Deriving the Eigenstate Thermalization Hypothesis from Eigenstate Typicality and Kinematic Principles

TL;DR

This paper derives the eigenstate thermalization hypothesis (ETH) from fundamental principles like eigenstate typicality and entropy maximization, clarifying its validity in quantum-chaotic systems without relying on randomness assumptions.

Contribution

It provides a conceptual derivation of ETH from minimal dynamical and kinematic principles, linking eigenstate typicality to quantum thermalization.

Findings

ETH follows from eigenstate typicality and entropy principles.

Diagonal ETH results from measure concentration in chaotic systems.

Off-diagonal ETH structure is determined by entropic scaling and local correlations.

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.