  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.
