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 fji2=Dji+gji, where Dji is the diagonal-overlap baseline and gji=∑r≥2gji(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=κ4/⟨ c2⟩2; normalization further fixes the cross-channel covariance C̅i, 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.
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BibTeX
@misc{huang2026multiresolvent,
title = {{A Multi-Resolvent Hierarchy for the ETH Smooth Function}},
author = {Zhiqiang Huang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.19861},
eprint = {2607.19861},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.19861},
}
RIS
TY - UNPB
TI - A Multi-Resolvent Hierarchy for the ETH Smooth Function
AU - Zhiqiang Huang
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2607.19861
UR - https://arxiv.org/abs/2607.19861
AB - 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=∑_r≥2g_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=κ_4/⟨ c^2⟩^2; normalization further fixes the cross-channel covariance C̅_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.
ER -