Universality of the microcanonical entropy at large spin

TL;DR

This paper demonstrates that in certain two-dimensional conformal field theories, the density of high-spin operators grows rapidly, becoming dense at large spins, with implications for the spectral structure and gaps in the spectrum.

Contribution

It provides rigorous bounds on the growth of spectral density and the behavior of gaps for high-spin operators in non-rational 2D CFTs, based on modular invariance.

Findings

Spectral density grows like rac{rac{ ext{pi} \u221a{2/3}(c-1)J}}{ ext{2J}} at large J.

Density becomes dense without averaging over spins at high J.

Growth rate of the spectral density is slower below a certain twist threshold.

Abstract

We consider rigorous consequences of modular invariance for two-dimensional unitary non-rational CFTs with $c > 1$. 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 $\exp\left( \pi \sqrt{\frac{2}{3}(c-1) J} \right)/\sqrt{2J}$ in any twist interval at or above $(c-1)/12$, with a known twist-dependent prefactor. This proves that the large $J$ spectrum becomes dense even without averaging over spins. For twists below $(c-1)/12$ 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.