  Non-Abelian anyons can exist as point-like particles in two-dimensional
systems, and have particle exchange statistics which are neither bosonic nor
fermionic. Like in spin systems, the role of fusion (Heisenberg-like)
interactions between anyons has been well studied. However, unlike our
understanding of the role of bosonic and fermionic statistics in the formation
of different quantum phases of matter, little is known concerning the effect of
non-Abelian anyonic statistics. We explore this physics using an anyonic
Hubbard model on a two-legged ladder which includes braiding and nearest
neighbour Heisenberg interactions among anyons. We study two of the most
prominent non-Abelian anyon models: the Fibonacci and Ising type. We discover
rich phase diagrams for both anyon models, and show the different roles of
their fusion and braid statistics.
