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Note on ETH of descendant states in 2D CFT

Wu-zhong Guo, Feng-Li Lin, Jiaju Zhang

TL;DR

This work probes eigenstate thermalization in 2D large-$c$ CFTs by comparing highly excited descendant states to the canonical ensemble using short-interval entanglement entropy and relative entropy via twist-operator OPE, deriving constraints on the first vacuum quasiprimary operators and KdV charges. It provides universal all-orders-in-$c$ relations and fixed-order expansions in $c$ for these observables, highlighting nontrivial distinctions between EE, relative entropy, and KdV constraints. Checks on three descendant classes show that only descendant states slightly excited atop a heavy primary can satisfy leading-order ETH, while heavily excited descendants generally violate ETH, posing questions about the holographic interpretation and the role of GGE. The results suggest that thermality in high-energy 2D CFTs is selective among states and has potential implications for black hole microstate descriptions in AdS$_3$/CFT.

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.

Note on ETH of descendant states in 2D CFT

TL;DR

This work probes eigenstate thermalization in 2D large- CFTs by comparing highly excited descendant states to the canonical ensemble using short-interval entanglement entropy and relative entropy via twist-operator OPE, deriving constraints on the first vacuum quasiprimary operators and KdV charges. It provides universal all-orders-in- relations and fixed-order expansions in for these observables, highlighting nontrivial distinctions between EE, relative entropy, and KdV constraints. Checks on three descendant classes show that only descendant states slightly excited atop a heavy primary can satisfy leading-order ETH, while heavily excited descendants generally violate ETH, posing questions about the holographic interpretation and the role of GGE. The results suggest that thermality in high-energy 2D CFTs is selective among states and has potential implications for black hole microstate descriptions in AdS/CFT.

Abstract

We investigate the eigenstate thermalization hypothesis (ETH) of highly excited descendant states in two-dimensional large central charge 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 up to order . Using these results to ensure ETH of a heavy state when compared with the canonical ensemble state up to various orders of , we get the constraints on the expectation values of the first few quasiprimary operators in the vacuum conformal family at the corresponding order of . 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.

Paper Structure

This paper contains 14 sections, 110 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Constraints of expectation values from ETH
  3. Constraints for all orders of large $c$
  4. Constraints for fixed orders of large $c$
  5. Checks for primary and various descendant eigenstates
  6. Conclusion and discussion
  7. Quasiprimary operators and their correlation functions
  8. Vacuum conformal family
  9. Non-vacuum conformal family
  10. Short interval expansions of EE and relative entropy
  11. OPE of twist operators
  12. EE
  13. Relative entropy
  14. Proof of $a_\mathcal{X}=0$ for $h_\mathcal{X}>2$

Figures (1)

  • Figure 1: Relations of the three sets of constraints from the EE (\ref{['genconEE']}), relative entropy (\ref{['genconRE']}) and expectation values of KdV charges (\ref{['genconKdV']}).