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arXiv:2610.00261

Exact replica-sector hierarchy of multi-resolvent correlations in random free fermions

Zhiqiang Huang

quant-phSubmitted Sep 24, 2026• Announced Sep 25, 2026• 0 citations
v1 · Sep 24, 2026v2 · Oct 29, 2026
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Abstract

The spectral weight with which a many-body eigenstate contributes to a given channel is not fixed by the smooth one-point ETH envelope, and its higher-order correlations---the connected cumulants of three and more channel-resolved spectral functions---have lacked a systematic organization. We construct one. In random free fermions, where every channel overlap is a squared Slater minor of a Haar-orthogonal one-body eigenvector matrix, the $r$-resolvent moment admits an exact replica-sector decomposition: a sum over the double-coset sectors of the orthogonal Weingarten algebra, in which the permutation sectors are the replica contraction classes and each sector value is determined by the overlap pattern of the channels and of the eigenstates. The sector coefficients follow from an exact occupancy-matrix assembly---a signed convolution over the slot permutations, a decomposition into connected blocks, and a finite-state chain transfer matrix whose local transition rules are independent of $r$ and $k$---so the sector theorem holds at arbitrary finite multiplicity, realized exactly at $r=3$ ($k=2,3,4$), $r=4$ and $r=5$ ($k=2$). The structure carries the physical hierarchy: the fully coincident sector is the projection that yields the Selberg moment ratios $q_r$, the levels are nested by exact marginalization, and the sectors are the multi-energy spectral cumulants behind the resolvent cumulants, with the irreducible self-energy vertices of the Feshbach ladder as the connected kernel. The multi-resolvent correlations of random free fermions are therefore organized, level by level, by one replica-sector algebra---an exact microscopic benchmark for the higher-order fluctuation sector of eigenstate thermalization, with its microscopic sector geometry made explicit.

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22 pages, 1 figure

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Eigenstate thermalization fixes the smooth energy-resolved envelope of few-body observables but not the microscopic overlap statistics that realize it. In the multi-resolvent hierarchy, this missing information appears in a two-resolvent fluctuation sector. We solve that sector exactly in a tractable benchmark, random free fermions, obtaining a non-Porter-Thomas intensity hierarchy, a structured channel-distance covariance, and an exact overlap kernel that closes the two-resolvent covariance. We then reconstruct the same sector without using its exact solution: a projected self-consistency scheme fixes the irreducible vertex through independently computable projections, while unconstrained residuals identify the sectors missed by a minimal ansatz and are completed exactly by a finite-size Weingarten evaluation, without fitting. For interacting deformations, we prove a model-independent moment identity showing that the deep-plane coefficients are fixed by Hamiltonian-moment covariances. This yields an exact rigidity law: pair-hopping interactions freeze the sector, dense perturbations add only an isotropic layer, and density-density interactions produce an explicit polynomial deformation in channel geometry. By contrast, the bulk sector requires the interacting eigenstate-overlap kernel and exhibits a logarithmically divergent perturbative response. The fluctuation sector therefore splits into a rigid moment-determined part and an intrinsically nonperturbative eigenstate-resolved part.
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We develop a nonperturbative framework for generic nonintegrable many-body systems that reorganizes the expansion of diagonal Green's functions. Starting from exact projection identities and the spectral representation of the resolvent, we derive a recursive hierarchy for the self-energy in which cross-correlated propagation processes are systematically rewritten in terms of diagonal resolvents. Under a diagonal closure approximation, the hierarchy becomes formally closed yet remains systematically improvable. The framework combines two nonperturbative mechanisms. First, a Lanczos continued-fraction representation provides a recursive single-resolvent structure that naturally produces non-Lorentzian spectral features beyond self-consistent Born approximations. Second, an exact projected multi-resolvent hierarchy introduces nonlocal frequency couplings through products of resolvents and their Hilbert transforms. These contributions mix parity sectors under energy reflection and generate spectral skewness, which is absent in single-resolvent closures. To solve the resulting equations, we employ a hierarchy of Lorentzian, Gaussian, and Voigt-type ansätze together with an effective Faddeeva self-energy representation ensuring analyticity and causality. Spectral broadening, distribution tails, and higher-order fluctuations emerge from the interplay between continued-fraction recursion and multi-resolvent correlations. The framework requires no small expansion parameters or diagrammatic truncations, relying instead on ETH-type statistical assumptions appropriate for dense chaotic spectra. It provides a unified route from microscopic interactions to emergent spectral structure, revealing a progression from single-pole self-consistent dynamics to continued-fraction renormalization and finally to multi-resolvent interference effects.