  There are many interesting parallels between systems of interacting
non-Abelian anyons and quantum magnetism, occuring in ordinary SU(2) quantum
magnets. Here we consider theories of so-called su(2)_k anyons, well-known
deformations of SU(2), in which only the first k+1 angular momenta of SU(2)
occur. In this manuscript, we discuss in particular anyonic generalizations of
ordinary SU(2) spin chains with an emphasis on anyonic spin S=1 chains. We find
that the overall phase diagrams for these anyonic spin-1 chains closely mirror
the phase diagram of the ordinary bilinear-biquadratic spin-1 chain including
anyonic generalizations of the Haldane phase, the AKLT construction, and
supersymmetric quantum critical points. A novel feature of the anyonic spin-1
chains is an additional topological symmetry that protects the gapless phases.
Distinctions further arise in the form of an even/odd effect in the deformation
parameter k when considering su(2)_k anyonic theories with k>4, as well as for
the special case of the su(2)_4 theory for which the spin-1 representation
plays a special role. We also address anyonic generalizations of spin-1/2
chains with a focus on the topological protection provided for their gapless
ground states. Finally, we put our results into context of earlier
generalizations of SU(2) quantum spin chains, in particular so-called (fused)
Temperley-Lieb chains.
