Due to the retarded nature of the phonon-mediated electron-electron interaction, neither fast Fourier transform (FFT) nor previously developed NRG methods for Hubbard-type purely electronic models are applicable, while brute-force solutions are limited by the demands on computational time and storage which increase rapidly at low temperature T. A new numerical renormalization group (NRG) approach to the solution of self-consistent-field equations for correlated lattice electrons is illustrated for the two-dimensional Hubbard-type model. This approach allows us to handle the fermion frequency matrix inversions in the FLEX equations efficiently. Several orders of magnitude of CPU time and storage can be saved at the temperature scales relevant to electronic phase transitions.
Momentum- and frequency-dependent self-consistent field theories provide a potentially powerful tool for analyzing condensed-matter models for interacting electrons. Such theories generally require the solution of multidimensional integral equations. Solutions are limited by demands on computational time and storage, which increase rapidly for calculation of low-temperature properties. A renormalization-group technique which allows the sequential elimination of regions of high frequency and momentum is presented in detail, and is then applied to the solution of the fluctuation exchange approximation for the two-dimensional Hubbard model. Effective meshes with as many as
104 frequencies and
5122 k points can be analyzed on a vector supercomputer using this approach. As a sample application, it is demonstrated that the fluctuation exchange approximation exhibits non-Fermi liquid behavior near half filling for sufficiently large interaction strengths.