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

• URL: http://arxiv.org/abs/2603.09107v1
• Date: Tue, 10 Mar 2026 02:39:46 GMT
• Title: Topological phase transition of deformed ${\mathbb Z}_3$ toric code
• Abstract summary: We investigate the topological phase transitions of the deformed $mathbbZ_3$ toric code.<n>We map the wavefunction norm to classical partition functions.<n>Unlike the $mathbbZ_3$ case, the absence of a sign duality leads to a richer phase structure.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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