Eigenstate Thermalization Hypothesis in Conformal Field Theory

TL;DR

The paper proves that in conformal field theories, a weak form of ETH for local primary operators (local ETH) automatically enforces a strong form for finite subregions (subsystem ETH) by bounding the trace distance between the eigenstate’s reduced density matrix and a universal $\rho_B(E)$. It uses conformal mapping, OPE data, and a relative-entropy framework in the Rindler frame to derive exponential suppression $\eta_a=e^{-O(S(E))}$ of off-diagonal and perturbative corrections. In 1+1D CFTs, the local ETH implies that heavy primary states’ finite-size density matrices are captured by the Virasoro identity block, enabling explicit finite-$R/\lambda_T$ expansions for entanglement quantities and revealing a finite discrepancy in Renyi entropies despite entanglement entropy matching thermal results. The work suggests chaotic CFTs satisfy local ETH while integrable ones do not, and it outlines a program to connect ETH with quantum chaos diagnostics and bootstrap techniques in QFT.

Abstract

We investigate the eigenstate thermalization hypothesis (ETH) in d+1 dimensional conformal field theories by studying reduced density matrices in energy eigenstates. We show that if local probes of high energy primary eigenstates satisfy ETH, then any finite energy observable with support on a subsystem of finite size satisfies ETH. In two dimensions, we discover that if ETH holds locally, the finite size reduced density matrix of states created by heavy primary operators is well-approximated by a projection to the Virasoro identity block.

Figures (3)

• Figure 1: (a) The cylinder $S^d\times R_t$ conformal frame (b) Radial quantization $\mathbb{R}^{d+1}$ conformal frame. (c) The conformal frame convenient for the study of the density matrix on subsystem $B$.
• Figure 2: (a) and (b) The Rindler frame path-integrals that prepare, respectively, $\tilde{\rho}_B^a$ and $\tilde{\rho}_B(E)$ (c) The path-integral for $\rho_B(E)$ in the radial quantization frame.
• Figure 3: The relative entropy replica trick.