Eigenstate Thermalization Hypothesis correlations via non-linear Hydrodynamics

TL;DR

This paper predicts a universal hydrodynamic scaling for correlations in many-body quantum systems described by ETH, supported by numerical simulations of spin models.

Contribution

It introduces a hydrodynamic approach to determine the form of smooth functions in ETH, linking them to universal low-frequency behavior.

Findings

Universal scaling of late-time free cumulants predicted by non-linear hydrodynamics.

Numerical simulations confirm the hydrodynamic predictions in spin models.

Good agreement observed across different temperatures and local observables.

Abstract

The thermalizing dynamics of many-body systems is often described through the lens of the Eigenstate Thermalization Hypothesis (ETH). ETH postulates that the statistical properties of observables, when expressed in the energy eigenbasis, are described by smooth functions, that also describe correlations among the matrix elements. However, the form of these functions is usually left undetermined, constituting a key missing component of the ETH framework. In this work, we investigate the structure of such smooth functions by focusing on their Fourier transform, recently identified as free cumulants. Using non-linear hydrodynamics, we provide a prediction for the universal scaling of the late-time behavior of time-ordered free cumulants in the thermodynamic limit. The prediction is further corroborated by large-scale numerical simulations of several non-integrable one-dimensional spin models which exhibit diffusive transport behavior. Good agreement is observed in both infinite and finite-temperature regimes and for a collection of local observables. Our results indicate that the smooth multi-point correlation functions within the ETH framework admit a universal hydrodynamic description at low frequencies.