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We study the phase diagram of a topological string-net type lattice model inthe presence of geometrically frustrated interactions. These interactions driveseveral phase transitions that reduce the topological order, leading to a richphase diagram including both Abelian ($\mathbb{Z}_2$) and non-Abelian($\text{Ising}\times \overline{\text{Ising}}$) topologically ordered phases, aswell as phases with broken translational symmetry. Interestingly, one of thesephases simultaneously exhibits (Abelian) topological order and long-rangedorder due to translational symmetry breaking, with non-trivial interactionsbetween excitations in the topological order and defects in the long-rangedorder. We introduce a variety of effective models, valid along certain lines inthe phase diagram, which can be used to characterize both topological andsymmetry-breaking order in these phases, and in many cases allow us tocharacterize the phase transitions that separate them. We use exactdiagonalization and high-order series expansion to study areas of the phasediagram where these models break down, and to approximate the location of thephase boundaries.

Frustrated topological symmetry breaking: geometrical frustration and anyon condensation