Monitored dynamical quantum matter of free fermions: symmetry classification and field theory description

Quantum dynamical systems monitored by measurements exhibit novel physics beyond those subject to unitary time evolutions. In particular, measurement-induced phase transitions have attracted tremendous attention recently. While a rich set of exotic phases and phase transitions have been identified numerically in monitored dynamical systems, their analytical understanding remains challenging. In this talk, I will present a general theoretical framework to study the universal behavior of monitored free-fermion dynamics. This framework unifies the description of monitored free-fermion dynamics and that of the classic Anderson localization problems via the Altland-Zirnbauer (AZ) 10-fold symmetry classification. As an application of this framework, I will show that the 1+1d monitored free-fermion dynamics on a Majorana chain and the disordered time-reversal-invariant superconductors in 2 spatial dimensions are described by a pair of closely related continuum theories. Both continuum theories are non-linear sigma models whose target space SO(R) is dictated by their shared AZ symmetry class, which is DIII in this case. The only difference is the "replica limit": the monitored dynamical system requires R->1 while the disordered system requires R->0. The similarity of these continuum theories allows us to identify the counterparts of the monitored 1+1d novel dynamic phases and transition in the 2d disordered system. Despite the similarities between the two systems, the measurement-induced phase transition in the monitored Majorana chains belongs to a novel universality class beyond Anderson localization due to the different replica limits. The results of extensive numerical simulations of the universal critical behavior at the transition will be presented.