Semiclassical Quantum Trajectories in the Monitored Lipkin-Meshkov-Glick Model

TL;DR

This paper explores measurement-induced phase transitions in a monitored Lipkin-Meshkov-Glick model using semiclassical stochastic equations, revealing a non-trivial transition in the thermodynamic limit that is observable experimentally.

Contribution

It introduces a semiclassical stochastic framework for analyzing monitored LMG models, uncovering a non-commuting limit behavior and a measurable phase transition.

Findings

MIPT appears in the thermodynamic limit but not for finite N.

Semiclassical equations accurately describe the dynamics and phase transition.

Transition is observable at the ensemble level, independent of post-selection.

Abstract

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 the quantum trajectories associated with different measurement outcomes. 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. We derive a set of semiclassical stochastic equations describing the evolution of the expectation values of global spin observables, which become exact in the thermodynamic limit. Our results shows that the limit $N\to\infty$ does not commute with the long-time limit: while for any finite $N$ the esamble average over the noise is expected to converge towards a trivial steady state, in the thermodynamic limit a MIPT appears. The transition is not affected by post-selection issues, as it is already visible at the level of ensemble averages, thus paving the way for experimental observations. We derive a quantitative theoretical picture explaining the nature of the transition within our semiclassical picture, finding an excellent agreement with the numerics.