Entanglement Transition in a Monitored Free-Fermion Chain: From Extended Criticality to Area Law

We analyze the quantum trajectory dynamics of free fermions subject to continuous monitoring. For weak monitoring, we identify a novel dynamical regime of subextensive entanglement growth, reminiscent of a critical phase with an emergent conformal invariance. For strong monitoring, however, the dynamics favors a transition into a quantum Zeno-like area-law regime. Close to the critical point, we observe logarithmic finite size corrections, indicating a Berezinskii-Kosterlitz-Thouless mechanism underlying the transition. This uncovers an unconventional entanglement transition in an elementary, physically realistic model for weak continuous measurements. In addition, we demonstrate that the measurement aspect in the dynamics is crucial for whether or not a phase transition takes place.

Figure 1

(a) Free fermions hopping on a chain of length $L$ subject to continuous monitoring with dimensionless rate $\gamma$. (b) Schematic “phase diagram” showing the different regimes of entanglement scaling with $L$ (the dashed line denotes a finite size crossover). (c) At small monitoring rate, a subextensive growth of the entropy $\sim \log (L)$ at sufficiently large $L$ is reminiscent of a critical, conformally invariant phase. For small $\gamma$, $L$, extensive growth $\sim L$ is observed (inset), approaching a volume law as $\gamma \to 0$. (d),(e) The effective central charge and residual entropy obtained by fitting the data to Eq. (2). The blue lines in (d) and (e) correspond to the nonunitary circuit evolution (QSDc), for which the transition is absent. The insets show the same data on a linear (d) and logarithmic (e) scale [35].

Figure 2

The conformal invariance at weak monitoring is confirmed (a) by a large, nonzero mutual information $l_A=l_B=L/2$, which rapidly decays to zero in the area-law regime, and (b) by a scaling collapse of the mutual information as a function of the cross ratio $\eta$, i.e., $\mathcal{I}(\eta )\sim \eta$ ($L=400$). (c) In the area-law regime no collapse is observed. (d) Equal-time correlations $\overline{C}(l,0)$ decay algebraically $\sim l^{-2}$ (exponentially) with the distance $l$ in the conformally invariant (area-law) regime ($L=800$). The inset shows a data collapse for different system sizes $L=200$, 400, 600, 800 (axes range as in main plot).

Figure 3

(a) The entanglement entropy as a function of the bipartition size $l$ reveals a clear, asymptotic logarithmic growth for weak monitoring and shows a transition to an area law for stronger monitoring $\gamma \ge {\gamma }_c$ (inset). (b) Finite size scaling collapse of the entanglement entropy, assuming BKT scaling of the correlation length and ${\gamma }_c=0.31$ (the inset shows the unrescaled data). (c) Finite size scaling of the effective central charge, predicting a jump of $c(\gamma )$ from $c({\gamma }_c-0^{+})\approx 2/{\gamma }_c$ to $c({\gamma }_c+0^{+})=0$ in the limit $L\to \infty$ (dotted lines are guides to the eye). The parameters are $\alpha =3.99$, ${\gamma }_c=0.21$, $g(L)=\{1+1/[2\log (L)-4.37]{\}}^{-1}$, and the legend from (b) applies in (c).