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arXiv:2604.00606

Resolvent-Based Self-Consistent Framework with Hierarchical Correlation Expansion for Strongly Correlated Many-Body Systems

Zhi-qiang Huang, Qing-yu Cai

quant-phSubmitted Apr 1, 2026• Announced Apr 2, 2026• 0 citations• Revised May 7, 2026 (v3)
v1 · Apr 1, 2026v2 · Apr 2, 2026v3 · May 7, 2026v4 · Jun 2, 2026
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Abstract

We develop a resolvent-based self-consistent framework for strongly correlated many-body systems by reorganizing many-body expansions at the level of the resolvent rather than through perturbative expansions in a small parameter. Starting from the spectral representation of the diagonal Green's function, we derive an exact recursive hierarchy for the self-energy in terms of correlated multi-resolvent propagation processes. The resulting hierarchy remains formally closed in terms of diagonal resolvents and provides a systematically improvable description of fluctuations beyond mean-field theory. The framework contains two complementary nonperturbative structures. The Lanczos continued-fraction representation governs recursive single-resolvent renormalization and generates non-Lorentzian spectral broadening beyond conventional self-consistent Born approximations (SCBA). By contrast, the multi-resolvent hierarchy introduces correlated frequency mixing through products of resolvents and Hilbert-transform couplings, providing a microscopic mechanism for spectral asymmetry and skewness absent in parity-preserving single-resolvent closures. To solve the hierarchy, we introduce Lorentzian, Gaussian, and hybrid Voigt-type closure schemes together with an effective Faddeeva self-energy representation preserving analyticity and causality. Spectral broadening, tail structures, and higher-order fluctuation effects then emerge naturally from the interplay between recursive renormalization and multi-resolvent correlations. Unlike conventional diagrammatic resummations, the present approach does not rely on finite-order truncations or small expansion parameters. Instead, correlations are organized through an exact resolvent hierarchy combined with ETH-type statistical assumptions, making the framework particularly suitable for nonintegrable many-body systems with dense spectra.

