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 $c=4\mathrm{/}5$ (${\mathbb{Z}}_3$ parafermion conformal field theory) and $c=8\mathrm{/}5$, 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.