Measurement induced phase transition with an extended log-law phase in an integrability-broken transverse field Ising model

Measurement induced entanglement phase transitions in many-body quantum systems are an important aspect of study in monitored quantum circuits due to their fundamental role in any quantum device. The unitary dynamics of these many-body systems competes against the localization of the wavefunction due to repeated measurements. Without any measurement, integrable systems stay in entangled or trivial volume-law phase of entanglement entropy. Upon repeated measurements, the system transitions into a disentangled Zeno-like area-law phase. However recently an extended critical phase with a logarithmic scaling of the entanglement entropy has been identified in a class of integrable models where the dynamics becomes non-unitary due to dissipation. We explore this further and study the critical transition in a non-integrable system – a one dimensional transverse field Ising model, in presence of an integrability-breaking field and no-click dissipation. First, 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. Then we show how it is connected to the transition from a critical log-law phase to the area-law phase via the integrability breaking field. We also identify the same phase transitions from the correlation function exponents and calculation of mutual information in each phase, and present the phase diagram for this non-integrable system.