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26 pages

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We develop a resolvent-based self-consistent framework for strongly correlated many-body systems by reorganizing many-body expansions at the level of the resolvent rather than through perturbative expansions in a small parameter. Starting from the spectral representation of the diagonal Green's function, we derive an exact recursive hierarchy for the self-energy in terms of correlated multi-resolvent propagation processes. The resulting hierarchy remains formally closed in terms of diagonal resolvents and provides a systematically improvable description of fluctuations beyond mean-field theory.
The framework contains two complementary nonperturbative structures. The Lanczos continued-fraction representation governs recursive single-resolvent renormalization and generates non-Lorentzian spectral broadening beyond conventional self-consistent Born approximations (SCBA). By contrast, the multi-resolvent hierarchy introduces correlated frequency mixing through products of resolvents and Hilbert-transform couplings, providing a microscopic mechanism for spectral asymmetry and skewness absent in parity-preserving single-resolvent closures.
To solve the hierarchy, we introduce Lorentzian, Gaussian, and hybrid Voigt-type closure schemes together with an effective Faddeeva self-energy representation preserving analyticity and causality. Spectral broadening, tail structures, and higher-order fluctuation effects then emerge naturally from the interplay between recursive renormalization and multi-resolvent correlations. Unlike conventional diagrammatic resummations, the present approach does not rely on finite-order truncations or small expansion parameters. Instead, correlations are organized through an exact resolvent hierarchy combined with ETH-type statistical assumptions, making the framework particularly suitable for nonintegrable many-body systems with dense spectra.
The exact multi-resolvent hierarchy of dense many-body spectra encodes higher-order spectral-weight correlations, but closing it predictively for interacting Hamiltonians remains a challenge. We introduce the multi-resolvent self-consistency (MRSC) equations, which separate an exact algebraic hierarchy from an approximate finite-parameter closure. The exact Feshbach geometric identity organizes r-resolvent cumulants into a ladder over irreducible vertices Gamma^(r), and the projected hierarchy expresses those vertices as functionals of the cumulants. The closure works in finite parameters: a connectivity basis organizes a general Hamiltonian's seed into class amplitudes, and a causal r-fold Stieltjes representation with a moment-field identity reduce moment matching to a finite-dimensional problem closed at a predictive fixed point. On the free-fermion benchmark the tested exact identities are reproduced and the Faddeeva closure is recovered at r=1. In a blind test on the interacting Ising chain, with the exact two-resolvent data withheld, the Hamiltonian-derived closure predicts the tested probes to 0.35%-0.97%. At the observable level the exact two-resolvent sector supplies a sign-correct, energy-resolved correction to the single-resolvent baseline for the off-diagonal eigenstate-thermalization-hypothesis (ETH) variance, of order one relative to it. Closure errors organize hierarchically: a level-by-level deficit decomposition, anchor measurements of the isolatable terms, and an explicit propagation bound. The same moments further yield a diagnostic chain for the window conditions of local rotational invariance -- an exact drift-kernel identity, a proven single-state identifiability boundary, and a transferability census over couplings and realizations -- a finite-order framework, not a proof of universal emergence.
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+gjif_{ji}^2=D_{ji}+g_{ji}fji2​=Dji​+gji​, where DjiD_{ji}Dji​ is the diagonal-overlap baseline and gji=∑r≥2gji(r)g_{ji}=\sum_{r\ge2}g_{ji}^{(r)}gji​=∑r≥2​gji(r)​ is a systematically improvable hierarchy of multi-channel interference processes. The leading r=3r=3r=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⟩2q-3=κ_4/\langle c^2\rangle^2q−3=κ4​/⟨c2⟩2; normalization further fixes the cross-channel covariance Cˉi\bar C_iCˉi​, 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.
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.
We develop a self-consistent spectral quadrature (sc-SQ) framework for the calculation of many-body Green functions. The method approximates the Källén--Lehmann spectral measure by Gauss--Christoffel (GC) quadrature, yielding a rational Green function representation with guaranteed spectral positivity that exactly reproduces the first 2N2N2N spectral moments at pole order NNN. A key component is an SVD-based rank-selection criterion on the Hankel matrix, which identifies the numerically resolvable pole rank N∗N^*N∗ from the singular-value gap and acts as a precision-guided diagnostic of correlation complexity. The scheme is made self-consistent by requiring that the spectral function used to evaluate expectation values coincides with the spectral function generated by the quadrature reconstruction. This defines a fixed-point hierarchy that connects systematically to established approximations, including Hartree--Fock and Hubbard-I, and incorporates non-perturbative features such as multi-peak spectral structure. We benchmark the approach for the Anderson impurity model against numerical renormalization group (NRG) results and apply it within dynamical mean-field theory for the Hubbard model on the Bethe lattice. The method captures the three-peak Anderson impurity spectrum and the suppression of quasiparticle weight in the half-filled Hubbard model on the Bethe lattice, including Mott-gap formation on the insulating branch for N⩾5N\geqslant 5N⩾5, in qualitative agreement with NRG references.
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 rrr-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 rrr and kkk---so the sector theorem holds at arbitrary finite multiplicity, realized exactly at r=3r=3r=3 (k=2,3,4k=2,3,4k=2,3,4), r=4r=4r=4 and r=5r=5r=5 (k=2k=2k=2). The structure carries the physical hierarchy: the fully coincident sector is the projection that yields the Selberg moment ratios qrq_rqr​, 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.
We investigate the generic emergence of entropy in isolated quantum systems from the spectral structure of their long-time dephased states. Using a resolvent-based hierarchy, we separate the smooth energy-shell envelope of eigenstate intensities from their intra-shell fluctuations. The single-resolvent sector determines the coarse-grained intensity envelope and yields an entropy contribution that is quantitatively captured by a Lorentzian-Gaussian form. We then show that the difference between this envelope entropy and the von Neumann entropy is governed exactly by the conditional statistics of the residual intensity fluctuations. The two-resolvent covariance sector identifies their leading variance, while higher correlation sectors control the full non-Gaussian correction. A Gaussian-amplitude closure provides a benchmark entropy deficit, whereas exact diagonalization of an interacting Ising chain exhibits substantial finite-size deviations from this benchmark. These results establish a hierarchical spectral framework in which entropy beyond the smooth envelope is directly tied to multi-resolvent correlations, providing a systematic route toward its microscopic characterization.