Thermality and excited state Rényi entropy in two-dimensional CFT

TL;DR

This work computes the one-interval Rényi and entanglement entropy for excited states of a 2D CFT on a cylinder using twist-operator OPE blocks, focusing on short-interval expansions. The authors show that entanglement entropy can reproduce thermal behavior for highly excited (heavy) states after identifying h_φ with cε_φ and β with L/√(24ε_φ−1), aligning with ETH expectations, while Rényi entropy does not admit a universal temperature identification and remains sensitive to the full entanglement spectrum. They further analyze contributions from non-vacuum OPE blocks and descendants of the vacuum family, finding nonuniversal and potentially non-thermality-indicating effects that challenge the universality of ETH beyond the entanglement entropy. The results imply thermality as captured by average entanglement can hide richer, nonthermal structure revealed by higher moments, thereby highlighting caveats in extending ETH and typicality notions to Rényi entropy in holographic CFTs.

Abstract

We evaluate one-interval Rényi entropy and entanglement entropy for the excited states of two-dimensional conformal field theory (CFT) on a cylinder, and examine their differences from the ones for the thermal state. We assume the interval to be short so that we can use operator product expansion (OPE) of twist operators to calculate Rényi entropy in terms of sum of one-point functions of OPE blocks. We find that the entanglement entropy for highly excited state and thermal state behave the same way after appropriate identification of the conformal weight of the state with the temperature. However, there exists no such universal identification for the Rényi entropy in the short-interval expansion. Therefore, the highly excited state does not look thermal when comparing its Rényi entropy to the thermal state one. As the Rényi entropy captures the higher moments of the reduced density matrix but the entanglement entropy only the average, our results imply that the emergence of thermality depends on how refined we look into the entanglement structure of the underlying pure excited state.

Figures (2)

• Figure 1: The CFT setup for our calculation of the one-interval excited state Rényi entropy and entanglement entropy on the interval $A=[0,\ell]$. The CFT is defined on a cylinder with spatial size $L$ in the excited state $|\phi\rangle$, which is created by acting on the vacuum state with an operator $\phi$.
• Figure 2: When we tune the parameter $\epsilon_\phi$ from $\epsilon_\phi<1/24$ to $\epsilon_\phi>1/24$, the entanglement entropy of the short interval on cylinder in excited state $|\phi\rangle$ with $h_\phi=c h_\phi$ behaves as the entanglement entropies of the same interval length on different manifolds. When $\epsilon_\phi<1/24$ there are cylinders with periodic boundary condition in the spatial direction, and the cylinder becomes 'fatter' when $\epsilon_\phi$ becomes larger. When $\epsilon_\phi=0$, it is the complex plane. When $\epsilon_\phi>1/24$ there are cylinders with periodic boundary condition in the temporal direction, and the cylinder becomes 'thinner' when $\epsilon_\phi$ becomes larger.