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arXiv:2502.09401 · Physical review. B./Physical review. B

Entanglement behavior and localization properties in monitored fermion systems

G. Piccitto, Giuliano Chiriacò, Davide Rossini, Angelo Russomanno

Published Nov 18, 2025• 3 citations
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Abstract

We study the stationary bipartite entanglement in various integrable and nonintegrable models of monitored fermions evolving along quantum trajectories. 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 also be captured by the same fit with a very small power-law exponent. For the nonintegrable cases, such as the staggered $t\text{\ensuremath{-}}V$ and the Sachdev-Ye-Kitaev (SYK) models, the numerical limitations prevent us from spanning different orders of magnitude in the system size. Here we fit the asymptotic entanglement versus the measurement strength with a generalized Lorentzian, finding a very good agreement, and then look at the scaling with the size of the fitting parameters. We find two different behaviors: for the SYK we observe a linear increase with the system size, while for the $t\text{\ensuremath{-}}V$ model we see the emergence of traces of an entanglement crossover. In the latter models, we study the localization properties in the Hilbert space through the inverse participation ratio, finding an anomalous-delocalization behavior with no relation with the entanglement properties. Finally, we show that our function also fits very well the system-size dependence of the fermionic logarithmic negativity of a quadratic model in a two-leg ladder geometry, with stroboscopic projective measurements.

