Non-Abelian statistics in one dimension: Topological momentum spacings and SU(2) level- k fusion rules

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 2n (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-12 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.