Transition of a $Z_3$ 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 topological phase transition under a string tension $g$ on a $Z_3$ topological state. This is first studied numerically in terms of the gauge-symmetry preserved quantum state renormalization group proposed by He, Moradi and Wen (arXiv:1401.5557). Modular matrices $S$ and $T$ can be obtained and used as order parameters to determine the critical string tension $g_c$. Then from a mapping to a classical 2D three-state Potts model on square lattice we obtain analytically the transition $g_c$ via the transition temperature of the three-state Potts model. We find the numerically determined $g_c$ agrees well with the analytic result via the mapping.