Semiclassical quantum trajectories in the monitored Lipkin-Meshkov-Glick model

Monitored quantum system have sparked great interest in recent years due to the possibility of observing measurement-induced phase transitions (MIPTs) in the full-counting statistics of quantum trajectories. Here, we investigate the dynamics of the Lipkin-Meshkov-Glick model, composed of N all-to-all interacting spins 1/2, under a weak external monitoring. In the thermodynamic limit, we derive a set of semiclassical stochastic equations describing the evolution of the expectation values of global spin observables. Our results show that the limit N→∞ does not commute with the long-time limit: while for any finite N the average over trajectories is expected to converge towards a trivial steady state, in the thermodynamic limit a MIPT appears. The transition is not affected by postselection issues, as it is already visible at the level of ensemble averages. We derive a quantitative theoretical picture explaining the nature of the transition within our semiclassical approach.