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arXiv:2605.05306

Measurement-induced phase transitions in disordered fermions

Yunxiang Liao, Max Matheussen, Xinghai Zhang

cond-mat.stat-mechcond-mat.dis-nnquant-phSubmitted May 6, 2026• Announced May 8, 2026• 0 citations
v1 · May 6, 2026v2 · Oct 5, 2026
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Abstract

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 $d$-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 dimensions d>1, a transition occurs between an area x log law phase and an area law phase, whereas in d=1, the system exhibits only an area law phase and no transition. Numerical results further demonstrate that both clean and disordered one-dimensional free fermions exhibit area-law behavior when the system size is large enough.

Comments
18 pages, 3 figures

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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 dimensions d>1, a transition occurs between an area x log law phase and an area law phase, whereas in d=1, the system exhibits only an area law phase and no transition. Numerical results further demonstrate that both clean and disordered one-dimensional free fermions exhibit area-law behavior when the system size is large enough.
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 (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.
Continuous monitoring of one-dimensional free fermionic systems can generate phenomena reminiscent of quantum criticality, such as logarithmic entanglement growth, algebraic correlations, and emergent conformal invariance, but in a nonequilibrium setting. However, whether these signatures reflect a genuine phase of nonequilibrium quantum matter or persist only over finite length scales is an active area of research. We address this question in a free fermionic chain subject to continuous monitoring of lattice-site occupations. An unraveling phase φ interpolates between measurement schemes, corresponding to different stochastic unravelings of the same Lindblad master equation; for φ = 0 , measurements disentangle lattice sites, while for φ = π / 2 they act as unitary random noise, yielding volume-law steady-state entanglement. Using replica Keldysh field theory, we obtain a nonlinear sigma model describing the long-wavelength physics. This analysis shows that for 0 ≤ φ < π / 2 , entanglement ultimately obeys an area law, but only beyond the exponentially large scale ln ( l φ , * ) ∼ J / [ γ cos ( φ ) ] , where J is the hopping amplitude and γ the measurement rate. Resolving l φ , * in numerical simulations is difficult for γ / J → 0 or φ → π / 2 . However, the theory also predicts that critical-like behavior appears below a crossover scale that grows only algebraically in J / γ , making it numerically accessible. Our simulations confirm these predictions, establishing the absence of measurement- or unraveling-induced entanglement transitions in this model.
The competition between quantum many-particle dynamics and continuous monitoring can lead to measurement-induced phase transitions (MIPTs). So far, MIPTs have been extensively explored in fermionic or spin systems. To examine the possibility of an MIPT in bosonic systems, we study the entanglement structure in continuously monitored free bosons with long-range couplings. When the measurement is local, we find that no MIPTs occur because the substantial entanglement generated by the long-range coupling overcomes the entanglement destruction due to the measurement. In contrast, we show that the nonlocal measurement can efficiently suppress the entanglement generation, leading to an MIPT where the bipartite entanglement entropy exhibits the subvolume-to-area law phase transition as the measurement strength is increased. Our numerical results indicate that the critical point should be described by a certain conformal field theory, while the transition does not belong to a conventional universality class such as Berezinskii-Kosterlitz-Thouless class.
The competition between unitary quantum dynamics and dissipative stochastic effects, as emerging from continuous-monitoring processes, can culminate in measurement-induced phase transitions. Here, a many-body system abruptly passes, when exceeding a critical measurement rate, from a highly entangled phase to a low-entanglement one. We consider a different perspective on entanglement phase transitions and explore whether these can emerge when the measurement process itself is modified, while keeping the measurement rate fixed. To illustrate this idea, we consider a noninteracting fermionic system and focus on diffusive detection processes. Through extensive numerical simulations, we show that, upon varying a suitable unraveling parameter---interpolating between measurements of different quadrature operators---the system displays a transition from a phase with area-law entanglement to one where entanglement scales logarithmically with the system size. Our findings may be relevant for tailoring quantum correlations in noisy quantum devices and for conceiving optimal classical simulation strategies.
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.