# Non-Abelian Statistics in one dimension: topological momentum spacings and SU(2) level $k$ fusion rules

## TL;DR

This paper demonstrates that non-Abelian SU(2) level k anyon statistics in one-dimensional spin chains manifest through topological momentum spacings, linking fusion rules to domain wall configurations in critical spin models.

## Contribution

It introduces a novel connection between topological momentum spacings in 1D spin chains and SU(2) level k fusion rules, extending the understanding of non-Abelian anyons to one dimension.

## Key findings

- Topological momentum spacings encode non-Abelian statistics.
- Fusion rules correspond to domain wall configurations.
- Derived rules for Ising and SU(2) level k anyons.

## Abstract

We use a family of critical spin chain models discovered recently by one of us [M. Greiter, Mapping of Parent Hamiltonians, Springer, Berlin/Heidelberg 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, 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 \half 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 Bratelli diagrams. Finally, we show that the choices in the fusion trees may be interpreted as the choices between different domain walls between the $2S+1$ possible, degenerate dimer configurations of the spin $S$ chains at the multicritical point.