Numerical renormalization-group approach to fluctuation exchange in the presence of electron-phonon coupling: Pairing in the Holstein-Hubbard model
C.-H. Pao, H.-B. Schüttler
The fluctuation exchange (FLEX) approximation is applied to study the Holstein-Hubbard model. Due to the retarded nature of the phonon-mediated electron-electron interaction, neither fast Fourier transform nor previously developed 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 equations efficiently. Several orders of magnitude of CPU time and storage can be saved at low T $(\ensuremath{\sim}80\mathrm{K})$. To test our approach, we compare our NRG results to brute force calculations on small lattices at elevated temperatures. Both s-wave and d-wave superconducting phase diagrams are then obtained by applying the NRG approach at low T. The isotope effect for s-wave pairing is BCS-like in a realistic phonon frequency range, but vanishes at unphysically large phonon frequency $(\ensuremath{\sim}\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h})$. For d-wave pairing, the isotope exponent is negative and small compared to the typical observed values in nonoptimally doped cuprates.
arXiv:cond-mat/9809321•Physical review. B, Condensed matter•Jul 1, 1999•5 citations