Fibonacci anyons have been studied in the context of topological
quantum computation. After introducing these anynos, we show that
chains of interacting Fibonacci anyons can support a variety of
collective ground states ranging from extended critical phases to
gapped phases with ground-state degeneracy and quasiparticle
excitations. In particular, we generalize the Majumdar-Ghosh
Hamiltonian to anyonic degrees of freedom by extending recently
studied pairwise anyonic interactions to three-anyon interactions.
The energetic competition between two- and three-anyon interactions
leads to a rich phase diagram that harbours multiple critical and
gapped phases.
