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Study of boundary phase transition in toric code under cylinder geometry via bulk projective measurement is reported. As the frequency of local measurement for bulk qubits is increased, spin-glass type long-range order on the boundaries emerges indicating spontaneous-symmetry breaking (SSB) of Z₂ symmetry. From the lattice gauge theory viewpoint, this SSB is a signal of a transition to Higgs phase with symmetry-protected topological order. We numerically elucidate the properties of this phase transition in detail, especially its criticality, and give a physical picture using nonlocal gauge-invariant symmetry operators. Phase transition in the bulk is also studied and its relationship to the boundary transition is discussed.
Kuno et al. (2024) studied this question.