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arXiv:2503.23807 · Physical review. B./Physical review. B

Measurement-induced phase transitions for free fermions in a quasiperiodic potential

Toranosuke Matsubara, Kazuki Yamamoto, Akihisa Koga

Published Aug 1, 2025• 4 citations
arXivHTMLPDF

Abstract

We study the dynamics under continuous measurements for free fermions in a quasiperiodic potential by using the Aubry-Andr\'e-Harper model with hopping rate $J$ and potential strength $V$. On the basis of the quantum trajectory method, we obtain the phase diagram for the steady-state entanglement entropy and demonstrate that robust logarithmic system-size scaling emerges up to a critical potential strength ${V}_{c}/J\ensuremath{\sim}2.3$. Moreover, we find that the measurement induces entanglement phase transitions from the logarithmic-law phase to the area-law phase for the potential strength $V<{V}_{c}$, while any finite measurement stabilizes the area-law phase for $V>{V}_{c}$. This result is distinct from the entanglement scaling in the unitary limit, where volume-law and area-law phases undergo a transition at $V/J=2$. To further support the phase diagram, we analyze the connected correlation function and find that it shows algebraic decay in the logarithmic-law phase, while it decays quickly in the area-law phase. Our results can be tested in ultracold atoms by introducing quasiperiodic potentials and continuously monitoring the local occupation number with an off-resonant probe light.

DOI
10.1103/3zfd-3hqt

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We study the dynamics under continuous measurements for free fermions in a quasiperiodic potential by using the Aubry-Andr'e-Harper model with hopping rate JJJ and potential strength VVV. On the basis of the quantum trajectory method, we obtain the phase diagram for the steady-state entanglement entropy and demonstrate that robust logarithmic system-size scaling emerges up to a critical potential strength Vc/J\ensuremath∼2.3{V}_{c}/J\ensuremath{\sim}2.3Vc​/J\ensuremath∼2.3. Moreover, we find that the measurement induces entanglement phase transitions from the logarithmic-law phase to the area-law phase for the potential strength V<VcV<{V}_{c}V<Vc​, while any finite measurement stabilizes the area-law phase for V>VcV>{V}_{c}V>Vc​. This result is distinct from the entanglement scaling in the unitary limit, where volume-law and area-law phases undergo a transition at V/J=2V/J=2V/J=2. To further support the phase diagram, we analyze the connected correlation function and find that it shows algebraic decay in the logarithmic-law phase, while it decays quickly in the area-law phase. Our results can be tested in ultracold atoms by introducing quasiperiodic potentials and continuously monitoring the local occupation number with an off-resonant probe light.
It is well known that unitary evolution tends to increase entanglement, whereas continuous monitoring counteracts this growth by pinning the wavefunction trajectories to the eigenstates of the measurement operators. In this work, we investigate the fate of the measurement-induced phase transition in a periodically driven free-fermionic quantum system, where the hopping amplitude is modulated periodically in time using a square pulse. In the high-frequency limit, a renormalization group analysis of the non-Hermitian quantum sine-Gordon model [as proposed in {Phys. Rev. X 11, 041004 (2021)}] reveals that if the hopping amplitude is varied symmetrically around zero, the system always favors the area-law phase, where the steady-state entanglement entropy is independent of subsystem size. In contrast, asymmetry in the drive amplitudes tends to promote entanglement growth. Furthermore, numerical evidence for the system sizes accessible to us suggests that decreasing the drive frequency typically favors entanglement growth. For such driven systems, at least for reasonably small frequency regimes, as a function of measurement strength, we observe a potential signature of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition between a gapless critical phase, characterized by logarithmic growth of entanglement entropy with subsystem size, and a gapped area-law phase. However, it is almost impossible to rule out the possibility that the transition observed here is not an actual thermodynamic transition, but a finite-size crossover between logarithmic to area law entanglement phase. Even in that scenario, the critical length scale beyond which the area law phase prevails increases with the increasing time period of driving. On the other hand, for a symmetric drive, the system consistently exhibits an area-law phase, regardless of the driving frequency.
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.
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The authors investigate here the dynamics of free fermions in a random potential system under continuous monitoring. Repeated measurements are known to induce an entanglement phase transition from a logarithmically entangled critical state to area law, while disorder leads to Anderson localization. They show that their interplay results in a stable critical phase at finite disorder strength and dissipation, which drive a transition consistent with Berezinskii-Kosterlitz-Thouless universality. This work is important for testing such phase transitions in quantum dot arrays and nanowires.
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.
We investigate entanglement phase transitions from volume-law to area-law entanglement in a quantum many-body state under continuous position measurement on the basis of the quantum trajectory approach. We find the signatures of the transitions as peak structures in the mutual information as a function of measurement strength, as previously reported for random unitary circuits with projective measurements. At the transition points, the entanglement entropy scales logarithmically and various physical quantities scale algebraically, implying emergent conformal criticality, for both integrable and nonintegrable one-dimensional interacting Hamiltonians; however, such transitions have been argued to be absent in noninteracting regimes in some previous studies. With the aid of U(1)U(1)U(1) symmetry in our model, the measurement-induced criticality exhibits a spectral signature resembling a Tomonaga-Luttinger liquid theory from symmetry-resolved entanglement. These intriguing critical phenomena are unique to steady-state regimes of the conditional dynamics at the single-trajectory level and are absent in the unconditional dynamics obeying the Lindblad master equation, in which the system ends up with the featureless, infinite-temperature mixed state. We also propose a possible experimental setup to test the predicted entanglement transition based on the subsystem particle-number fluctuations. This quantity should readily be measured by the current techniques of quantum gas microscopy and is in practice easier to obtain than the entanglement entropy itself.