Note on ETH of descendant states in 2D CFT

TL;DR

This paper examines the eigenstate thermalization hypothesis (ETH) for descendant states in 2D large central charge conformal field theories, deriving constraints on operator expectation values and testing ETH validity for various excited states.

Contribution

It introduces a method to analyze ETH for descendant states in 2D CFTs using operator product expansion and entanglement entropy calculations, revealing limitations of ETH for heavily excited states.

Findings

ETH constraints are satisfied only for leading order descendant states.

Heavy descendant states generally violate ETH constraints.

Constraints derived from Korteweg-de Vries charges support the ETH analysis.

Abstract

We investigate the eigenstate thermalization hypothesis (ETH) of highly excited descendant states in two-dimensional large central charge $c$ conformal field theory. We use operator product expansion of twist operators to calculate the short interval expansions of entanglement entropy and relative entropy for an interval of length $\ell$ up to order $\ell^{12}$. Using these results to ensure ETH of a heavy state when compared with the canonical ensemble state up to various orders of $c$, we get the constraints on the expectation values of the first few quasiprimary operators in the vacuum conformal family at the corresponding order of $c$. Similarly, we also obtain the constraints from the expectation values of the first few Korteweg-de Vries charges. We check these constraints for some types of special descendant excited states. Among the descendant states we consider, we find that at most only the leading order ones of the ETH constraints can be satisfied for the descendant states that are slightly excited on top of a heavy primary state. Otherwise, the ETH constraints are violated for the descendant states that are heavily excited on top of a primary state.