We study a 2D measurement-only random circuit motivated by the Bacon-Shor error correcting code. We find a rich phase diagram as one varies the relative probabilities of measuring nearest-neighbor Pauli XX and ZZ check operators. In the Bacon-Shor code, these checks commute with a group of stabilizer and logical operators, which therefore represent conserved quantities. Described as a subsystem symmetry, these conservation laws lead to a continuous phase transition between an $X$-basis and $Z$-basis spin-glass order. The two phases are separated by a critical point where the entanglement entropy between two halves of an $L\times L$ system scales as $LlnL$, a logarithmic violation of the area law. We generalize to a model where the check operators break the subsystem symmetries (and the Bacon-Shor code structure). In tension with established heuristics, we find that the phase transition is replaced by a smooth crossover, and the $X$- and $Z$-basis spin-glass orders spatially coexist. Additionally, if we approach the line of subsystem symmetries away from the critical point in the phase diagram, some spin-glass order parameters jump discontinuously.

• Received 4 March 2023
• Accepted 21 July 2023

DOI:https://doi.org/10.1103/PhysRevB.108.024205