Fluctuation exchange theory for superconductivity of $f$-electron systems with multipole degrees of freedom

Recently, an interesting possibility of superconductivity induced by multipole fluctuations has been discussed in some $f$-electron materials. In order to clarify such exotic superconductivity from a microscopic viewpoint, we apply fluctuation exchange (FLEX) approximation to an $f$-electron model with active orbital degree of freedom on the basis of a $j$-$j$ coupling scheme. In this study, we consider a square lattice, for simplicity. First, we evaluate orbital dependent fluctuations and effective paring interactions within the FLEX approximation. Then, we determine the symmetry of the gap function among possible superconducting states by solving the Eliashberg equation. In particular, we pay our attention to the effect of crystalline electric field on the appearance of superconductivity. By further decomposing complex orbital dependent fluctuations into multipole components, we also discuss possible relevance of multipole fluctuations to exotic superconductivity.