Universality of Quantum Information in Chaotic CFTs

TL;DR

We study the Eigenstate Thermalization Hypothesis (ETH) in chaotic conformal field theories across dimensions, focusing on the reduced density matrix of a ball-shaped subsystem in highly excited eigenstates. The local ETH framework yields the ETH density matrix $\psi_{\rm ETH}(B,E)$, which approximates the thermal reduced state with a trace-distance bound $\|\psi_{\rm ETH}(B,E)-\rho_{\rm th}(B,E)\| \sim e^{-O(S(E))}$ and reduces to a standard thermal ensemble in the appropriate limits. In two dimensions, the ETH density matrix is universal for fixed central charge $c$, formed from Virasoro identity-block descendants with coefficients fixed by $c$, and it matches thermal (Gibbs or large-$c$-GGE) expectations for local observables. For higher dimensions, entanglement entropies calculated from the ETH density matrix agree with holographic predictions via the Ryu-Takayanagi formula in black-hole backgrounds, supporting universality; the analysis extends to time-dependent coherent states and arbitrary initial states, establishing local equilibrium in slowly varying settings and clarifying the role of conserved charges in 2d via the Generalized Gibbs Ensemble.

Abstract

We study the Eigenstate Thermalization Hypothesis (ETH) in chaotic conformal field theories (CFTs) of arbitrary dimensions. Assuming local ETH, we compute the reduced density matrix of a ball-shaped subsystem of finite size in the infinite volume limit when the full system is an energy eigenstate. This reduced density matrix is close in trace distance to a density matrix, to which we refer as the ETH density matrix, that is independent of all the details of an eigenstate except its energy and charges under global symmetries. In two dimensions, the ETH density matrix is universal for all theories with the same value of central charge. We argue that the ETH density matrix is close in trace distance to the reduced density matrix of the (micro)canonical ensemble. We support the argument in higher dimensions by comparing the Von Neumann entropy of the ETH density matrix with the entropy of a black hole in holographic systems in the low temperature limit. Finally, we generalize our analysis to the coherent states with energy density that varies slowly in space, and show that locally such states are well described by the ETH density matrix.

Figures (4)

• Figure 1: (a) The cylinder $\mathbb{S}^d\times \mathbb{R}_t$ frame and the Euclidean path-integral that prepares the the density matrix in the eigenstate corresponding to $\Psi$ on subsystem $B$ (b) The same path-integral in the radial quantization $\mathbb{R}^{d+1}$ conformal frame (c) The path-integral for $\psi_{ETH}$ in the radial quantization frame.
• Figure 2: (a) The cylinder $\mathbb{S}^d\times \mathbb{R}_t$ conformal frame (b) The radial quantization $\mathbb{R}^{d+1}$ conformal frame. (c) The Rindler frame: the conformal frame convenient for the study of the density matrix on subsystem $B$.
• Figure 3: Renyi entropies correspond to $2n$-point function of the operator that creates the state.
• Figure 4: (a) The path-integral over complexified $\tau$ picks up $n$ poles at $\tau=2\pi j$. (b) The contour $C$ is deformed to run over $(-\infty+i(2\pi n-\epsilon),\infty+i(2\pi n-\epsilon))$ and $(\infty+i\epsilon,-\infty+i\epsilon)$ .