The superconducting order parameter of Sr₂RuO₄ has again become a topic of great interest after recent NMR experiments have contradicted the once popular p_x±ip_y spin-triplet superconducting state. We have previously explored a microscopic route to inter-orbital spin-triplet superconducting pairing with spin-orbit coupling to explain recent experimental data in Sr₂RuO₄ [1-3]. Here, we extend this microscopic theory to Sr₃Ru₂O₇, the bilayer version of Sr₂RuO₄ to explain why Sr₃Ru₂O₇ does not exhibit superconductivity, but spin-density wave order under a magnetic field.

[1] C. M. Puetter and H.-Y. Kee, EPL 98, 27010 (2012).
[2] A. W. Lindquist and H.-Y. Kee, PRR 2, 032055 (2020).
[3] J. Clepkens, A. W. Lindquist, and H.-Y. Kee, arXiv:2009.08597