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

TL;DR

This paper explores SU(3) spin-orbit coupling in ultracold atoms, revealing topologically non-trivial states and edge states, distinct from SU(2) systems, through theoretical models and algebraic calculations.

Contribution

It introduces the concept of SU(3) spin-orbit coupling in ultracold atoms and demonstrates its unique topological properties and edge states, expanding the understanding of non-Abelian gauge fields.

Findings

SU(3) spin-orbit coupling enables topologically non-trivial states on simple lattices.

An exact equivalence between Hofstadter and SU(N) models is established.

Three gapless edge states are found in the SU(3) topological insulator.

Abstract

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 non-trivial state to exist, while such SU(2) systems always have trivial topology. In deriving this result, we first establish an exact 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.