Transition of a three-dimensional $Z_N$ topologically ordered phase to a trivial phase

Topologically ordered quantum systems have robust physical properties, such as quasiparticle statistics and ground-state degeneracy, which do not depend on the microscopic details of the Hamiltonian. We consider a three-dimensional $Z_N$ topological phase under a string tension $g$. First we calculate the modular matrices S and T using tensor network methods and these matrices can serve as order parameters to determine the critical string tension $g_c$. The obtained transition agrees with results from a mapping to a three-dimensional classical N-state Potts model.