Refinements of the Eigenstate Thermalization Hypothesis under Local Rotational Invariance via Free Probability

TL;DR

This paper refines the Eigenstate Thermalization Hypothesis (ETH) by incorporating local rotational invariance using free probability, providing analytical predictions for matrix element correlations and validating them with numerical simulations.

Contribution

It introduces a refined ETH framework accounting for local rotational invariance and subleading corrections using free probability techniques.

Findings

Analytical characterization of subleading corrections to ETH.

Quantitative predictions validated by numerical simulations.

Linking statistical properties of matrix elements to empirical energy window averages.

Abstract

The Eigenstate Thermalization Hypothesis (ETH) was developed as a framework for understanding how the principles of statistical mechanics emerge in the long-time limit of isolated quantum many-body systems. Since then, ETH has shifted the attention towards the study of matrix elements of physical observables in the energy eigenbasis. In this work, we revisit recent developments leading to the formulation of full ETH, a generalization of the original ETH ansatz that accounts for multi-point correlation functions. Using tools from free probability, we explore the implications of local rotational invariance, a property that emerges from the statistical invariance of observables under random basis transformations induced by small perturbations of the Hamiltonian. This approach allows us to make quantitative predictions and derive an analytical characterization of subleading corrections to matrix-element correlations, thereby refining the ETH ansatz. Moreover, our analysis links the statistical properties of matrix elements under random basis changes to the empirical averages over energy windows that are usually considered when dealing with a single instance of the ensemble. We validate our analytical predictions through comparison with numerical simulations in non-integrable Floquet systems.