Realization of a period-3 coplanar state in one-dimensional spin-orbit coupled optical lattice

In ultracold atoms, achieving a period-$3$ structure poses a significant challenge. In this work, we propose a three-sublattice spin-flop transition mechanism, differing from the two-sublattice counterpart used to explain the emergence of ferrimagnetic orders in higher dimensions. Guided by this mechanism, we design a setup of alkaline-earth-metal atoms to create a spin-orbit coupled optical lattice, where we identify a triplet-fold degenerate $YX\bar{Y}$ state with a period-$3$ coplanar spin ordering within the deep Mott-insulating phase region of the ground-state phase diagram. The $YX\bar{Y}$ state is protected by a finite gap, and its characteristic angle can be finely tuned by specific setup parameters. Moreover, we use the Rabi spectroscopy technique to detect the $YX\bar{Y}$ state. Our work not only shows the feasibility of achieving a period-$3$ structure \textit{via} the new mechanism but also suggests its potential applications for exploring other periodic structures in optical lattices.