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arXiv:2407.08703 · Physical review. B./Physical review. B

Measurement-induced entanglement transition in chaotic quantum Ising chain

Manali Malakar, Marlon Brenes, Dvira Segal, Alessandro Silva

Published Oct 22, 2024• 6 citations
arXivHTMLPDF

Abstract

We numerically investigate the robustness against various perturbations of measurement-induced phase transition in monitored quantum Ising models in the no-click limit, where the dynamics is described by a non-Hermitian Hamiltonian. We study perturbations that break the integrability and/or the symmetry of the model, as well as modifications in the measurement protocol, characterizing the resulting chaos and lack of integrability through the dissipative spectral form factor. We show that while the measurement-induced phase transition and its properties appear to be broadly insensitive to lack of integrability and breaking of the ${\mathbb{Z}}_{2}$ symmetry, a modification of the measurement basis from the transverse to the longitudinal direction makes the phase transition disappear altogether.

DOI
10.1103/physrevb.110.134316

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We numerically investigate the robustness against various perturbations of measurement-induced phase transition in monitored quantum Ising models in the no-click limit, where the dynamics is described by a non-Hermitian Hamiltonian. We study perturbations that break the integrability and/or the symmetry of the model, as well as modifications in the measurement protocol, characterizing the resulting chaos and lack of integrability through the dissipative spectral form factor. We show that while the measurement-induced phase transition and its properties appear to be broadly insensitive to lack of integrability and breaking of the Z2{\mathbb{Z}}_{2}Z2​ symmetry, a modification of the measurement basis from the transverse to the longitudinal direction makes the phase transition disappear altogether.
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Monitored quantum many-body systems display a rich pattern of entanglement dynamics, which is unique to this nonunitary setting. This work studies the effect of quantum jumps on the entanglement dynamics beyond the no-click limit corresponding to a deterministic non-Hermitian evolution. To this aim, we introduce a new tool that looks at the statistics of entanglement-entropy gain and loss after and in between quantum jumps. This insight allows us to build a simple stochastic model of a random walk with partial resetting, which reproduces the entanglement dynamics, and to dissect the mutual role of jumps and non-Hermitian evolution on the entanglement scaling. We apply these ideas to the study of measurement-induced transitions in monitored fermions. We demonstrate that significant deviations from the no-click limit arise whenever quantum jumps strongly renormalize the non-Hermitian dynamics, as in the case of models with U ( 1 ) symmetry at weak monitoring. On the other hand, we show that the weak-monitoring phase of the Ising chain leads to a robust subvolume logarithmic phase due to weakly renormalized non-Hermitian dynamics. Published by the American Physical Society 2024
We study dynamical phase transitions occurring in the stationary state of the dynamics of integrable many-body non-hermitian Hamiltonians, which can be either realized as a no-click limit of a stochastic Schrödinger equation or using spacetime duality of quantum circuits. In two specific models, the Transverse Field Ising Chain and the Long Range Kitaev Chain, we observe that the entanglement phase transitions occurring in the stationary state have the same nature as that occurring in the vacuum of the non-hermitian Hamiltonian: bounded entanglement entropy when the imaginary part of the quasi-particle spectrum is gapped and a logarithmic growth for gapless imaginary spectrum. This observation suggests the possibility to generalize the area-law theorem to non-Hermitian Hamiltonians.
We study an entanglement phase transition in a class of chaotic non-Hermitian spin chains whose spin-spin coupling terms commute with the non-Hermitian contributions. Two representative models are investigated: the transverse-field Ising model with a complex longitudinal field and the non-Hermitian XX model with a transverse field. By analyzing their complex spectra, we find that both models undergo a dissipation-induced gapless-gapped phase transition when the transverse field exceeds a model-dependent threshold. Interestingly, the complex gap does not vary monotonically with the dissipation rate; instead, it exhibits pronounced oscillations before entering the gapped phase. By simulating their non-unitary dynamics, we show that the steady-state entanglement entropy undergoes a transition from volume-law to area-law scaling as the dissipation rate increases. Moreover, several unexpected features emerge within the volume-law regime: a larger complex gap or dissipation rate may lead to a more entangled steady state. We trace these unusual behaviors of the complex gap and the steady-state entanglement to level crossings between the maximal imaginary level and other spectral levels. Our work uncovers an exotic entanglement transition in chaotic non-Hermitian many-body systems.
We numerically investigate measurement-induced phase transitions in monitored free fermions through the spectral and eigenstate properties of the entanglement Hamiltonian. By analyzing entanglement scaling, we identify three nontrivial fixed points, for which the entanglement follows exact analytical scaling forms: chaotic unitary dynamics at infinitesimal monitoring, characterized by a Gaussian Page law; a Fermi-liquid fixed point at moderate monitoring, defining a metallic phase with logarithmic entanglement growth and emergent space-time invariance; and a quantum Lifshitz fixed point marking the measurement-induced phase transition into a localized area-law phase. Adopting a random-matrix perspective on the entanglement Hamiltonian, we show that short-range spectral correlations, such as the adjacent gap ratio $\ensuremath{\langle}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{r}\ensuremath{\rangle}$ and the Kullback-Leibler divergence KL1{\text{KL}}_{1}KL1​, sharply detect an ergodic to nonergodic phase transition at the quantum Lifshitz fixed point, and yield precise estimates of the critical point and correlation length exponent. Long-range probes, including the spectral form factor and the associated Thouless time, corroborate this picture, while the variant KL2{\text{KL}}_{2}KL2​ uncovers signatures of a possible nonergodic extended (multifractal) regime at intermediate monitoring strengths. Together, these results establish the entanglement Hamiltonian as a powerful framework for diagnosing metallic, localized, and multifractal regimes in monitored quantum dynamics, and highlight unexpected scaling structures, Fermi liquid, and Lifshitz criticality, that lie beyond current field-theoretic approaches.