Quantum dynamics of monitored free fermions: Evolution of quantum correlations and scaling at measurement-induced phase transition

TL;DR

This paper investigates the quantum dynamics of monitored free fermions, analyzing how quantum correlations evolve and scale during measurement-induced phase transitions through analytical and numerical methods.

Contribution

It extends the nonlinear sigma-model mapping to finite times and different initial states, providing new insights into correlation development and critical scaling at phase transitions.

Findings

Analytical predictions for correlation scaling are confirmed numerically.

The measurement-induced transition point and critical exponent are determined.

Long-time dynamics reveal quasi-one-dimensional behavior and localization scaling.

Abstract

We explore, both analytically and numerically, the quantum dynamics of a many-body free-fermion system subjected to local density measurements. We begin by extending the mapping to the nonlinear sigma-model (NLSM) field theory for the case of finite evolution time $T$ and different classes of initial states, which lead to different NLSM boundary conditions. The analytical formalism is then used to study how quantum correlations gradually develop, with increasing $T$, from those determined by the initial state towards their steady-state form. The analytical results are confirmed by numerical simulations for several types of initial states. We further consider the long-time limit, when the system in $d+1$ space-time dimensions becomes quasi-one-dimensional, and analyze the scaling of the ``localization'' time (which is simultaneously the purification time and the charge-sharpening time for this class of problems). The analytical predictions for scaling properties are fully confirmed by numerical simulations in a $d=2$ model around the measurement-induced phase transition. We use this dynamical approach to determine numerically the measurement-induced transition point and the associated correlation-length critical exponent.