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

Measurement induced entanglement phase transitions (MIPT) are a class of recently discovered theoretical dynamical transitions in quantum many-body systems, in which a unitary quantum circuit's evolution is interspersed with measurements. The unitary dynamics competes against the localization of the wavefunction due to repeated measurements, resulting in a transition from the quantum entangled (volume-law) phase into a disentangled Zeno-like (area-law) phase at strong measurements, that’s unsuitable for further quantum operations. Recently an extended critical phase with a logarithmic scaling of the entanglement entropy has been identified in a class of integrable models with dissipative dynamics. We extend this 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 these transitions can be connected via the integrability breaking field. We also identify the same phase transitions from the correlation function exponents in each phase, and present the complete phase diagram for this non-integrable system.