{document}(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.