The measurement-induced phase transition in a strongly disordered spin chain

In this work, we investigate the dynamics of a strongly disordered spin chain subject to random local measurements. Using a microscopic and effective many-body localized (MBL) Hamiltonian without conserved quantities beyond energy, we demonstrate that both the prethermal and MBL regimes are unstable under local measurements, resulting in a volume-law entangled steady state. As the measurement rate increases, the system undergoes a measurement-induced phase transition (MIPT) to an area-law entangled phase. We identify the critical measurement rate, marking the transition, and find that it scales exponentially with the disorder strength W and the average overlap O between the measurement operator and the local integrals of motion. Finally, we analyze dynamical time scales in the measurement-induced volume-law phase and show that they exhibit no smooth crossover to the MBL limit, highlighting the fundamentally distinct nature of entanglement dynamics in measured disordered systems.