From spin to anyon notation: The XXZ Heisenberg model as a $D_{3}$ (or $su(2)_{4}$) anyon chain

We discuss a relationship between certain one-dimensional quantum spin chains and anyon chains. In particular we show how the XXZ Heisenberg chain is realised as a $D_{3}$ (alternately $su(2)_{4}$) anyon model. We find the difference between the models lie primarily in choice of boundary condition.