Deriving the Eigenstate Thermalization Hypothesis from Eigenstate Typicality and Kinematic Principles
Yucheng Wang
TL;DR
The paper derives the ETH from minimal premises by combining a kinematic entropy-based framework with the eigenstate typicality principle (ETP). It shows that canonical typicality (concentration of measure on a high-dimensional microcanonical shell) implies diagonal ETH, while the sparse, local nature of observables together with ETP fixes off-diagonal elements as entropy-suppressed and statistically Haar-like, yielding A_{mn} = e^{-S(E)/2} f(E,ω) R_{mn}. The smooth envelopes 𝒜(E) and f(E,ω) follow from thermodynamic continuity and finite-time decay of correlations, respectively, without relying on random-matrix statistics. Altogether, ETH emerges from entropy, Hilbert-space geometry, and chaos-driven eigenstate typicality, clarifying its regime of validity and deepening the link between quantum chaos and thermalization.
Abstract
The eigenstate thermalization hypothesis (ETH) provides a powerful framework for understanding thermalization in isolated quantum many-body systems, yet a complete and conceptually transparent derivation has remained elusive. In this work, we derive the structure of ETH from a minimal dynamical principle, which we term the eigenstate typicality principle (ETP), together with general kinematic ingredients arising from entropy maximization, Hilbert-space geometry, and locality. ETP asserts that in quantum-chaotic systems, energy eigenstates are statistically indistinguishable, with respect to local measurements, from states drawn from the Haar measure on a narrow microcanonical shell. Within this framework, diagonal ETH arises from concentration of measure, provided that eigenstate typicality holds. The structure of off-diagonal matrix elements is then fixed by entropic scaling and the finite-time dynamical correlations of local observables, with ETP serving as the dynamical bridge to energy eigenstates, without invoking random-matrix assumptions. Our results establish ETH as a consequence of entropy, Hilbert-space geometry, and chaos-induced eigenstate typicality, and clarify its regime of validity across generic quantum-chaotic many-body systems, thereby deepening our understanding of quantum thermalization and the emergence of statistical mechanics from unitary many-body dynamics.