Ch. Pao et Hb. Schuttler, Numerical renormalization-group approach to fluctuation exchange in the presence of electron-phonon coupling: Pairing in the Holstein-Hubbard model, PHYS REV B, 60(2), 1999, pp. 1283-1295
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].