Eigenstate thermalization hypothesis

Statement

Let $H$ be the Hamiltonian of an isolated quantum many-body system of size $N$ that is non-integrable (chaotic), with eigenvalues $E_n$ and orthonormal eigenvectors $\lvert E_n \rangle$. Let $A$ be a local, or few-body, self-adjoint operator. The eigenstate thermalization hypothesis concerns eigenstates in the bulk of the spectrum. For them it asserts that $$\langle E_m \rvert A \lvert E_n \rangle = \mathcal{A}(\bar E)\, \delta_{mn} + e^{-S(\bar E)/2} f_A(\bar E, \omega)\, R_{mn},$$ where $\bar E = (E_m + E_n)/2$ and $\omega = E_n - E_m$. Here $\mathcal{A}$ and $f_A$ are smooth functions, and $S(E)$ is the thermodynamic entropy at energy $E$. The numbers $R_{mn}$ behave like random variables of mean $0$ and variance $1$. Moreover, $\mathcal{A}(E)$ equals the microcanonical ensemble average of $A$ at energy $E$, up to corrections that vanish as $N \to \infty$.