The Eigenstate Thermalization Hypothesis and Out of Time Order Correlators

TL;DR

This paper shows that the traditional ETH, which treats off-diagonal matrix elements as independent random variables, is incomplete for chaotic quantum systems. By introducing higher-order correlations through functions ${\bf F}^{(n)}_{e_+}$ and entropy weights $e^{-(n-1)S(e_+)/2}$, and employing a typicality-based energy-window framework, the authors connect multi-point matrix-element statistics to n-time correlation functions and OTOCs. They derive a structured, diagrammatic description where all-indices-different contributions drive the Lyapunov growth, while lower-point correlators encode remaining dynamics, and provide numerical evidence in a nonintegrable 1D model. The findings imply that capturing chaos signatures like the Lyapunov exponent and dynamic heterogeneity requires extending ETH to include non-Gaussian, joint distributions of matrix elements, with potential implications for understanding quantum chaotic dynamics and thermalization.

Abstract

The Eigenstate Thermalization Hypothesis (ETH) implies a form for the matrix elements of local operators between eigenstates of the Hamiltonian, expected to be valid for chaotic systems. Another signal of chaos is a positive Lyapunov exponent, defined on the basis of Loschmidt echo or out-of-time-order correlators. For this exponent to be positive, correlations between matrix elements unrelated by symmetry, usually neglected, have to exist. The same is true for the peak of the dynamic heterogeneity length, relevant for systems with slow dynamics. These correlations, as well as those between elements of different operators, are encompassed in a generalized form of ETH.

Figures (9)

• Figure 1: A rotation group with three independent $\Delta*\Delta$ unitary transformation matrices
• Figure 2: A dotted line in the diagram on the left column means that the indices are in the same energy interval, in a diagram on the right column that they are in the same interval but are not equal. In the diagrams of the right columns a full line stands for a Kronecker delta, imposing the equality of indices. A double line imposes the same equality twice (e.g. $\delta_{ij} \delta_{ji}$), while a blued dot a delta with equal indices (e.g. $\delta_{ii}$): they are indicated only for counting purposes, to show that there are three Kronecker symbols in each diagram.
• Figure 3: The final situation: products as in diagram (a), diagrams (c,d,e) and diagram (b) have different scaling functions.
• Figure 4: Products corresponding to three-point functions (an alternative representation to Fig. \ref{['diag3']}) : a loop and two cacti.
• Figure 5: Products corresponding to four-point functions.