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Superconductivity arises mostly at energy and temperature scales that aremuch smaller than the typical bare electronic energies. Since the computationaleffort of diagrammatic many-body techniques increases with the number ofrequired Matsubara frequencies and thus with the inverse temperature, phasetransitions that occur at low temperatures are typically hard to addressnumerically. In this work, we implement a fluctuation exchange (FLEX) approachto spin fluctuations and superconductivity using the "intermediaterepresentation basis" (IR) [Shinaoka et al., PRB 96, 2017] for Matsubara Greenfunctions. This FLEX+IR approach is numerically very efficient and enables usto reach temperatures on the order of $10^{-4}$ in units of the electronic bandwidth in multi-orbital systems. After benchmarking the method in the dopedrepulsive Hubbard model on the square lattice, we study the possibility ofspin-fluctuation-mediated superconductivity in the hydrated sodium cobaltmaterial Na$_x$CoO$_2\cdot y$H$_2$O reaching the scale of the experimentaltransition temperature $T_{\mathrm{c}}=4.5$ K and below.

physical review. b./physical review. b

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