Full Eigenstate Thermalization via Free Cumulants in Quantum Lattice Systems

The Eigenstate-Thermalization-Hypothesis (ETH) has been established as the general framework to understand quantum statistical mechanics. Only recently has the attention been paid to so-called full ETH, which accounts for higher-order correlations among matrix elements, and that can be rationalized theoretically using the language of Free Probability. In this work, we perform the first numerical investigation of the full ETH in physical many-body systems with local interactions by testing the decomposition of higher-order correlators into thermal free cumulants for local operators. We perform exact diagonalization on two classes of local non-integrable (chaotic) quantum many-body systems: spin chain Hamiltonians and Floquet brickwork unitary circuits. We show that the dynamics of four-time correlation functions are encoded in fourth-order free cumulants, as predicted by ETH. Their dependence on frequency encodes the physical properties of local many-body systems and distinguishes them from structureless, rotationally invariant ensembles of random matrices.