  We study a system of dipolar bosons in a one-dimensional optical lattice
using exact diagonalization and density matrix renormalization group methods.
In particular, we analyze low energy properties of the system at an average
filling of 3/2 atoms per lattice site. We identify the region of the parameter
space where the system has non-Abelian Fibonacci anyon excitations that
correspond to fractional domain walls between different charge-density-waves.
When such one-dimensional systems are combined into a two-dimensional network,
braiding of Fibonacci anyon excitations has potential application for fault
tolerant, universal, topological quantum computation. Contrary to previous
calculations, our results also demonstrate that super-solid phases are not
present in the phase diagram for the discussed 3/2 average filling. Instead,
decreasing the value of the nearest-neighbor tunneling strength leads to a
direct, Berezinskii-Kosterlitz-Thouless, super-fluid to charge-density-wave
quantum phase transition.
