Entanglement transitions with free fermions

We use Majorana operators to study entanglement dynamics under random free fermion unitary evolution and projective measurements in one dimension. For certain choices of unitary evolution, namely, those which swap neighboring Majorana operators, and measurements of neighboring Majorana bilinears, one can map the evolution to the statistical model of completely packed loops with crossings (CPLC) and study the corresponding phase diagram. 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. We find that both the Goldstone and area-law phases persist in this new phase diagram, but with a shifted phase boundary. One important qualitative aspect of the new phase boundary is that even for the case of commuting measurements, the Goldstone phase persists up to a finite nonzero measurement rate. This is in contrast with the CPLC, in which noncommuting measurements are necessary for realizing the Goldstone phase. We also numerically compute the correlation length critical exponent at the transition, which we find to be near to that of the CPLC, and give a tentative symmetry-based explanation for some differences in the phase transition line between the CPLC and generalized models.