Boundary and Symmetry Breaking in a Deformed Toric Code

TL;DR

This work analyzes a solvable deformation of the Kitaev toric code that drives a transition from a $\mathbb{Z}_{2}$ topologically ordered phase to a trivial phase via a tunable string tension controlled by $\beta$. By placing the model on a cylinder, it reveals how bulk $1$-form symmetries split into boundary operators and how boundary condensation and symmetry breaking reorganize under deformation, with a key role played by a degraded $t'\text{Hooft}$ anomaly between electric and magnetic line operators. A holographic (1+1)D boundary theory is used to extract an effective central charge $c(\beta)$ from entanglement scaling, which is strongly suppressed near the bulk critical region $\beta_c$ and recovered at large $\beta$, indicating that boundary entanglement responds primarily to bulk criticality rather than bulk topological order. The findings point to a symmetry-structure collapse mechanism for destroying long-range entanglement and suggest a possible intermediate gapless regime, with implications for understanding transitions out of topological order and for boundary-bulk correspondences in deformed topological phases.

Abstract

This work explores a deformation of the Kitaev toric code that induces a phase transition out of the topologically ordered phase. By placing the model on a cylinder, the bulk global 1-form symmetries separate into distinct boundary operators, allowing us to show that the transition is accompanied by the breaking of one higher-form symmetry. Using a holographic $(1+1)$-dimensional boundary Hamiltonian, we extract an effective central charge and find a pronounced suppression near $β_c$, followed by its restoration at strong coupling, indicating sensitivity to bulk criticality rather than topological order.

Figures (8)

• Figure 1: Electric and magnetic degrees of freedom on the toric code, color online.
• Figure 2: Ratio of the partition functions against the coupling $\beta$.
• Figure 3: Smooth boundary on the cylinder. $B_{p}^{\oslash}$ should be thought at just one site.
• Figure 4: Order parameter $\mathcal{O}_{e}(\beta)$ for $L=64$ on a square lattice for the $\ket{0}$ ground state with varying line distance. Notice that the green and blue points are almost coincident.
• Figure 5: Order parameter at fixed separation $x=L/2$ and varying system size $L=8,16,64$ for a square lattice.