DOI
10.1103/3sv9-2tkw

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We study the stationary bipartite entanglement in various integrable and nonintegrable models of monitored fermions evolving along quantum trajectories. 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 also be captured by the same fit with a very small power-law exponent. For the nonintegrable cases, such as the staggered t\ensuremath-Vt\text{\ensuremath{-}}Vt\ensuremath-V and the Sachdev-Ye-Kitaev (SYK) models, the numerical limitations prevent us from spanning different orders of magnitude in the system size. Here we fit the asymptotic entanglement versus the measurement strength with a generalized Lorentzian, finding a very good agreement, and then look at the scaling with the size of the fitting parameters. We find two different behaviors: for the SYK we observe a linear increase with the system size, while for the t\ensuremath-Vt\text{\ensuremath{-}}Vt\ensuremath-V model we see the emergence of traces of an entanglement crossover. In the latter models, we study the localization properties in the Hilbert space through the inverse participation ratio, finding an anomalous-delocalization behavior with no relation with the entanglement properties. Finally, we show that our function also fits very well the system-size dependence of the fermionic logarithmic negativity of a quadratic model in a two-leg ladder geometry, with stroboscopic projective measurements.
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On finite system sizes, monitored one-dimensional free-fermion chains display a broad crossover in entanglement scaling as the measurement rate increases, from sub-extensive behavior at weak monitoring toward an area law at strong monitoring. Analytical field theory and recent large-scale simulations indicate that this apparent change is not a finite-rate transition in the thermodynamic limit. I test, using trajectory-resolved correlation-matrix simulations (chains up to L=96L=96L=96 and up to 128 Born-rule trajectories per parameter point), whether two quantities built from the single-particle entanglement spectrum are useful finite-size diagnostics: the bipartite particle-number fluctuation FA=∑kνk(1−νk)F_A=\sum_kν_k(1-ν_k)FA​=∑k​νk​(1−νk​) and a participation-style effective number of entangling modes, MA=exp⁡[−∑kwkln⁡wk]\mathcal{M}_A=\exp[-\sum_k w_k\ln w_k]MA​=exp[−∑k​wk​lnwk​], with wk∝νk(1−νk)w_k\proptoν_k(1-ν_k)wk​∝νk​(1−νk​). Both quantities track the crossover. However, for trajectory- and time-averaged steady-state values, M\mathcal{M}M is nearly a deterministic function of FFF: a pooled curve explains 99.7\% of its variance across all rates, so M\mathcal{M}M carries little independent information. Its main advantage is statistical: its trajectory-to-trajectory coefficient of variation is up to a factor of ∼2\sim2∼2 smaller than that of the entropy or FFF under strong monitoring. Subsystem-scaling comparisons at L=48L=48L=48--969696 also show that a pure logarithmic law is not statistically preferred under weak monitoring; a small residual quasi-extensive component and a drifting logarithmic coefficient are instead consistent with a crossover rather than a critical phase. Thus, FFF remains the natural experimentally motivated companion to the entropy, whereas M\mathcal{M}M is best viewed as a variance-reduced numerical summary of essentially the same information.
We study the entanglement dynamics of a one-dimensional chain of monitored non-interacting complex fermions with random power-law hopping characterized by a decay exponent ααα. For α≲1α\lesssim 1α≲1, in stark contrast with the case of hopping to nearest neighbors, the scaling of the entanglement entropy (EE) of the steady state with system size LLL is faster than logarithmic for any monitoring or disorder strength and it tends towards a linear (volume-law) scaling for sufficiently small α≲1/2α\lesssim 1/2α≲1/2. For α>3/2α> 3/2α>3/2, the EE is in the area-law phase, namely, no scaling with LLL, for any monitoring strength. For 1<α≲3/21 < α\lesssim 3/21<α≲3/2, we identify an ααα-dependent measurement-induced phase transition (MIPT) at a critical value of the monitoring strength separating the mentioned area-law and sub-volume-law phases. At this critical point, the EE scales logarithmically with system size, and the density-density correlation function, closely related to the EE, exhibits multifractal features. These results highlight the importance of superdiffusive classical hopping in the entanglement dynamic of quantum many-body systems and also help differentiate its role with respect to conventional sources of entanglement such as genuine quantum non-locality.
We study the asymptotic bipartite entanglement entropy of the quantum trajectories of a free-fermionic system, when subject to a continuous nonlocal monitoring. The measurements are described by Gaussian-preserving two-point operators, whose strength decays as a power law with exponent $\ensuremath{\alpha}$. Different behaviors of the entanglement entropy with the system size emerge: for $\ensuremath{\alpha}$ below a given threshold value a volume-law behavior sets in, while for larger $\ensuremath{\alpha}$ we observe a transition from subvolume to area law, whose exact location depends on the measurements rate and on the presence of a Hamiltonian dynamics. We also consider the expectation probability distribution of the measurement operators, and find that this distribution features a transition from a unimodal to a bimodal shape. We discuss the possible connections between this qualitative change of the distribution and the entanglement transition points.
Understanding how quantum chaotic systems generate entanglement can provide insight into their microscopic chaotic dynamics and can help distinguish between different classes of chaotic behavior. Using von Neumann entanglement entropy, we study a nonentangled state evolved under three variants of the Sachdev-Ye-Kitaev (SYK) model with a finite number of Majorana fermions N. All the variants exhibit linear entanglement growth at early times, which at late times saturates to a universal value consistent with random matrix theory (RMT), but their growth rates differ. We interpret this as a large-N effect, arising from the enhanced nonlocality of fermionic operators in SYK and binary SYK, absent in spin operators of the spin-SYK model. Numerically, we find that these differences emerge gradually with increasing N. Although all variants are quantum chaotic, their entanglement dynamics reflect varying scrambling rates and indicate that the entanglement production rate serves as a fine-grained probe of scrambling beyond conventional measures. To probe its effect on the thermalization properties of these models, we study the two-point autocorrelation function, finding no differences between the SYK variants, but deviations from RMT predictions for N≥24, particularly near the crossover from exponential decay to saturation regime.
We investigate the nonequilibrium dynamics of entanglement entropy and out-of-time-order correlator (OTOC) of noninteracting fermions at half-filling starting from a product state to distinguish the delocalized, multifractal (in the limit of nearest neighbor hopping), localized and mixed phases hosted by the quasiperiodic Aubry-André-Harper (AAH) model in the presence of long-range hopping. For sufficiently long-range hopping strength a secondary logarithmic behavior in the entanglement entropy is found in the mixed phases whereas the primary behavior is a power-law the exponent of which is different in different phases. The saturation value of entanglement entropy in the delocalized, multifractal and mixed phases depends linearly on system size whereas in the localized phase (in the short-range regime) it is independent of system size. The early-time growth of OTOC shows very different power-law behaviors in the presence of nearest neighbor hopping and long-range hopping. The late time decay of OTOC leads to noticeably different power-law exponents in different phases. The spatial profile of OTOC and its system-size dependence also provide distinct features to distinguish phases. In the mixed phases the spatial profile of OTOC shows two different dependences on space for small and large distances respectively. Interestingly the spatial profile contains large fluctuations at the special locations related to the quasiperiodicity parameter in the presence of multifractal states.