Effects of longer-range interaction on superconductivity in Sr₂RuO₄

The superconducting symmetry of Sr₂RuO₄ (SRO) remains a puzzle. Currently, none of the proposed order parameters (OP) can account for all the key experiments, highlighting the importance of exploring different alternatives. Recently, among others, d_x²-y² + ig_xy(x²-y²) and s+id_xy pairings have been proposed as OP candidates. However, their stability remains unclear. In this work, with effects of the sizable spin-orbit coupling accounted for, we theoretically study the superconducting instabilities in SRO in the presence of both local and longer-range interactions. We show that inclusion of a second nearest neighbor repulsion or orbital-anisotropy of the longer-range interaction can have a significant impact on the stability of the g-wave pairing. Different pairing states, including d_x²-y² + ig, s+id_xy, mixed helical, and d_x²-y² + ip, are found to be stabilized in different regimes of the interaction parameter space. We further investigate their properties such as Knight shift and spontaneous edge current. Lastly, we make a connection to the orbital-basis classification of superconducting states that has been actively discussed recently.