Measurement induced entanglement phase transitions have attracted immense attention in recent years, especially in the context of monitored quantum circuits. In such systems, the dynamics due to unitary evolution competes with the localizing effects of measurements. The quantum phase transition between a trivial volume-law phase of entanglement entropy into an area-law obeying quantum Zeno-like phase for frequent and/or strong measurements is well known in many integrable quantum models with unitary dynamics. However recently a critical phase with a logarithmic scaling of the entanglement entropy has been identified in a class of integrable models in the presence of dissipation.
We take this one step further and study the critical transition in non-integrable systems – a one dimensional transverse field Ising model, in presence of an integrability-breaking field and dissipation. We show that the measurement induced transitions in this system is qualitatively different from the trivial volume-law to area-law transition of the entanglement entropy in integrable systems, and present the phase diagram for this non-integrable system.