A Multi-Resolvent Hierarchy for the ETH Smooth Function

Abstract

The eigenstate thermalization hypothesis (ETH) parametrizes off-diagonal matrix elements by a smooth function whose microscopic origin remains largely phenomenological. We develop a multi-resolvent hierarchy that derives this smooth structure from the microscopic Hamiltonian. Starting from exact projection identities, we express the ETH variance as $f_{ji}^2=D_{ji}+g_{ji}$, where $D_{ji}$ is the diagonal-overlap baseline and $g_{ji}={\sum }_{r\ge 2}{g}_{ji}^{(r)}$ is a systematically improvable hierarchy of multi-channel interference processes. The leading $r=3$ sector generates an odd-parity component in the energy difference, inaccessible to parity-preserving single-resolvent closures. The same resolvent construction yields an exact covariance representation of eigenstate fluctuations. Under amplitude isotropy, decorrelation, and regularity, it reduces to the Gaussian limit, with $q-3={\kappa }_4/⟨c^2{⟩}^2$; normalization further fixes the cross-channel covariance ${\bar{C}}_i$, including its energy-resolved form. Exact diagonalization verifies these relations in random-matrix and structured systems, while the latter exhibit controlled breakdown of the isotropic Gaussian closure. The framework thus provides a microscopic hierarchy for both the smooth and fluctuation sectors of subsystem ETH.

36 pages