# SU(3) Spin-Orbit Coupling in Systems of Ultracold Atoms

Motivated by the recent experimental success in realizing synthetic spin-orbit coupling in ultracold atomic systems, we consider *N*-component atoms coupled to a non-Abelian SU(*N*) gauge field. More specifically, we focus on the case, referred to here as “SU(3) spin-orbit-coupling,” where the internal states of three-component atoms are coupled to their momenta via a matrix structure that involves the Gell-Mann matrices (in contrast to the Pauli matrices in conventional SU(2) spin-orbit-coupled systems). It is shown that the SU(3) spin-orbit-coupling gives rise to qualitatively different phenomena and in particular we find that even a homogeneous SU(3) field on a simple square lattice enables a topologically nontrivial state to exist, while such SU(2) systems always have trivial topology. In deriving this result, we first establish an equivalence between the Hofstadter model with a 1/*N* Abelian flux per plaquette and a homogeneous SU(*N*) non-Abelian model. The former is known to have a topological spectrum for *N*>2, which is thus inherited by the latter. It is explicitly verified by an exact calculation for *N*=3, where we develop and use a new algebraic method to calculate topological indices in the SU(3) case. Finally, we consider a strip geometry and establish the existence of three gapless edge states—the hallmark feature of such an SU(3) topological insulator.

1. Barnett, R., Boyd, G. R. & Galitski, V. SU(3) Spin-Orbit Coupling in Systems of Ultracold Atoms. *Phys. Rev. Lett.* **109,** 235308 (2012).