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Transition of a $\mathbb{Z}_3$ topologically ordered phase to trivial and critical phases

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 topological phase transitions under a deformation such as an effective string tension on a $\mathbb{Z}_3$ topological state. This is studied in terms of the gauge-symmetry preserved quantum state renormalization group, first proposed by He, Moradi and Wen [Phys. Rev. B {\bf 90}, 205114 (2014)]. In this approach modular matrices $S$ and $T$ can be obtained and used as order parameters to characterize the topological properties of the phase and determine phase transitions. From a mapping to a classical 2D Potts model on the square lattice, the critical string tension, at which the transition to a topologically trivial phase takes place, can be obtained analytically and agrees with the numerically determined value. Such a transition can be generalized to a $\mathbb{Z}_N$ topological model under a string tension and determined in the same way. With different deformations, the $\mathbb{Z}_3$ topological phase can also be driven to a critical phase which contains, in the large deformation limit, a point analogous to Rokhsar-Kivelson point in the quantum dimer model.