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 Holstein-Hubbard model. 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 $T$. Here,we describe a new numerical renormalization group (NRG) technique to solve the FLEX equations efficiently. Several orders of magnitude of CPU time and storage can be saved at low $T$ ($\sim 80K$). 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 ($\sim $ band width). For d-wave pairing, the isotope exponent is negative and small compared to the typical observed values in non-optimally doped cuprates.