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

Abstract

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

arXiv commentv2: revised in response to referee reports at SciPost Physics; additions and replacements highlighted in blue