Exotic 3D Spin-Orbit Couplings

We describe a scheme for creating an isotropic three-dimensional spin-orbit coupling, dubbed Weyl spin-orbit coupling, in systems of ultracold atoms. This coupling is induced by Raman transitions that link four internal atomic states with a tetrahedral geometry. This spin-orbit coupling gives rise to a Dirac point that is robust against environmental perturbations. We then propose a general procedure for generating exotic three-dimensional spin-orbit couplings with degenerate ground states on more complex manifolds. The procedure is applied to produce a spin-orbit coupling with a toroidal ground state manifold. Finally, we discuss the many-body implications of the exotic spin-orbit couplings.