| Summary: | We revisit the phase diagram of spin-1 $su(2)_k$ anyonic chains, originally
studied by Gils {\it et. al.} [Phys. Rev. B, {\bf 87} (23) (2013)]. These
chains possess several integrable points, which were overlooked (or only
briefly considered) so far.
Exploiting integrability through a combination of algebraic techniques and
exact Bethe ansatz results, we establish in particular the presence of new
first order phase transitions, a new critical point described by a $Z_k$
parafermionic CFT, and of even more phases than originally conjectured. Our
results leave room for yet more progress in the understanding of spin-1 anyonic
chains.
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