Numerical renormalization-group approach to fluctuation exchange in the presence of electron-phonon coupling: Pairing in the Holstein-Hubbard model

The fluctuation exchange (FLEX) approximation is applied to study the Holst ein-Hubbard model. Due to the retarded nature of the phonon-mediated electr on-electron interaction, neither fast Fourier transform nor previously deve loped numerical renormalization-group (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. Here, we describe a NRG technique to solve the FLEX equa tions efficiently. Several orders of magnitude of CPU time and storage can be saved at low T (similar to 80 K). To test our approach, we compare our N RG results to brute force calculations-on small lattices at elevated temper atures. Both s-wave and d-wave superconducting phase diagrams are then obta ined by applying the NRG approach at low T. The isotope effect for s-wave p airing is BCS-like in a realistic phonon frequency range, but vanishes at u nphysically large phonon frequency (similar to bandwidth). For d-wave pairi ng, the isotope exponent is negative and small compared to the typical obse rved values in nonoptimally doped cuprates. [S0163-1829(99)11325-0].