, sharply detect an ergodic to non-ergodic 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

KL_2

uncovers signatures of a possible non-ergodic 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.

DOI: 10.48550/arxiv.2509.08584

Related papers

Spectral transitions of the entanglement Hamiltonian in monitored free fermions

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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 ${\text{KL}}_{1}$ , 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 ${\text{KL}}_{2}$ 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.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 ${\text{KL}}_{1}$, 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 ${\text{KL}}_{2}$ 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.

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\langle \tilde r\rangle

and the Kullback-Leibler divergence

KL_1

KL_2

{\text{KL}}_{1}

, 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

{\text{KL}}_{2}

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.

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

α

. For

α\lesssim 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

L

is faster than logarithmic for any monitoring or disorder strength and it tends towards a linear (volume-law) scaling for sufficiently small

α\lesssim 1/2

. For

α> 3/2

, the EE is in the area-law phase, namely, no scaling with

L

, for any monitoring strength. For

1 < α\lesssim 3/2

, we identify an

α

-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 investigate the entanglement structure and wave function characteristics of continuously monitored free fermions with

\mathrm{U}(1)

symmetry in two spatial dimensions. By deriving the exact fermion replica-quantum master equation, we line out two approaches: (i) a nonlinear sigma model analogous to disordered free fermions, resulting in an

\mathrm{SU}(R)

-symmetric field theory of symmetry class AIII in (

2+1

) space-time dimensions, or (ii) for bipartite lattices, third quantization leading to a non-Hermitian

\mathrm{SU}(2R)

-symmetric Hubbard model. Using exact numerical simulations, we explore the phenomenology of the entanglement transition in two-dimensional (2D) monitored fermions, examining entanglement entropy and wave function inverse participation ratio. At weak monitoring, we observe characteristic

LlogL

entanglement growth and multifractal dimension

{D}_{q}=2

, resembling a metallic Fermi liquid. Under strong monitoring, wave functions localize and the entanglement saturates towards an area law. Between these regimes, we identify a high-symmetry point exhibiting both entanglement growth indicative of emergent conformal invariance and maximal multifractal behavior. While this multifractal behavior aligns with the nonlinear sigma model of the Anderson transition, the emergent conformal invariance is an unexpected feature not typically associated with Anderson localization. These discoveries add a new dimension to the study of 2D monitored fermions and underscore the need to further explore the connection between nonunitary quantum dynamics in

D

dimensions and quantum statistical mechanics in

D+1

dimensions.