Non-Abelian anyons can exist as point-like particles in two-dimensionalsystems, and have particle exchange statistics which are neither bosonic norfermionic. Like in spin systems, the role of fusion (Heisenberg-like)interactions between anyons has been well studied. However, unlike ourunderstanding of the role of bosonic and fermionic statistics in the formationof different quantum phases of matter, little is known concerning the effect ofnon-Abelian anyonic statistics. We explore this physics using an anyonicHubbard model on a two-legged ladder which includes braiding and nearestneighbour Heisenberg interactions among anyons. We study two of the mostprominent non-Abelian anyon models: the Fibonacci and Ising type. We discoverrich phase diagrams for both anyon models, and show the different roles oftheir fusion and braid statistics.
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