Efficient fluctuation exchange approach to low-temperature spin fluctuations and superconductivity: from the Hubbard model to Na$_x$CoO$_2\cdot y$H$_2$O

Superconductivity arises mostly at energy and temperature scales that are much smaller than the typical bare electronic energies. Since the computational effort of diagrammatic many-body techniques increases with the number of required Matsubara frequencies and thus with the inverse temperature, phase transitions that occur at low temperatures are typically hard to address numerically. In this work, we implement a fluctuation exchange (FLEX) approach to spin fluctuations and superconductivity using the "intermediate representation basis" (IR) [Shinaoka et al., PRB 96, 2017] for Matsubara Green functions. This FLEX+IR approach is numerically very efficient and enables us to reach temperatures on the order of $10^{-4}$ in units of the electronic band width in multi-orbital systems. After benchmarking the method in the doped repulsive Hubbard model on the square lattice, we study the possibility of spin-fluctuation-mediated superconductivity in the hydrated sodium cobalt material Na$_x$CoO$_2\cdot y$H$_2$O reaching the scale of the experimental transition temperature $T_{\mathrm{c}}=4.5$ K and below.