Quantum dynamics of monitored free fermions

TL;DR

The paper develops a dynamical, field-theoretic framework for monitored free-fermion systems by mapping their evolution to a $(d+1)$-dimensional nonlinear sigma model (NLSM) and identifying boundary conditions that encode different initial Gaussian states. In the diffusive regime, analytical expressions for density correlations $\mathcal{C}(\boldsymbol{q},T)$ and subsystem charge fluctuations $\mathcal{C}_A^{(2)}(T)$ are derived for maximally mixed, maximally disentangled, and volume-law initial states, with numerical simulations in $d=1$ corroborating the theory. For long times, the dynamics reveal a purification/charge-sharpening time scale $T^{\ast}$ that behaves like a time-direction localization length; finite-size scaling then yields the measurement-induced transition point $\gamma_c$ and the correlation-length exponent $\nu$ in $d=2$, consistent with steady-state analyses. The results establish a consistent link between dynamical development of correlations and steady-state MIPT characteristics, and suggest broad applicability of the NLSM approach to monitored quantum systems. Potential extensions include other symmetry classes, non-Gaussian initial states, stochastic continuous measurements, and spatially non-uniform monitoring protocols.

Abstract

We explore, both analytically and numerically, the quantum dynamics of a many-body free-fermion system subjected to local density measurements. We begin by extending the mapping to the nonlinear sigma-model (NLSM) field theory for the case of finite evolution time $T$ and different classes of initial states, which lead to different NLSM boundary conditions. The analytical formalism is then used to study how quantum correlations gradually develop, with increasing $T$, from those determined by the initial state towards their steady-state form. The analytical results are confirmed by numerical simulations for several types of initial states. We further consider the long-time limit, when the system in $d+1$ space-time dimensions becomes quasi-one-dimensional, and analyze the scaling of the ``localization'' time (which is simultaneously the purification time and the charge-sharpening time for this class of problems). The analytical predictions for scaling properties are fully confirmed by numerical simulations in a $d=2$ model around the measurement-induced phase transition. We use this dynamical approach to determine numerically the measurement-induced transition point and the associated correlation-length critical exponent.

Figures (7)

