Topological phase transition of deformed {mathbb Z}₃ toric code

We investigate the topological phase transitions of the deformed mathbb{Z}₃ toric code, constructed by applying local deformations to the mathbb{Z}₃ cluster state followed by projective measurements. Using the loop-gas and net configuration framework, we map the wavefunction norm to classical partition functions: the Q=3 Potts model for single-parameter deformations and a novel mathbb{Z}₃ generalization of the Ashkin-Teller model AT$₃$ for the general two-parameter case. The phase diagram, obtained via the projected entangled pair state (PEPS) representation and the variational uniform matrix product state (VUMPS) method, exhibits three phases - the toric code phase, an e-confined phase, and an e-condensed phase - separated by critical lines with central charges c=4/5 $mathbb{Z}₃$ parafermion conformal field theory and c=8/5, along with isolated antiferromagnetic critical points at c=1 $mathbb{Z}₄$ parafermion conformal field theory. At these critical points, the system reduces to a square ice model with an emergent U(1) 1-form symmetry, exhibiting Hilbert space fragmentation and quantum many-body scar states. Unlike the mathbb{Z}₂ case, the absence of a sign-change duality leads to a richer phase structure.