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

摘要

The fluctuation exchange (FLEX) approximation is applied to study theHolstein-Hubbard model. Due to the retarded nature of the phonon-mediatedelectron-electron interaction, neither fast Fourier transform (FFT) norpreviously developed NRG methods for Hubbard-type purely electronic models areapplicable, while brute force solutions are limited by the demands oncomputational time and storage which increase rapidly at low temperature $T$.Here,we describe a new numerical renormalization group (NRG) technique to solvethe FLEX equations efficiently. Several orders of magnitude of CPU time andstorage can be saved at low $T$ ($\sim 80K$). To test our approach, we compareour NRG results to brute force calculations on small lattices at elevatedtemperatures. Both s-wave and d-wave superconducting phase diagrams are thenobtained by applying the NRG approach at low $T$. The isotope effect for s-wavepairing is BCS-like in a realistic phonon frequency range, but vanishes atunphysically large phonon frequency ($\sim $ band width). For d-wave pairing,the isotope exponent is negative and small compared to the typical observedvalues in non-optimally doped cuprates.