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Rigorous proof of the Eigenstate Thermalization Hypothesis (ETH) ansatz

Establish a rigorous proof of the Eigenstate Thermalization Hypothesis (ETH) ansatz for few-body observables in quantum chaotic many-body systems: prove that, in the eigenbasis of the Hamiltonian H with eigenvalues E_j and eigenstates |E_j>, the observable matrix elements satisfy O_ij = <E_i|O|E_j> = O(E_i) δ_ij + f(E_i, E_j) r_ij, where O(E_i) and f(E_i, E_j) are smooth functions and r_ij are random variables with zero mean and unit variance.

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Background

The paper recalls the Eigenstate Thermalization Hypothesis (ETH) ansatz, which posits a universal structure for matrix elements of few-body observables in the energy eigenbasis, separating smooth envelopes from random fluctuations. This structure underlies thermalization in isolated quantum chaotic systems and connects observable statistics to Random Matrix Theory.

While the ETH ansatz is widely supported numerically, the text explicitly notes the lack of a rigorous proof, indicating that the ansatz remains a conjectural assumption. The authors’ paper focuses on deviations from Random Matrix Theory via the envelope function f(E_i, E_j), but the foundational question of a rigorous derivation of the ETH ansatz itself is left unresolved.

References

The ETH ansatz conjectures that \begin{equation}\label{ETH} O_{ij}=\bra{E_i}O\ket{E_j}=O(E_i)\delta_{ij}+f(E_i,E_j)r_{ij}, \end{equation} where $E_j$ and $\ket{j}$ denote the eigenvalue and eigenstate of $H$, respectively. $O(E_i)$ and $f(E_i,E_j)$ are smooth functions of their arguments, $\delta_{ij}$ is the Kronecker Delta function, and $r_{ij}=r*_{ji}$ are random variables with a normal distribution (zero mean and unit variance). Although ETH remains a hypothesis due to the lack of rigorous proof, most aspects of the ETH have been validated through numerical simulations .

ETH:

Oij=⟨Ei∣O∣Ej⟩=O(Ei)δij+f(Ei,Ej)rij,O_{ij}=\bra{E_i}O\ket{E_j}=O(E_i)\delta_{ij}+f(E_i,E_j)r_{ij},

— Observable-manifested correlations in many-body quantum chaotic systems (2502.16885 - Wang et al., 24 Feb 2025) in Section 1 (Introduction), around Eq. (ETH)