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arXiv:2508.18468 · Physical review. B./Physical review. B

Entanglement dynamics of monitored noninteracting fermions on graphics processing units

Bo Fan, Can Yin, Antonio M. Garcı́a-Garcı́a

Published Feb 27, 2026• 3 citations
arXivHTMLPDF

Abstract

The description of the entanglement dynamics of monitored noninteracting fermions, including the existence of measurement-induced phase transitions (MIPTs), is a challenging problem with conflicting results in the literature. The mapping of the problem onto a non-linear sigma model (NLSM) indicates that relatively large lattice sizes are required to determine the nature of the entanglement entropy (EE) in the thermodynamics limit. Here we address this problem numerically for monitored noninteracting fermions with $U(1)$ symmetry. The use of graphics processing unit (GPU) techniques, even with outdated hardware, makes it possible to reach much larger lattice sizes ($L = 16384$ and $160\times160$ in one (1d) and two (2d) dimensions respectively) than in previous studies which enables us to characterize quantitatively the entanglement dynamics. In 1d, we show that in order to confirm the absence of a MIPT, for both projective and homodyne measurements, predicted by the NLSM it is necessary to reach $L \sim 10000$. In 2d, also as predicted by the NLSM, we observe for both protocols a MIPT at finite monitoring rate characterized by a scale invariant mutual information. The critical monitoring strength depends on the protocol while the critical exponent $ν\approx 1.3$ governing the approach to the MIPT is similar in both cases. These features are not correctly predicted by the NLSM. Our results paves the way for a fully quantitative description of the entanglement dynamics of monitoring quantum systems.

