Shadowed Triplet Pairings in Hund's Metals with Spin-Orbit Coupling

Hund's coupling in multi-orbital systems allows for the possibility of even-parity orbital-antisymmetric spin-triplet pairing which can be stabilized by spin-orbit coupling (SOC). While this pairing expressed in the orbital basis is uniform and spin-triplet, it appears at the Fermi surface (FS) as a pseudospin-singlet, with the momentum-dependence determined by the SOC and the underlying triplet character remaining in the form of interband pairing active away from the FS. Here, we examine the role of momentum-dependent SOC ($\boldsymbol{k}$-SOC) in generating non-trivial pairing symmetries at the FS, as well as the hidden triplet nature associated with inter-orbital pairing, which we dub a "$\textit{shadowed triplet}$". Applying this concept to Sr$_{2}$RuO$_{4}$, we first derive several forms of $\boldsymbol{k}$-SOC with d-wave form-factors from a microscopic model and subsequently show that for a reasonable range of SOC parameters, a pairing state with $s + id_{xy}$ symmetry at the FS can be stabilized. Such a pairing state is distinct from pure spin-singlet and -triplet pairings due to its unique character of pseudospin-singlet pairing on the FS with underlying pseudospin-triplet nature away from the FS. We discuss experimental probes to differentiate the shadowed triplet pairing from pure spin-triplet and -singlet pairings.