Triggering boundary phase transitions through bulk measurements in two-dimensional cluster states

We investigate the phase diagram at the boundary of an infinite two-dimensional cluster state subject to bulk measurements using tensor network methods. The state is subjected to uniform measurements M=cosθZ+sinθX on the lower boundary qubits and in all bulk qubits. Our results show that the boundary of the system exhibits volume-law entanglement at the measurement angle θ=π/2 and area-law entanglement for any θ<π/2. Within the area-law phase, a phase transition occurs at θc=1.371. The phase with θ∈(θc,π/2) is characterized by a noninjective matrix product state, which cannot be realized as the unique ground state of a one-dimensional local, gapped Hamiltonian. Instead, it resembles a cat state with spontaneous symmetry breaking. These findings demonstrate that the phase diagram of the boundary of a two-dimensional system can be more intricate than that of a standard one-dimensional system.