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Previous theoretical studies [W. Cho, C. Platt, R. H. McKenzie, and S. Raghu,Phys. Rev. B 92, 134514 (2015); N. Lera and J. V. Alvarez, Phys. Rev. B 92,174523 (2015)] have suggested that Li$_{0.9}$Mo$_6$O$_{17}$, a quasi-onedimensional "purple bronze" compound, exhibits spin-triplet superconductivityand that the gap function changes sign across the two nearly degenerate Fermisurface sheets. We investigate the role of spin-orbit coupling (SOC) indetermining the symmetry and orientation of the $d$-vector associated with thesuperconducting order parameter. We propose that the lack of local inversionsymmetry within the four-atom unit cell leads to a staggered spin-orbitcoupling analogous to that proposed for graphene, MoS$_2$, or SrPtAs. Inaddition, from a weak-coupling renormalization group treatment of an effectivemodel Hamiltonian, we find that SOC favors the odd parity $A_{1u}$ state with$S_z = \pm 1$ over the $B$ states with $S_z=0$, where $z$ denotes theleast-conducting direction. We discuss possible definitive experimentalsignatures of this superconducting state.

Spin-orbit coupling and odd-parity superconductivity in the quasi-one-dimensional compound Li$_{0.9}$Mo$_6$O$_{17}$