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

TL;DR

This paper explores the topological phase transitions in a deformed $ ext{Z}_3$ toric code, revealing a complex phase diagram with multiple phases and critical points characterized by conformal field theories and emergent symmetries.

Contribution

It introduces a novel $ ext{Z}_3$ generalization of the Ashkin-Teller model and maps the wavefunction norm to classical partition functions, advancing understanding of topological phase transitions in $ ext{Z}_3$ systems.

Findings

Identifies three distinct phases: toric code, $e$-confined, and $e$-condensed.

Maps phase transitions to classical models with known critical behaviors.

Discovers emergent $U(1)$ 1-form symmetry and Hilbert space fragmentation at critical points.

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.