Strong Eigenstate Thermalization within a Generalized Shell in Noninteracting Integrable Systems

Integrable systems do not obey the strong eigenstate thermalization hypothesis (ETH), which has been proposed as a mechanism of thermalization in isolated quantum systems. It has been suggested that an integrable system reaches a steady state described by a generalized Gibbs ensemble (GGE) instead of thermal equilibrium. We prove that a generalized version of the strong ETH holds for noninteracting integrable systems with translation invariance. Our generalized ETH states that any pair of energy eigenstates with similar values of local conserved quantities looks similar with respect to local observables, such as local correlations. This result tells us that an integrable system relaxes to a GGE for any initial state that has subextensive fluctuations of macroscopic local conserved quantities. Contrary to the previous derivations of the GGE, it is not necessary to assume the cluster decomposition property for an initial state.