  Motivated by a recent experiment [L. F. Livi, et al., Phys. Rev. Lett. 117,
220401(2016)], we study the ground-state properties of interacting fermions in
a one-dimensional optical lattice clock with spin-orbit coupling. As the
electronic and the hyperfine-spin states in the clock-state manifolds can be
treated as effective sites along distinct synthetic dimensions, the system can
be considered as multiple two-leg ladders with uniform magnetic flux
penetrating the plaquettes of each ladder. As the inter-orbital spin-exchange
interactions in the clock-state manifolds couple individual ladders together,
we show that exotic interaction-induced vortex states emerge in the
coupled-ladder system, which compete with existing phases of decoupled ladders
and lead to a rich phase diagram. Adopting the density matrix renormalization
group approach, we map out the phase diagram, and investigate in detail the
currents and the density-density correlations of the various phases. Our
results reveal the impact of interactions on spin-orbit coupled systems, and
are particularly relevant to the on-going exploration of spin-orbit coupled
optical lattice clocks.
