Description
Despite the unitary evolution of closed quantum systems, long-time expectation of local observables are well described by thermal ensembles, providing the foundation of quantum statistical
mechanics. A promising route to understanding this quantum thermalization is the eigenstate thermalization hypothesis (ETH), which posits that individual energy eigenstates already appear locally
thermal. Subsequent studies have extended this concept to the full ETH, which captures higherorder correlations among matrix elements through nontrivial relations. In this work, we perform
a detailed exact-diagonalization study of finite-size corrections to these relations in the canonical
ensemble. We distinguish two distinct sources of corrections: those arising from energy fluctuations,
which decay polynomially with system size, and those originating from fluctuations within each
energy window, which decay exponentially with system size. In particular, our analysis resolves the
puzzle that, for certain observables, finite-size corrections exhibit anomalous growth with increasing
system size even in chaotic systems. Our results provide a systematic and practical methodology
for validating the full ETH in quantum many-body systems.
