Triggering Boundary Phase Transitions through Bulk Measurements in 2D 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 $M=\cos \theta Z+\sin \theta X$ 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 $\theta =\pi \mathrm{/}2$ and area-law entanglement for any $\theta <\pi \mathrm{/}2$. Within the area-law phase, a phase transition occurs at ${\theta }_c=1.371$. The phase with $\theta \in ({\theta }_c,\pi \mathrm{/}2)$ 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.