  We consider the string-net model on the honeycomb lattice for Ising anyons in
the presence of a string tension. This competing term induces a nontrivial
dynamics of the non-Abelian anyonic quasiparticles and may lead to a breakdown
of the topological phase. Using high-order series expansions and exact
diagonalizations, we determine the robustness of this doubled Ising phase which
is found to be separated from two gapped phases. An effective quantum dimer
model emerges in the large tension limit giving rise to two different
translation symmetry-broken phases. Consequently, we obtain four transition
points, two of which are associated with first-order transitions whereas the
two others are found to be continuous and provide examples of recently proposed
Bose condensation for anyons.
