2026/07/22 by Zhiqiang Huang
#quant-ph
The eigenstate thermalization hypothesis (ETH) provides a statistical description of thermalization in isolated quantum many-body systems, yet the phenomenological smooth function fO(E,ω) -- which controls the energy dependence of off-diagonal matrix elements -- lacks a systematic microscopic foundation. We develop a multi-resolvent hierarchy for the correlation corrections entering the ETH smooth function. Using recursive projection identities together with a diagonal closure approximation (DCA), the hierarchy organizes multi-channel interference processes by the number of interacting bath channels, replacing the uncontrolled neglect of higher-order correlations with a systematically improvable expansion. The ETH smooth function is thereby obtained as fji2 = Dji + ∑r≥ 2 gji(r), where the diagonal baseline Dji and each correlation level gji(r) are expressed entirely through diagonal spectral functions and microscopic interaction couplings, providing a unified, closed, and systematically improvable microscopic theory. A rigorous projector sum rule constrains the entire hierarchy: the integrated off-diagonal correlation carries a negative bias of order unity, a consequence of projector idempotency. The hierarchy further reveals a parity structure in which even-r sectors carry even parity under ω→-ω while the r=3 sector generates the first odd-parity (skewness) contribution -- absent from all single-resolvent closures -- suggesting experimentally testable signatures in quantum many-body systems.