Charge fluctuation and charge-resolved entanglement in a monitored quantum circuit with $U\left(1\right)$ symmetry

We study a ($1+1$)-dimensional quantum circuit consisting of Haar-random unitary gates and projective measurements, both of which conserve a total $U\left(1\right)$ charge and thus have $U\left(1\right)$ symmetry. In addition to a measurement-induced entanglement transition between a volume-law and an area-law entangled phase, we find a phase transition between two phases characterized by bipartite charge fluctuation growing with the subsystem size or staying constant. At this charge-fluctuation transition, steady-state quantities obtained by evolving an initial state with a definitive total charge exhibit critical scaling behaviors akin to Tomonaga-Luttinger-liquid theory for equilibrium critical quantum systems with $U\left(1\right)$ symmetry, such as logarithmic scaling of bipartite charge fluctuation, power-law decay of charge correlation functions, and logarithmic scaling of charge-resolved entanglement whose coefficient becomes a universal quadratic function in a flux parameter. These critical features, however, do not persist below the transition, in contrast to a recent prediction based on replica field theory and mapping to a classical statistical mechanical model.

Figure 1

Schematic picture of the hybrid quantum circuit. The horizontal and vertical axes correspond to a spatial and temporal direction, respectively. An initial Néel state $|1010\cdots 10\rangle$ is evolved by two-site Haar-random unitary gates preserving the $U\left(1\right)$ symmetry (blue rectangles) and by projective measurements of qubits in the Pauli $Z$ basis (red squares).

Figure 2

Time evolution of the von Neumann entanglement entropy $S_A\left(t\right)$ for $L=20, \left|A\right|=L/2$, and $\overline{n}=1/2$. Error bars indicate the standard errors on trajectory average.

Figure 3

Steady-state value of the von Neumann entanglement entropy $S_A$ as a function of the subsystem size $\left|A\right|$ for $L=24$ and $\overline{n}=1/2$.

Figure 4

Steady-state values of (a) the von Neumann mutual information $I(i:j)$, (b) the square of the connected correlation function for Pauli-$X$ operators ${\langle X_i{X}_j\rangle }_c^2$, and (c) that for charge operators ${\langle n_i{n}_j\rangle }_c^2$ between two antipodal sites with $|i-j|=L/2$ for $\overline{n}=1/2$.

Figure 5

Steady-state values of (a) bipartite mutual information $I(A:C)$ and (b) tripartite mutual information $I_3(A:B:C)$ for $\overline{n}=1/2$ under the partition of the system into four subsystems $A, B, C$, and $D$ of the length $L/4$. The insets show scaling collapses of $I(A:C)$ and $I_3(A:B:C)$ against $(p-p_c)L^{1/\nu }$ for which we have used $(p_c,\nu )=(0.151,1.32)$ and $(p_c,\nu )=(0.133,1.59)$, respectively.

Figure 6

(a) Steady-state value of von Neumann entanglement entropy is plotted against the chord length $x_A$ of the subsystem $A$ for $L=8$ to 24. Steady-state values of (b) two-site mutual information and squared correlation functions (c) ${\langle X_i{X}_j\rangle }_c^2$ and (d) ${\langle n_i{n}_j\rangle }_c^2$ are plotted against the chord distance $x_{ij}$ between two sites for $L=8$ to 26. All data are obtained for $\overline{n}=1/2$ at $p=0.13$. The solid lines are fitting functions for data points with $x_A,x_{ij}\ge 5$.

Figure 7

Steady-state value of bipartite charge fluctuation $F_A$ as a function of the subsystem size $\left|A\right|$ for the $L=26$ and $\overline{n}=1/2$.

Figure 8

Steady-state value of subsystem-charge correlation function ${\langle n_A{n}_C\rangle }_c$ between two antipodal regions $A$ and $C$ with $\left|A\right|=\left|C\right|=L/4$ for $\overline{n}=1/2$. The inset shows scaling collapse of ${\langle n_A{n}_C\rangle }_c$ against $(p-p_t)L^{1/{\nu }_t}$ with $(p_t,{\nu }_t)=(0.121,1.85)$.

Figure 9

Steady-state values of (a) bipartite charge fluctuation and (b) charge correlation function $-{\langle n_i{n}_j\rangle }_c$ are plotted against the chord length $x_A$ of the subsystem $A$ and the chord distance $x_{ij}$ between two sites, respectively. We have used data at $p=0.12$ close to the charge-fluctuation transition for $\overline{n}=1/2$ and $L=8$ to 26. The solid lines in both panels are fitting functions obtained by fitting data points with $x_A,x_{ij}\ge 5$.

Figure 10

Steady-state values of (a) von Neumann entanglement entropy and (b) bipartite charge fluctuation plotted against the chord length $x_A$ of the susbsystem $A$ for $\overline{n}=1/2$ at $p=0.0$, 0.03, 0.06, 0.09, and 0.12. Data points for different system sizes are distinguished by markers and those with different measurement rates are distinguished by colors.

Figure 11

Steady-state values of charge correlation function ${\langle n_i{n}_j\rangle }_c$ for $\overline{n}=1/2$ at (a) $p=0.06$ and (b) $p=0.09$. The solid lines are fitting functions of the form $C/x_{ij}^a$, whereas the dashed lines are quadratic functions $C^{\prime }/x_{ij}^2$ shown as reference. (c) Critical exponent $a$ of ${\langle n_i{n}_j\rangle }_c$ as a function of the measurement rate $p$ extracted by assuming a power-law form for data points with $x_{ij}\ge 5$. The horizontal-solid line shows the exponent $a=2$ predicted by TLL theory. The error bars are least squares uncertainties detailed in the main text.