DOI
10.1103/x7vx-tqr4

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The description of the entanglement dynamics of monitored noninteracting fermions, including the existence of measurement-induced phase transitions (MIPTs), is a challenging problem with conflicting results in the literature. The mapping of the problem onto a non-linear sigma model (NLSM) indicates that relatively large lattice sizes are required to determine the nature of the entanglement entropy (EE) in the thermodynamics limit. Here we address this problem numerically for monitored noninteracting fermions with U(1)U(1)U(1) symmetry. The use of graphics processing unit (GPU) techniques, even with outdated hardware, makes it possible to reach much larger lattice sizes (L=16384L = 16384L=16384 and 160×160160\times160160×160 in one (1d) and two (2d) dimensions respectively) than in previous studies which enables us to characterize quantitatively the entanglement dynamics. In 1d, we show that in order to confirm the absence of a MIPT, for both projective and homodyne measurements, predicted by the NLSM it is necessary to reach L∼10000L \sim 10000L∼10000. In 2d, also as predicted by the NLSM, we observe for both protocols a MIPT at finite monitoring rate characterized by a scale invariant mutual information. The critical monitoring strength depends on the protocol while the critical exponent ν≈1.3ν\approx 1.3ν≈1.3 governing the approach to the MIPT is similar in both cases. These features are not correctly predicted by the NLSM. Our results paves the way for a fully quantitative description of the entanglement dynamics of monitoring quantum systems.
We show that the entanglement entropy (EE) of one-dimensional (1d) non-interacting fermions with U(1)U(1)U(1) symmetry in the presence of a disordered or quasi-periodic potential in which the occupation number is being monitored by homodyne or projection protocols is always in an area-law phase so no measurement induced phase transition (MIPT) occurs. The reason for the previously claimed MIPT in these systems was a finite size effect related to the fact that the maximum lattice size L≃500L \simeq 500L≃500 was of the order of the correlation length. By increasing the system size up to L≃18000L \simeq 18000L≃18000, employing Graphics Processing Units (GPUs), and performing a careful finite-size scaling analysis, we find that the critical monitoring strength is consistent with zero, so no MIPT occurs. For the disordered case, these numerical results are fully supported by an analytical calculation based on mapping the problem onto a nonlinear sigma model (NLSM) that confirms the absence of the MIPT for any monitoring or disorder strength. The effect of disorder is captured by a change of symmetry from BDI to AIII, which results in an enhanced correlation length in the weak disorder limit, and by an effective monitoring strength that increases linearly with disorder.
Measurement-induced phase transitions are nonequilibrium transitions between phases characterized by distinct entanglement scaling behaviors, driven by the competition between unitary dynamics and measurements. Despite recent numerical efforts, how quenched disorder affects these transitions remains unclear. In this work, we study a ddd-dimensional noninteracting fermionic system subject to both quenched disorder and continuous monitoring of the local particle density, and derive an effective field theory describing its long-time universal behaviors. We find that the system is governed by the same nonlinear sigma model as in the case of clean monitored fermions, with disorder entering only through a modification of model parameters. This result suggests that the presence or absence of a measurement-induced phase transition is unaffected by the introduction of disorder. In spatial dimension d=1d=1d=1, the system exhibits only an area law phase and no transition, whereas in d>1d>1d>1, a transition persists between an area×\times×log law phase and an area law phase, with the critical measurement rate suppressed by disorder. Numerical simulation confirms the absence of MIPT in d=1d=1d=1, and the linear shift of the critical measurement rate by disorder scattering rate in higher dimensions predicted by analytical theory.
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.
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.
Measurement-induced phase transitions (MIPTs) in monitored quantum dynamics are nonequilibrium phase transitions between quantum-chaotic (volume-law entangled) and entanglement-suppressed, area-law phases. Here we reveal how monitored dynamics are situated within the framework of general far-from-equilibrium, quantum condensed-matter physics. Measurement-induced heating effects scramble the distribution function in generic (interacting) monitored fermion systems, and this enables a simplified symmetry-based description of the dynamics. We demonstrate the equivalence of the Keldysh technique with the conventional Statistical-Mechanics Model for circuits, resulting from a doubled Hilbert-space (Choi-Jamio\l{}kowski) mapping. We illustrate this using the monitored dynamics of interacting fermions with a conserved charge, deriving a unified effective-field theory that captures all phases and phase transitions. The noninteracting counterpart in one-dimensional space only has an area-law phase, with no MIPT. This was explained via an effective nonlinear sigma model replica field theory possessing a very large symmetry. We show that other phases and phase transitions emerge when the replica symmetry is reduced by interactions. The reduced symmetry combines a replica permutation symmetry and charge-conservation within each replica. The former and its spontaneous breaking govern the MIPT, which can be recognized via a separatrix in the renormalization-group flow. The replica-resolved charge conservation dictates the ``charge-sharpening'' transition between two kinds of dynamics, where the global charge information is either hidden or reconstructible from the measurements. The field theory explains why the charge-sharpening transition should occur only in the volume-law phase. Our framework provides a template for other classes of MIPTs and situates these within the arena of nonequilibrium condensed-matter physics.
A theory of the measurement-induced entanglement phase transition for free-fermion models in d>1d>1d>1 dimensions is developed. The critical point separates a gapless phase with ${\ensuremath{\ell}}^{d\ensuremath{-}1}\mathrm{ln}\ensuremath{\ell}$ scaling of the second cumulant of the particle number and of the entanglement entropy and an area-law phase with ${\ensuremath{\ell}}^{d\ensuremath{-}1}$ scaling, where $\ensuremath{\ell}$ is a size of the subsystem. The problem is mapped onto an SU(R)\mathrm{SU}(R)SU(R) replica nonlinear sigma model in d+1d+1d+1 dimensions, with $R\ensuremath{\rightarrow}1$. Using renormalization-group analysis, we calculate critical indices in one-loop approximation justified for $d=1+\ensuremath{\epsilon}$ with $\ensuremath{\epsilon}\ensuremath{\ll}1$. Further, we carry out a numerical study of the transition for a d=2d=2d=2 model on a square lattice, determine numerically the critical point, and estimate the critical index of the correlation length, $\ensuremath{\nu}\ensuremath{\approx}1.4$.