Boundary and Symmetry Breaking in a Deformed Toric Code

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.

💡 Research Summary

The paper investigates a β‑deformation of Kitaev’s toric code that drives a transition from a Z₂ topologically ordered phase to a trivial phase. The authors place the model on a cylinder, which splits the bulk global 1‑form symmetries into distinct boundary operators. In the undeformed toric code two non‑commuting 1‑form symmetries exist: an electric loop operator Γ_e and a magnetic loop operator Γ_m. Their non‑trivial commutation (a ’t Hooft anomaly) forces one of them to be spontaneously broken, giving rise to the fourfold ground‑state degeneracy on a torus.

The deformation modifies only the star terms, adding a factor e^{‑β²∑Z_i}. This introduces a string tension that suppresses large loop configurations. At a critical coupling β_c = ½ ln(1+√2) the topological entanglement entropy drops abruptly from the toric‑code value to zero, signalling loss of topological order without any explicit symmetry breaking in the bulk Hamiltonian.

By cutting the torus into a cylinder, the two loop operators separate into a set Γ_d that winds around the periodic direction and a set Y_d that runs along the length. The cylinder hosts two degenerate ground states |GS,0⟩ and |GS,1⟩ = Y_d|GS,0⟩, which are eigenstates of Γ_d′ with eigenvalues (−1)^ω. The authors construct two complementary boundary Hamiltonians:

1. Magnetic boundary H_M^∂M = Σ_{s∈∂M} e^{‑β²∑Z_i}(1‑A_s^⊥)e^{‑β²∑Z_i}.
   Here the electric 1‑form symmetry Γ_e remains unbroken, while the deformed magnetic symmetry ˜Γ_m acts non‑unitarily and exchanges the two ground states. As β passes β_c, ˜Γ_m ceases to be a genuine symmetry.

2. Electric boundary H_E^∂M = Σ_{p∈∂M}(1‑B_p^⊥).
   In this case the magnetic symmetry ˜Γ_m is preserved, whereas the electric symmetry Γ_e is spontaneously broken, leading to a doublet |±⟩ that flips under the electric line operator Y_e.

Both boundary terms commute with the bulk Hamiltonian but not with each other, thereby defining two distinct boundary phases (electric‑condensed vs. magnetic‑condensed). By tuning the coefficients J_p and J_s in the full Hamiltonian H = H_blk + J_p H_E^∂M + J_s H_M^∂M, the authors explore boundary phase transitions that accompany the bulk 1‑form symmetry breaking.

To probe the bulk criticality from the boundary, they map the quantum ground‑state expectation values onto correlators of a classical 2D Ising model at inverse temperature β. This mapping enables Monte‑Carlo evaluation of quantities such as the ratio of partition functions and the normalization factors of the magnetic loop operators. Near β_c the magnetic symmetry operator loses its unitarity, confirming the spontaneous breaking of the bulk 1‑form symmetry.

Finally, the authors extract an effective central charge c from the entanglement scaling of the (1+1)‑dimensional boundary Hamiltonian. They find that c is strongly suppressed in the vicinity of β_c, reflecting the influence of the bulk critical point on the boundary conformal field theory, and that c recovers its value (≈½) at strong coupling where the bulk is trivially ordered. This demonstrates that the boundary CFT is sensitive to bulk criticality rather than to the presence of topological order per se.

In summary, the work provides a clear demonstration that (i) the β‑deformation introduces a tunable string tension leading to a topological‑trivial transition, (ii) on an open geometry the bulk 1‑form symmetries decompose into boundary operators whose spontaneous breaking signals the transition, and (iii) holographic analysis of the boundary Hamiltonian captures the bulk critical behavior via the suppression and restoration of the effective central charge. These insights deepen our understanding of the interplay between higher‑form symmetries, boundary conditions, and phase transitions in lattice gauge models.