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Majorana fermion models for interacting non-Abelian anyons

We introduce an effective free Majorana fermion lattice model to study collective states of interacting non-Abelian anyons. We apply it to Kitaev’s honeycomb lattice model and show that the Abelian phases emerging due to the presence of anyonic vortex lattices are fully determined by the anyon-anyon interactions when both nearest and next to nearest interactions are included. The necessity to consider also the next to nearest terms derives from oscillations in the interactions. These are present in most topological models such as p-wave superconductors or the fractional quantum Hall liquids. Our results provide a first demonstration of an anyon-anyon interaction induced transition in a microscopic model. Given the ability to measure the tunneling amplitudes, the introduced effective model is readily employed to study the phase diagram of any system that supports tunneling Majorana fermions.