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

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.