Measurement-protected quantum phases

We introduce a class of hybrid quantum circuits, with random unitaries and projective measurements, which host long-range order in the area-law entanglement phase of the steady state. Our primary example is circuits with unitaries respecting a global Ising symmetry and two competing types of measurements. The phase diagram has an area-law phase with spin-glass order, which undergoes a direct transition to a paramagnetic phase with volume-law entanglement, as well as a critical regime. Using mutual information diagnostics, we find that such entanglement transitions preserving a global symmetry are in universality classes different from those without symmetry. We analyze generalizations of such hybrid circuits to higher dimensions, which allow for coexistence of order and volume-law entanglement, as well as topological order without any symmetry restrictions.