Entanglement behavior and localization properties in monitored fermion systems

TL;DR

This work addresses how monitoring via weak measurements alters entanglement in fermionic many-body systems by analyzing the asymptotic bipartite entanglement entropy across integrable and nonintegrable models. It introduces a single phenomenological fit, f(L) = AL/(1 + C L^b), to describe the steady-state entanglement as a function of system size and demonstrates, through extensive numerics, that the fit captures transitions among area-law, logarithmic, subvolume, and volume-law regimes in diverse settings. For nonintegrable cases, the authors show that the entanglement as a function of measurement strength gamma can be described by a generalized Lorentzian, and they connect the gamma- and L-dependences through scaling relations of the fit parameters, revealing model-dependent trends and an apparent lack of direct correlation between localization (IPR) and entanglement transitions. They further extend the approach to the fermionic logarithmic negativity in a ladder geometry, confirming the broad applicability of the fitting paradigm to different entanglement measures. The results provide a practical framework for characterizing entanglement phases in monitored quantum systems and offer guidance for interpreting finite-size data in experiments and simulations.

Abstract

We study the asymptotic bipartite entanglement in various integrable and nonintegrable models of monitored fermions. We find that, for the integrable cases, the entanglement versus the system size is well fitted, over more than one order of magnitude, by a function interpolating between a linear and a power-law behavior. Up to the sizes we are able to reach, a logarithmic growth of the entanglement can be also captured by the same fit with a very small power-law exponent. We thus propose a characterization of the various entanglement phases using the fitting parameters. For the nonintegrable cases, as the staggered t-V and the Sachdev-Ye-Kitaev (SYK) models, the numerics prevents us from spanning different orders of magnitude in the size, therefore we fit the asymptotic entanglement versus the measurement strength and then look at the scaling with the size of the fitting parameters. We find two different behaviors: for the SYK we observe a volume-law growth, while for the t-V model some traces of an entanglement transition emerge. In the latter models, we study the localization properties in the Hilbert space through the inverse participation ratio, finding an anomalous delocalization with no relation with the entanglement properties. Finally, we show that our function fits very well the fermionic logarithmic negativity of a quadratic model in ladder geometry with stroboscopic projective measurements.

Figures (12)

• Figure 1: The asymptotic averaged EE for the model in Eqs. \ref{['U1_int:eqn']}. (a) Some examples with the behavior of $\overline{S}_{L/2}$ versus $L$ (circles) for different values of $\gamma$, together with the corresponding fits with Eq. \ref{['eq:fit_function']} (continuous lines) where we fix the parameter $b$ as $b=1$. (b) The length scale $L_0=1/C$ ---with $C$ obtained by fitting the curves in panel (a) with Eq. \ref{['eq:fit_function']}--- versus $\gamma$ in a double-logarithmic plot. The straight line results from the fit of the data for $\gamma < 0.1$ and corresponds to a power law of the form $L_0\sim \gamma^{-0.88}$. We simulate the time evolution until $t_f = 6 \times 10^6$, with a step $\delta t = 0.01$. Here and in the next figures, we work in units of $J=1$.
• Figure 2: The EE for the model in Eqs. \ref{['Z2_int:eqn']}. (a) $\overline{S}_{L/4}$ versus $L$ (circles) for different values of $h$ and fixed $\gamma=1.5$, together with the corresponding fits with Eq. \ref{['eq:fit_function']} (continuous lines). (b) The fitting parameter $b$ ---obtained by fitting the curves in panel (a) with Eq. \ref{['eq:fit_function']}--- versus $h$. (c) The length scale $L_0$ [see Eq. \ref{['l0:eqn']}] versus $h$ obtained with the numerical fit. We evolve up to $t_f = 60$ with a step $\delta t = 0.05$ and take $N_{\rm r}=100$.
• Figure 3: The EE for the model in Eqs. \ref{['eq:Ham_Z2_int']} and \ref{['Z2_int_lr:eqn']}. (a) $\overline{S}_{L/2}$ versus $L$ (circles) for different values of $\alpha$ and fixed $\gamma = 0.1$, $h = 0.5$, together with the corresponding fits with Eq. \ref{['eq:fit_function']} (continuous lines). (b) The fitting parameter $b$ ---obtained by fitting the curves in panel (a) with Eq. \ref{['eq:fit_function']}--- versus $\alpha$. (c) The length scale $L_0$ [see Eq. \ref{['l0:eqn']}] versus $\alpha$ obtained with the numerical fit. We simulate the time evolution until $t_f = 3 \times 10^3$, with a step $\delta t = 0.005$. The shaded areas locate the two crossover regions found in Ref. Russomanno2023_longrange for the same set of parameters used here.
• Figure 4: The EE for the model in Eqs. \ref{['U1_nonint:eqn']}. Some examples of $\overline{S}_{L/2}$ versus $\gamma$ (circles), for various system sizes up to $L=20$ (see legend), together with the corresponding fits with the generalized Lorentzian function in Eq. \ref{['eq:fit_function_nonint']} (continuous lines). We simulate the time evolution until $t_f = 2 \times 10^3$ with a step $\delta t = 0.01$, while we set $W = V = 1$.
• Figure 5: The EE for the model in Eqs. \ref{['eq:SYK_model']}. We show some examples of $\overline{S}_{L/2}$ versus $\gamma$ (circles), for various sizes up to $L=20$ (see legend), together with the corresponding fits with Eq. \ref{['eq:fit_function_nonint']} (continuous lines). We simulate the time evolution until $t_f = 3.4 \times 10^2$, with a step $\delta t = 0.01$, and fix $J=1$.