Measurement-induced entanglement transitions in free-fermion models

There has been recent activity on the entanglement dynamics under random unitary evolution and projective measurements in one dimensional quantum systems. Using Majorana operators, we can study this phenomenon with free-fermion unitary dynamics. For certain choices of unitary evolution, namely those which swap neighboring Majorana operators, we map the evolution to the statistical model of completely packed loops with crossings and study the corresponding phase diagram. This diagram exhibits a transition from an area law to a critical Goldstone phase with a logarithmic entropy scaling law. We generalize this model using the language of Fermionic Gaussian states to a general free fermion unitary evolution acting on neighboring Majorana operators, and numerically compute its phase diagram as well. We find that the phase transition persists, but the transition line occurs at a different location in the diagram.