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Distinguishing different topologically ordered phases and characterizingphase transitions between them is a difficult task due to the absence of localorder parameters. In this paper, we use a combination of analytical andnumerical approaches to distinguish two such phases and characterize a phasetransition between them. The "toric code" and "double semion" models are simplelattice models exhibiting Z2 topological order. Although both models expressthe same topological ground state degeneracies and entanglement entropies, theyare distinct phases of matter because their emergent quasi-particles obeydifferent statistics. For a 1D model, we tune a phase transition between thesetwo phases and obtain an exact solution to the entire phase diagram, finding asecond-order Ising x Ising transition. We then use exact diagonalization tostudy the 2D case and find indications of a first-order transition. We showthat the quasi-particle statistics provides a robust indicator of the distincttopological orders throughout the whole phase diagram.

Numerical study of a transition between Z2 topologically ordered phases