Topological stability of q-deformed quantum spin chains

Quantum mechanical systems, whose degrees of freedom are so-called $su(2)_k$ anyons, form a bridge between ordinary spin systems and systems of interacting non-Abelian anyons. Such a connection can be made for arbitrary spin-S systems, and we explicitly discuss spin-$1/2$ and spin-$1$ systems. Anyonic spin-$1/2$ chains exhibit a topological protection mechanism that stabilizes their gapless ground states and which vanishes only in the limit ($k \to \infty$) where the system turns into the ordinary spin-$1/2$ Heisenberg chain. For anyonic spin-$1$ chains we show that their phase diagrams closely mirror the one of the biquadratic spin-$1$ chain. This includes generalizations of the Haldane phase, of the AKLT point, and the appearance of several stable critical phases described by (super)conformal field theories.