Measurement-Induced Phase Transition for Free Fermions above One Dimension

Abstract

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 ${\ensuremath{\ell}}^{d\ensuremath{-}1}\mathrm{ln}\ensuremath{\ell}$ scaling of the second cumulant of the particle number and of the entanglement entropy and an area-law phase with ${\ensuremath{\ell}}^{d\ensuremath{-}1}$ scaling, where $\ensuremath{\ell}$ is a size of the subsystem. The problem is mapped onto an $\mathrm{SU}(R)$ replica nonlinear sigma model in $d+1$ dimensions, with $R\ensuremath{\rightarrow}1$. Using renormalization-group analysis, we calculate critical indices in one-loop approximation justified for $d=1+\ensuremath{\epsilon}$ with $\ensuremath{\epsilon}\ensuremath{\ll}1$. 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, $\ensuremath{\nu}\ensuremath{\approx}1.4$.