  Distinguishing different topologically ordered phases and characterizing
phase transitions between them is a difficult task due to the absence of local
order parameters. In this paper, we use a combination of analytical and
numerical approaches to distinguish two such phases and characterize a phase
transition between them. The "toric code" and "double semion" models are simple
lattice models exhibiting Z2 topological order. Although both models express
the same topological ground state degeneracies and entanglement entropies, they
are distinct phases of matter because their emergent quasi-particles obey
different statistics. For a 1D model, we tune a phase transition between these
two phases and obtain an exact solution to the entire phase diagram, finding a
second-order Ising x Ising transition. We then use exact diagonalization to
study the 2D case and find indications of a first-order transition. We show
that the quasi-particle statistics provides a robust indicator of the distinct
topological orders throughout the whole phase diagram.
