Self-Consistent Spectral Quadrature Approach to Many-Body Green Functions

We develop a self-consistent spectral quadrature (sc-SQ) framework for the calculation of many-body Green functions. The method approximates the K\"all\'en--Lehmann spectral measure by Gauss--Christoffel (GC) quadrature, yielding a rational Green function representation with guaranteed spectral positivity that exactly reproduces the first $2N$ spectral moments at pole order $N$. A key component is an SVD-based rank-selection criterion on the Hankel matrix, which identifies the numerically resolvable pole rank $N^*$ 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 $N\geqslant 5$, in qualitative agreement with NRG references.