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

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.

Figures (14)

• Figure 1: (Left panel) Vertex and plaquette stabilizers, $a_v$ and $b_p$, in the toric code model. (Right panel) The lattice and stabilizers following the dual transformation $\mathcal{D}_{e-m}$. The original lattice is shown with dotted lines, while the dualized lattice is depicted with solid lines.
• Figure 2: Expansion of the cluster state on the Lieb lattice for (a) the $\mathbb{Z}_2$ case and (b) the $\mathbb{Z}_3$ case. Each term in the product of Eq. \ref{['eq:z2-cluster']} (or Eq. \ref{['eq:z3-cluster']}) corresponds to a dimer configuration. In (a), empty circles and squares represent $|+\rangle$ and $|0\rangle$, while filled ones represent $|-\rangle$ and $|1\rangle$; orange lines denote links where $Z_{v_l}X_l Z_{v_l'}$ is applied. In (b), circles and squares with three shadings represent the $\mathbb{Z}_3$ eigenstates $|\omega^0\rangle$, $|\omega^1\rangle$, $|\omega^2\rangle$ and $|0\rangle$, $|1\rangle$, $|2\rangle$, respectively; blue arrows indicate the direction of the $ZXZ$ dimers.
• Figure 3: Visualization of the rotated basis states $|\theta^0\rangle$, $|\theta^1\rangle$, and $|\theta^2\rangle$ as vectors in the three-dimensional vector space. (Left) Perspective view showing the three vectors tilted at an angle $\theta$ from the $z$-axis. (Right) Top-down view along the $z$-axis, highlighting the $120^\circ$ rotational symmetry about the $[111]$-axis. As $\theta$ varies, the mutual overlap $\langle \theta^a | \theta^b \rangle$ interpolates between $-\frac{1}{2}$ (coplanar) and $1$ (collinear), with $0$ (orthonormal) at $\theta = 0$.
• Figure 4: Loop-gas representation of the $\mathbb{Z}_3$ toric code ground state $|{\rm TC}_3\rangle$ as an equal superposition of closed-loop configurations $|c_i\rangle$. The numbers at the center of each plaquette indicate how many times (mod 3) the operator $b_p$ has been applied. Blue arrows denote directed loops: links with rightward (downward) arrows carry $|1\rangle$, links with leftward (upward) arrows carry $|2\rangle$, and unoccupied links correspond to $|0\rangle$. Empty, gray, and black squares represent the local states $|0\rangle$, $|1\rangle$, and $|2\rangle$, respectively.
• Figure 5: Net configurations arising from the overlap $\langle \tilde{\Psi}(\beta_z)|\tilde{\Psi}(\beta_z)\rangle$ under the $\beta_z$ deformation. Each overlap $\langle \tilde{c}_i|\tilde{c}_i\rangle$ is computed from the corresponding loop configuration in Fig. \ref{['fig:z3-loop']}. The directional information of the loops is lost in the overlap, yielding undirected orange nets that may include branching points. The $\mathbb{Z}_3$ numbers on the plaquettes correspond to classical spin variables in the mapping to the $Q=3$ Potts model.