Full Eigenstate Thermalization in Integrable Spin Systems

The Eigenstate Thermalization Hypothesis(ETH) is a standard tool to understand the thermalization properties of an isolated quantum system. Its generalization to higher order correlations of matrix elements of local operators, dubbed the full ETH, predicts the decomposition of higher-order correlation function into thermal free cumulants. In this work, we numerically test these predictions of full ETH using exact diagonalization of two spin models: the Ising and the XXZ Heisenberg models. The differences from the behavior of full ETH prediction in chaotic systems are highlighted and contrasted along the way. We also show that although in these integrable spin models the dynamics of the four-time correlators, specifically the out-of-time-ordered correlator (OTOC), is encoded in the fourth order free cumulant, it exhibits late-time dynamics that is different from nonintegrable systems.