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 ${\ell }^{d-1}\ln \ell$ 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 $\mathrm{SU}(R)$ replica nonlinear 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, $\nu \approx 1.4$.

Figure 1

Correlation function $C(q)$ of an $L\times L$ system with $L=40$ across the transition. Shown are dependencies of $C(q)/g_0\tilde{q}$ on $\tilde{q}{l}_0$ [with $\tilde{q}=2\sin (q/2)$] for values of the measurement rate $\gamma /J$ from 0.5 to 4.0. Dashed lines: cubic polynomial extrapolation down to $q\to 0$ using five lowest momenta. Inset: the extrapolated value at $q\to 0$ as a function of $\gamma /J$. For $\gamma >{\gamma }_c$, where ${\gamma }_c/J\simeq 3.2$, the extrapolated value is zero within numerical precision, which is a manifestation of the measurement-induced phase transition.

Figure 2

Particle number covariance $\overline{G_{AB}}$ between two regions $A$ and $B$ in a geometrical setup shown in upper inset, as a function of ratio $\gamma /J$ for different system sizes $L$. Arrow marks the estimated position of the transition, manifesting itself as a crossing point of all curves,; see Eq. (24). Inset: collapse according to Eq. (24) with a scaling function $\tilde{f}(x^{\nu })$, where $x$ is given by the expression labeling the $x$ axis of the inset. Obtained values of ${\gamma }_c$ and $\nu$ are shown.