Topological phase transition of deformed ${\mathbb Z}_3$ toric code

We investigate topological phase transitions in a family of deformed \mathbb{Z}_3 ℤ 3 toric-code wavefunctions prepared from a cluster state by local deformations and projective measurements. Their norms map to the Q=3 Q = 3 Potts model for single-parameter deformations and to a three-state Ashkin–Teller-like (AT _3 3 ) construction with two independent four-spin couplings in the general case. Projected entangled-pair-state (PEPS) and variational uniform matrix-product-state (VUMPS) calculations identify the toric-code (TC) phase and phases in which electric ( e e ) anyons are confined or condensed. These phases are separated by critical structures with central charges c=4/5 c = 4 / 5 , 8/5 8 / 5 , and isolated c=1 c = 1 antiferromagnetic (AFM) endpoints. A normalized finite-distance e e -anyon pair-state norm provides a Fredenhagen–Marcu-type check of the confinement boundary, while the topological data of the quantum double D(\mathbb{Z}_3) D ( ℤ 3 ) imply a topological entanglement entropy \gamma=\log3 γ = log ⁡ 3 throughout the gapped toric-code phase. Relative to the \mathbb{Z}_2 ℤ 2 case, the absence of sign-change folding leaves the AFM endpoints unfolded, and the extreme deformation reaches square ice with an emergent U(1) U ( 1 ) one-form symmetry, Hilbert-space fragmentation, and exact scar configurations.