Phase Transitions without gap closing in monitored quantum mean-field systems

TL;DR

The paper addresses how alternating unitary dynamics and measurements of an extensive charge shape the nonequilibrium behavior of quantum many-body systems in the thermodynamic limit. It develops a mean-field, semiclassical framework that yields a diffusive drift for the magnetization on the Bloch ball and derives a corresponding Langevin dynamics, linking averaged dynamics to quantum trajectories via a drift-diffusion description. A key finding is the emergence of novel stationary states in the infinite-volume limit that are not tied to a closing Lindbladian gap, with a finite spectral gap Δ=1/2 for the associated Markov operator at the critical point; finite-size spectra do not anticipate these thermodynamic-limit stationary states. The results suggest that monitoring-induced phase transitions in many-body systems may require thermodynamic-limit diagnostics beyond finite-L Lindbladian gaps, and they motivate extensions to short-range systems and broader classes of globally measured observables.

Abstract

We investigate the monitored dynamics of many-body quantum systems in which projective measurements of extensive operators are alternated with unitary evolution. Focusing on mean-field models characterized by all-to-all interactions, we develop a general framework that captures the thermodynamic limit, where a semiclassical description naturally emerges. Remarkably, we uncover novel stationary states, distinct from the conventional infinite-temperature state, that arise upon taking the infinite-volume limit. Counterintuitively, this phenomenon is not linked to the closing of the Lindbladian gap in that limit. We provide analytical explanation for this unexpected behavior.

Figures (3)

• Figure 1: Monitoring via projective measurements of the total magnetization. Left (Averaged dynamics): A probability distribution with an initial given magnetization density $m^z$ spreads diffusively across the Bloch sphere. Right (Quantum trajectories): Quantum trajectories correspond to random walks of $m^z$; in any single realization, the state is always localized on the Bloch-sphere circumferences corresponding to the given $m^z$.
• Figure 2: Left: Finite-size spectral gap $\Delta$ of the Markov operator $W$ in Eq. \ref{['eq:Markov_Mat_el']} as a function of $1/L$, with $L\leq 400$. The data (green points) are consistent with a linear fit (black dashed line), and the extrapolation $\Delta \rightarrow 1/2$ for $L\rightarrow \infty$. Right: First eigenfunction above the gap computed at finite size ($L=300$), versus the corresponding prediction \ref{['eq:p_exc']} for $L\rightarrow \infty$: an irrelevant proportionality constant, necessary to compare the two curves, has been fitted.
• Figure 3: Left: Dynamics of the averaged magnetization $\langle m^z \rangle$ for different system sizes, starting from the fully polarized state. We observe a parametric slowdown as a function of $L$. Right: Relaxation time $t_{\text{rel}}$, identified by $\langle m^z \rangle=1/2$, as a function of $L$ (green markers). The data are consistent with a fit $t_{\text{rel}} = a \log L +b$ (black dashed line).