  The theory of topological quantum computation is underpinned by two important
classes of models. One is based on non-abelian Chern-Simons theory, which
yields the so-called $\rm{SU}(2)_k$ anyon models that often appear in the
context of electrically charged quantum fluids. The physics of the other is
captured by symmetry broken Yang-Mills theory in the absence of a Chern-Simons
term, and results in the so-called quantum double models. Extensive resources
have been invested into the search for $\rm{SU}(2)_k$ anyon quasi-particles; in
particular the so-called Ising anyons ($k=2$) of which Majorana zero modes are
believed to be an incarnation. In contrast to the $\rm{SU}(2)_k$ models,
quantum doubles have attracted little attention in experiments despite their
pivotal role in the theory of error correction. Beyond topological error
correcting codes, the appearance of quantum doubles has been limited to
contexts primarily within mathematical physics, and as such, they are of
seemingly little relevance for the study of experimentally tangible systems.
However, recent works suggest that quantum double anyons may be found in spinor
Bose-Einstein condensates. In light of this, the core purpose of this article
is to provide a self-contained exposition of the quantum double structure,
framed in the context of spinor condensates, by constructing explicitly the
quantum doubles for various ground state symmetry groups and discuss their
experimental realisability. We also derive analytically an equation for the
quantum double Clebsch-Gordan coefficients from which the relevant braid
matrices can be worked out. Finally, the existence of a particle-vortex duality
is exposed and illuminated upon in this context.
