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

We investigate topological phase transitions in a family of deformed

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.