Eigenstate Thermalisation Hypothesis for Translation Invariant Spin Systems

Abstract: We prove the Eigenstate Thermalisation Hypothesis (ETH) for local observables in a typical translation invariant system of quantum spins with mean field interaction. This mathematically verifies the observation made in [L.Santos and M.Rigol, Phys.Rev.E 82, 031130 (2010, https://journals.aps.org/pre/abstract/10.1103/PhysRevE.82.031130)] that ETH may hold for systems with additional translation symmetries for a naturally restricted class of observables. We also present numerical support for the same phenomenon for Hamiltonians with local interactions.