# On thermality of CFT eigenstates

## TL;DR

This paper investigates the conditions under which eigenstates in (1+1)-dimensional conformal field theories approximate thermal states, revealing universal and non-universal deviations influenced by central charge and structure constants.

## Contribution

It identifies a class of operators satisfying ETH at leading order in large central charge and quantifies deviations in reduced density matrices in terms of subsystem size and structure constants.

## Key findings

- Universal deviation scales as the eighth power of subsystem fraction.
- Deviations are suppressed by inverse powers of the central charge.
- Non-universal deviations relate to heavy-light-heavy structure constants.

## Abstract

The Eigenstate Thermalization Hypothesis (ETH) provides a way to understand how an isolated quantum mechanical system can be approximated by a thermal density matrix. We find a class of operators in (1+1)-$d$ conformal field theories, consisting of quasi-primaries of the identity module, which satisfy the hypothesis only at the leading order in large central charge. In the context of subsystem ETH, this plays a role in the deviation of the reduced density matrices, corresponding to a finite energy density eigenstate from its hypothesized thermal approximation. The universal deviation in terms of the square of the trace-square distance goes as the 8th power of the subsystem fraction and is suppressed by powers of inverse central charge ($c$). Furthermore, the non-universal deviations from subsystem ETH are found to be proportional to the heavy-light-heavy structure constants which are typically exponentially suppressed in $\sqrt{h/c}$, where $h$ is the conformal scaling dimension of the finite energy density state. We also examine the effects of the leading finite size corrections.