Eigenstate Typicality as the Dynamical Bridge to the Eigenstate Thermalization Hypothesis: A Derivation from Entropy, Geometry, and Locality

Abstract: The eigenstate thermalization hypothesis (ETH) provides a powerful framework for understanding thermalization in isolated quantum many-body systems, yet its physical foundations and minimal underlying assumptions remain actively debated. In this work, we develop a unified framework that clarifies the origin of ETH by separating kinematic typicality from dynamical input. We show that the characteristic ETH structure of local operator matrix elements follows from four ingredients: the maximum entropy principle, the geometry of high-dimensional Hilbert space, the locality of physical observables, and a minimal dynamical principle, which we term the eigenstate typicality principle (ETP). ETP asserts that in quantum-chaotic systems, energy eigenstates are statistically indistinguishable from typical states within a narrow microcanonical shell with respect to local measurements. Within this framework, diagonal ETH emerges from measure concentration, while the universal exponential suppression and smooth energy-frequency dependence of off-diagonal matrix elements arise from entropic scaling and local dynamical correlations, without invoking random-matrix assumptions. Our results establish ETH as a consequence of entropy, geometry, and chaos-induced typicality, and clarify its scope, thereby deepening our understanding of quantum thermalization and the emergence of statistical mechanics from unitary many-body dynamics.