  The Rashba spin-orbit coupling is equivalent to the finite Yang-Mills flux of
a static SU(2) gauge field. It gives rise to the protected edge states in
two-dimensional topological band-insulators, much like magnetic field yields
the integer quantum Hall effect. An outstanding question is which collective
topological behaviors of interacting particles are made possible by the Rashba
spin-orbit coupling. Here we addresses one aspect of this question by exploring
the Rashba SU(2) analogues of vortices in superconductors. Using the
Landau-Ginzburg approach and conservation laws, we classify the prominent
two-dimensional condensates of two- and three-component spin-orbit-coupled
bosons, and characterize their vortex excitations. There are two prominent
types of condensates that take advantage of the Rashba spin-orbit coupling.
Their vortices exist in multiple flavors whose number is determined by the spin
representation, and interact among themselves through logarithmic or linear
potentials as a function of distance. The vortices that interact linearly
exhibit confinement and asymptotic freedom similar to quarks in quantum
chromodynamics. One of the two condensate types supports small metastable
neutral quadruplets of vortices, and their tiles as metastable vortex lattices.
Quantum melting of such vortex lattices could give rise to non-Abelian
fractional topological insulators, SU(2) analogues of fractional quantum Hall
states. The physical systems in which these states could exist are trapped two-
and three-component bosonic ultra-cold atoms subjected to artificial gauge
fields, as well as solid-state quantum wells made either from Kondo insulators
such as SmB$_6$ or conventional topological insulators interfaced with
conventional superconductors.
