Synthetic spin-orbit coupling for the multi-spin models in optical lattices

The essential role of synthetic spin-orbit coupling in discovering new topological matter phases with cold atoms is widely acknowledged. However, the engineering of spin-orbit coupling remains unclear for arbitrary-spin models due to the complexity of spin matrices. In this work, we develop a more general but relatively straightforward method to achieve spin-orbit coupling for multi-spin models. Our approach hinges on controlling the coupling between distinct pseudo-spins through two intermediary states, resulting in tunneling with spin flips that have direction-dependent strength. The engineered spin-orbit coupling can facilitate topological phase transitions with Chern numbers over 1, a unique characteristic of multi-spin models compared to spin-1/2 models. By utilizing existing cold atom techniques, our proposed method provides an ideal platform for investigating topological properties related to large Chern numbers.