• Figure 1: Boundary conditions for NLSM field theory: Schematic space-time representation of a monitored evolution subject to different initial conditions. Red dots represent individual measurements. The system is evolved with some measurement rate $\gamma$ from the initial time $t=-T$ to time $t=0$ where the correlation function $\mathcal{C}_{\boldsymbol{x} \boldsymbol{x}^\prime}$ and related observables are studied. The boundary condition at the $t=0$ boundary is always absorbing. (a) Initial state is a maximally mixed state. Due to global current conservation $\hat{\mathcal{J}}(t=-T) = \hat{\mathcal{J}}(t=0)$, schematically represented by arrows, this corresponds to maximal "conductance" between $S$ and $S_0$, which is realized with the absorbing boundary condition for the NLSM at $t = -T$, Eq. \ref{['eq:BC:absorbing']}. (b) Initial state a maximally disentangled (random bitstring) state. It can be realized from an arbitrary random state at $t=-T^\prime < -T$ by evolving it to $t = -T$ with an infinite measurement rate $\gamma \to \infty$. This corresponds to an "insulating" system for $t \in [-T^\prime,-T]$; thus the boundary condition at $t = -T$ is reflecting, Eq. \ref{['eq:BC:reflecting']}. (c) A volume-law pure state can be realized from random bitstring at $t = -T^\prime \to -\infty$ (corresponding to the reflecting boundary, as in (b)) by evolving it unitarily ($\gamma \to 0$) up to time $t = -T$. This corresponds to a perfect conductor at $t \in [-T^\prime,-T]$, so that the boundary condition at $t=-T$ is absorbing. However, all the current that flows down through this boundary eventually gets reflected at $t = -T^\prime$ and returns back to the system homogeneously through the whole $t = -T$ surface. Thus, owing to the total current conservation, the $\boldsymbol{q} = 0$ mode gets reflected, see Eq. \ref{['eq:BC:mixed']}.
• Figure 2: Evolution of the density correlation function $\mathcal{C}(q)$ with time $T$ for a monitored system with a maximally mixed initial state (three top curves in each of the panels) and a random bitstring initial state (three bottom curves in each of the panels). The times $T$ are shown in the legend. (a) Analytical results in the diffusive approximation, Eqs. \ref{['eq:CqAbsorbing']} and \ref{['eq:CqReflecting']} respectively, with the coupling $g$ and the mean-free path $\ell_0$ corresponding to the parameters of numerical simulations. (b) Numerical results for a $d=1$ system of size $L = 2000$ with the Hamiltonian \ref{['eq:H']}, hopping constant $J = 1$ and measurement rate $\gamma = 0.1$. Numerical results for the volume-law pure initial state are indistinguishable from the maximally mixed state, except for the $q=0$ value, $\mathcal{C}(q=0) = 0$, and are not shown here.
• Figure 3: Evolution of the variance of charge fluctuation in a subsystem, $\mathcal{C}_A^{(2)}$, with time $T$ (shown in the legend). In all the panels, the variance $\mathcal{C}_A^{(2)}$ is shown as a function of the subsystem size $\ell_A$. Top panels: analytical, diffusive-approximation results for a maximally mixed initial state [Eq. \ref{['eq:CA2-T-max-mixed']}, panel (a)] and maximally disentangled initial state [Eq. \ref{['eq:CA2-disentangled']}, panel (c)], with parameters corresponding to those used in numerical simulations. Bottom panels: numerical results for a $d=1$ system of size $L = 2000$ with the Hamiltonian \ref{['eq:H']}, hopping constant $J = 1$ and measurement rate $\gamma = 0.1$, for a maximally mixed initial state [panel (b), full lines], maximally entangled pure initial state [panel (b), dotted lines], and maximally disentangled initial state [panel (d), full lines]. Dashed lines in panels (b) and (d) are best fits to Eqs. \ref{['eq:CA2-T-max-mixed']} and \ref{['eq:CA2-disentangled']} with fitting parameters $g$ and $\ell_0$. The values of $g$ obtained from the fits are $\sim 20 - 30\%$ below the diffusive-approximation analytical values used in panels (a,c). Insets in the panels (a) and (b) present the same data as in main plots but shown on a linear scale for $x$ axis, which highlights the volume-law behavior at $\ell_A > vT$.
• Figure 4: Dynamical scaling analysis of MIPT in a $d=2$ monitored system with $J = 1$ (i.e., $v = \sqrt{2}$) (a) Ratio $T^\ast(L,\gamma)/L$ of the purification time scale $T^\ast(L,\gamma)$ to the system size $L$, as a function of measurement rate $\gamma$ for system sizes from $L=20$ to $L=52$ (see legend). The crossing point marked with an arrow provides the position of the transition point $\gamma_c$. (b) Best single-parameter collapse according to Eq. \ref{['eq:TastScaling']}, which allows us to determine the critical measurement rate $\gamma_c$ and the correlation-length critical exponent $\nu$ (the values shown in the figure) with a good accuracy. Black dashed line marks the scaling function $\Psi(x)$ as a guide to the eye.
• Figure 5: Example of long-time quantum dynamics in a $d=2$ system with $J = 1$ on the delocalized side of the MIPT, $\gamma = 2.6$, for different system sizes $L$. (a) Numerical results for time dependence of $\overline{\ln\mathcal{C}^{(2)}(T)}$ characterizing the typical variance of charge fluctuations. Dashed lines show exponential fits $\mathcal{C}^{(2,\text{typ})}(T)\sim\exp\left(-T/T^{\ast}\right)$, from which the values of $T^{\ast}(L,\gamma)$ are obtained. (b) Purification time $T^{\ast}(L,\gamma)$ extracted from fitting of the exponential tail in panel (a) as a function of system size.