On thermality of CFT eigenstates

TL;DR

This work tests the Eigenstate Thermalization Hypothesis in 1+1D CFTs by analyzing a heavy energy eigenstate and its reduced density matrices for small subsystems. It develops a short-interval expansion of the trace-square distance between the eigenstate and the thermal state, separating universal contributions from the identity Virasoro module and non-universal contributions from non-vacuum primaries, and shows that universal deviations scale as $r^8$ with a $1/c$ suppression at leading order. In the thermodynamic limit and large central charge, identity-block quasi-primaries satisfy global ETH, while non-universal heavy-light contributions are exponentially suppressed in $\sqrt{h/c}$; finite-size corrections are even more suppressed, reinforcing the thermal-like behavior of finite-energy-density states. The results have holographic implications for dual BTZ black holes and suggest that ETH-like behavior emerges robustly in holographic CFTs, with integrability in RCFTs and finite-$c$ effects providing important caveats.

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.

Figures (2)

• Figure 1: Sewed Riemann surfaces corresponding to the path integral representation of the terms in the trace square distance, equation \ref{['all1']} and \ref{['glued-Riemann']}.
• Figure 2: Short interval expansion of $Z_{{\mathbb{R}\times \mathbb{S}^1_L},{\mathbb{T}^2}}$ : A small disk of radius of $\ell$ (corresponding to the subsystem ${\cal A}$) is cut, subsequently the sewing is performed by inserting a complete set of states i.e $|\mathcal{I}\rangle$