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The ruthenate family of layered perovskites has been a topic of intenseinterest with much work dedicated to understanding the superconducting state ofthe single layer, Sr$_2$RuO$_4$. Another longstanding puzzle is the lack ofsuperconductivity in its sister compound, Sr$_3$Ru$_2$O$_7$, which constrainsthe possible superconducting mechanisms of Sr$_2$RuO$_4$. Here we address amicroscopic mechanism that unifies superconducting and spin-density wave orderin this family of materials. Beginning from a model of Sr$_2$RuO$_4$ featuringintraband pseudospin-singlet superconductivity originating from interorbitalspin-triplet pairing via Hund's and spin-orbit couplings, the addition ofbilayer coupling and staggered rotations of the oxygen octahedra areinvestigated. We find that the bilayer coupling alone enhancessuperconductivity, while staggered rotations destroy interorbitalsuperconductivity, leaving a paramagnetic metal. The magnetic field then shiftsvan Hove singularities near the Fermi level, allowing intraorbital SDW order toform in the bilayer. Our theory predicts that bilayer Sr$_3$Ru$_2$O$_7$ withoutstaggered rotations exhibits interorbital superconductivity with a possiblyhigher transition temperature.

Evolution of Interorbital Superconductor to Intraorbital Spin-Density Wave in Layered Ruthenates