Figure 12

Luttinger parameter $K$ extracted from the asymptotic behaviors of the bipartite charge fluctuation $F_A$, charge correlation function ${\langle n_i{n}_j\rangle }_c$, and $\varphi$ dependence of the charged moments $Z_n\left(\varphi \right)$ with $n=1$, 2, and 3 for $\overline{n}=1/2$.

Figure 13

Steady-state values of the logarithms of the charged moments $M_n\left(\varphi \right)$ with $\varphi =\pi /4$ and (a) $n=1$, (b) $n=2$, and (c) $n=3$ are plotted against the chord length $x_A$ of the subsystem $A$ for $\overline{n}=1/2$ and $L=8$ to 24. For $M_n\left(\varphi \right)$ with $n=2$ and 3, the contribution from $M_n\left(0\right)$ is subtracted.

Figure 14

Steady-state values of ${\beta }_n\left(\varphi \right)$ for $n=1$, 2, and 3 are plotted against ${(\varphi /2\pi )}^2$ at $p=0.12$ for $\overline{n}=1/2$, which are estimated from linear fitting of $M_1\left(\varphi \right)$ and $M_n\left(\varphi \right)-M_n\left(0\right)$ for $n=2,3$ against $ln{x}_A$, using data points with $x_A\ge 5$. The error bars are the least squares uncertainties.

Figure 15

Steady-state values of (a) bipartite mutual information $I(A:C)$ and (b) tripartite mutual information $I_3(A:B:C)$ for $\overline{n}=1/4$ under partition of the system into four contiguous subsystems with $\left|A\right|=\left|B\right|=\left|C\right|=\left|D\right|=L/4$. The insets show their scaling collapses against $(p-p_c)L^{1/\nu }$. The critical point $p_c$ and the critical exponent $\nu$ of the entanglement transition is determined as $(p_c,\nu )=(0.095,1.18)$ from $I(A:C)$ and $(p_c,\nu )=(0.101,1.70)$ from $I_3(A:B:C)$.

Figure 16

(a) Steady-state values of von Neumann entanglement entropy are plotted against the chord length $x_A$ of the subsystem $A$ for $L=12$ to 28. Steady-state values of (b) mutual information and squared correlation functions (c) ${\langle X_i{X}_j\rangle }_c^2$ and (d) ${\langle n_i{n}_j\rangle }_c^2$ are plotted against the chord distance $x_{ij}$ between two sites for $L=12$ to 32. All data are obtained for $\overline{n}=1/4$ at $p=0.1$. The solid lines are fitting functions for data points with $x_A,x_{ij}\ge 7$.

Figure 17

(a) Steady-state values of subsystem-charge correlation function ${\langle n_A{n}_C\rangle }_c$ between two antipodal regions $A$ and $C$ with $\left|A\right|=\left|C\right|=L/4$ for $\overline{n}=1/4$. The inset shows scaling collapse with $p_t=0.071$ and ${\nu }_t=1.81$. Steady-state values of (b) bipartite charge fluctuation and (c) charge correlation function are plotted against $x_A$ and $x_{ij}$, respectively, for $\overline{n}=1/4$ at $p=0.07$. The solid lines are fitting functions obtained from data points with $x_A,x_{ij}\ge 7$.

Figure 18

Critical exponent $a$ of ${\langle n_i{n}_j\rangle }_c$ for $\overline{n}=1/4$ as a function of the measurement rate $p$ extracted by assuming the power-law form $C/x_{ij}^a$ for data points with $x_{ij}\ge 7$. The horizontal-solid line shows the exponent $a=2$ predicted by TLL theory. The error bars correspond to one standard deviation computed from least squares uncertainty.

Figure 19

Steady-state values of (a) tripartite mutual information $I_3(A:B:C)$ and (b) subsystem-charge correlation function for $\overline{n}=1/6$ under the partition of the system into four subsystems $A, B, C$, and $D$ with $\left|A\right|=\left|C\right|=L/6$ and $\left|B\right|=\left|D\right|=L/3$. The inset in (a) shows a magnified view in the vicinity of $I_3(A:B:C)=0$ (horizontal-solid line), and that in (b) shows scaling collapse of ${\langle n_A{n}_C\rangle }_c$ with $p_t=0.051$ and ${\nu }_t=1.78$.

Figure 20

Steady-state values of (a) bipartite charge fluctuation and (b) charge correlation function ${\langle n_i{n}_j\rangle }_c$ are plotted against the chord length $x_A$ of the subsystem $A$ and the chord distance $x_{ij}$ between two sites, respectively. We have used data at $p=0.05$ close to the charge-fluctuation transition for $\overline{n}=1/6$ and $L=12$ to 36. The solid lines in both panels are fitting functions obtained by using data points with $x_A,x_{ij}\ge 8$.

Figure 21

Estimated value of the critical measurement rate $p_c$ and the correlation length exponent $\nu$ and corresponding minimum of the objective function $Q_{\text{min}}$ for the tripartite mutual information $I_3(A:B:C)$ and the subsystem-charge correlation function ${\langle n_A{n}_C\rangle }_c$ and for each data set specified by the data range $w$ and minimal system length $L_{\text{min}}$. While we have used only $w=0.02,0.04,0.06,0.08,0.1,0.12$, and 0.14, the data points are slightly shifted from these values for visibility.