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

5+11 pages, 4 figures

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