Abstract
We use a family of critical spin chain models discovered recently by one
of us M. Greiter, Mapping of Parent Hamiltonians (Springer, Berlin,
2011) to propose and elaborate that non-Abelian, SU(2) level-k = 2S
anyon statistics manifests itself in one dimension through topological
selection rules for fractional shifts in the spacings of linear momenta,
which yield an internal Hilbert space of 2(n) (in the
thermodynamic-limit) degenerate states. These shifts constitute the
equivalent to the fractional shifts in the relative angular momenta of
anyons in two dimensions. We derive the rules first for Ising anyons,
and then generalize them to SU(2) level-k anyons. We establish a
one-to-one correspondence between the topological choices for the
momentum spacings and the fusion rules of spin-1/2 spinons in the SU(2)
level-k Wess-Zumino-Witten model, where the internal Hilbert space is
spanned by the manifold of allowed fusion trees in the Bratteli
diagrams. Finally, we show that the choices in the fusion trees may be
interpreted as the choices of different domain walls between the 2S + 1
possible, degenerate dimer configurations of the spin-S chains at the
multicritical point.
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