Eigenstate Thermalization Hypothesis

Random matrix elements of observables in chaotic vs integrable systems

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⟨m|Ô|n⟩ = O(Ē)δₘₙ + e^{-S(Ē)/2} f(Ē,ω) Rₘₙ  |  ETH: matrix elements smooth in energy
ETH (Deutsch 1991, Srednicki 1994) explains thermalization in isolated quantum systems. Chaotic systems (GOE random matrices) have smooth, nearly diagonal observable matrix elements. Integrable systems have erratic, non-smooth elements — they fail to thermalize. Left: |⟨m|Ô|n⟩|² as a heatmap. Right: diagonal elements vs energy eigenvalue index.