Erratum: Measurement-induced quantum criticality under continuous monitoring [Phys. Rev. B 102, 054302 (2020)]

Figure 2

Time evolutions of the local particle densities $\langle n_j\left(t\right)\rangle$ for a nonintegrable model with $V=0,J^{\prime }=1$, and $L=20$ at half-filling. Panels (a) and (b) show those for a single trajectory with $\gamma =0.05$ and $\gamma =0.5$, respectively. Panels (c) and (d) show their averages over 400 quantum trajectories with standard errors for sites $j=1$ and $j=2$.

Figure 3

Time evolutions of the local particle densities $\langle n_j\left(t\right)\rangle$ for an integrable model with $V=1,J^{\prime }=0$, and $L=20$ as shown in Fig. 2.

Figure 4

Steady-state values of the particle densities $\langle n_j\rangle$ for the $L=20$ systems of (a) a nonintegrable case with $V=0,J^{\prime }=1$ and (b) an integrable case with $V=1,J^{\prime }=0$ for $\gamma =0.05,0.5,5$. The averages are taken over 400 quantum trajectories and over the time interval (a) $t\in [50,100]$ or (b) $t\in [150,200]$.

Figure 5

Time evolutions of the von Neumann entanglement entropies $S(l_A=L/2,t)$ averaged over 800 quantum trajectories for several values of the measurement strength $\gamma$. The data are shown for the $L=20$ systems of nonintegrable models (a) $V=0,J^{\prime }=0.5$, (b) $V=0,J^{\prime }=1$, and (c) $V=0,J^{\prime }=0.5$, and an integrable model (d) $V=1,J^{\prime }=0$. The standard errors are smaller than the widths of the curves.

Figure 6

Steady-state values of the von Neumann entanglement entropies $S(l_A=L/2)$ plotted against half of the system size $L$ for several values of the measurement strength $\gamma$. The averages are taken over 800 quantum trajectories and over the time intervals (a) $t\in [50,100]$ for a nonintegrable model with $V=0,J^{\prime }=1$ and (b) $t\in [150,200]$ for an integrable model with $V=1,J^{\prime }=0$.

Figure 7

Steady-state values of the mutual information $I_{AB}(r_{AB}=L/2)$ and the absolute squares of the connected correlation functions ${\langle X_1{X}_{L/2+1}\rangle }_c$ and ${\langle n_1{n}_{L/2+1}\rangle }_c$ between antipodal sites are plotted as functions of $\gamma$. The data are shown for $L=8–20$ and for the parameter choices of nonintegrable models (a) $V=0,J^{\prime }=0.5$, (b) $V=0,J^{\prime }=1$, and (c) $V=1,J^{\prime }=0.5$, and an integrable model (d) $V=1,J^{\prime }=0$. The averages are taken over 800 quantum trajectories and over the time intervals $t\in [50,100]$ for (a)–(c) and $t\in [150,200]$ for (d).

Figure 8

Steady-state values of the von Neumann entanglement entropy $S\left(l_A\right)$ at the entanglement transitions $\gamma ={\gamma }_c$ are plotted against the logarithm of the chord length $x_A$ of the subsystem $A$ for $L=8–24$. The averages are taken over 400 quantum trajectories and over the time intervals $t\in [50,100]$ for nonintegbrable models (a)–(c) and $t\in [150,200]$ for an integrable model (d). The solid lines are fitting functions of the form ${\alpha }_{S}ln{x}_A+{\beta }_S$.

Figure 9

Data collapses of the steady-state entanglement entropy into the scaling form (23) in the original paper for given estimates of the critical measurement strength ${\gamma }_c$. The data are shown for nonintegrable models (a)–(c) and an integrable model (d). Each data point is the averaged value over 800 quantum trajectories.

Figure 10

Steady-state values of the mutual information and the absolute squares of the connected correlation functions of $X_j$ and $n_j$ at the entanglement transition $\gamma ={\gamma }_c$ are plotted against the chord distance $x_{AB}$ between two sites. The logarithmic scales are used for both axes. Each data point is averaged over 400 quantum trajectories. The solid lines are fitting functions of the form $D{x}_{AB}^{-2\mathrm{\Delta }}$. For the correlation functions, fitting is performed for data points with $ln{x}_{AB}\ge 1$.

Figure 11

Largest eigenvalues of the symmetry-resolved reduced density matrix ${\rho }_A^{\left(N_A\right)}$ for $l_A=L/2$ are shown in the form $-[ln{\lambda }_{\max }^{\left(N_A\right)}\left(l_A\right)+S\left(l_A\right)/2]ln{l}_A$ as functions of $m_A=N_A-l_A/2$. Each data point is taken at $t=100$ for nonintegrable models (a)–(c) and at $t=200$ for an integrable model (d) and then averaged over 400 quantum trajectories. The solid lines are fitting functions quadratic in $m_A$.

Figure 12

Bipartite particle number fluctuation $F_A\left(l_A\right)$ plotted against the logarithm of the chord distance $x_A$ of the subsystem $A$ for $L=8–24$. The data are shown for a nonintegrable model with $V=0$ and $J^{\prime }=0.5$ at the critical measurement strength $\gamma =0.9$. Each data point is the averaged value over 400 quantum trajectories and over the time interval $t\in [50,100]$. The solid line is the fitting function of the form ${\alpha }_{F}ln{x}_A+{\beta }_F$.

Figure 13

Absolute square of the interference amplitude $\langle |A_Q{|}^2\rangle$ defined in Eq. (29) of the original paper is plotted against the measurement strength $\gamma$ for the $L=20$ system of a nonintegrable model with $V=0$ and $J^{\prime }=0.5$. When the $j=k$ contribution ${\sum }_j{\langle n_j\rangle }^2$ is subtracted, $\langle |A_Q{|}^2\rangle$ shows a peak structure. Each data point is the averaged value over 800 quantum trajectories and over the time interval $t\in [50,100]$.

Figure 14

Data collapses of the steady-state entanglement entropy into the scaling form (23) of the original paper, obtained by the search algorithm presented in Ref. [9]. The data are shown for nonintegrable models (a)–(c) and an integrable model (d). Each data point is averaged over 800 quantum trajectories.