Measurement-Induced Phase Transition for Free Fermions above One Dimension

A theory of the measurement-induced entanglement phase transition for free-fermion models in $d>1$ dimensions is developed. The critical point separates a gapless phase with ^d-1ln scaling of the second cumulant of the particle number and of the entanglement entropy and an area-law phase with ^d-1 scaling, where is a size of the subsystem. The problem is mapped onto an SU(R) replica nonlinear sigma model in $d+1$ dimensions, with R→1. Using renormalization-group analysis, we calculate critical indices in one-loop approximation justified for d=1+ε with ε1. Further, we carry out a numerical study of the transition for a $d=2$ model on a square lattice, determine numerically the critical point, and estimate the critical index of the correlation length, ν≈1.4.