Spectral Transitions of the Entanglement Hamiltonian in Monitored Free Fermions

• URL: http://arxiv.org/abs/2509.08584v1
• Date: Wed, 10 Sep 2025 13:32:24 GMT
• Title: Spectral Transitions of the Entanglement Hamiltonian in Monitored Free Fermions
• Authors: Karim Chahine, Michael Buchhold,
• Abstract summary: We numerically investigate measurement-induced phase transitions in monitored free fermions through the spectral and eigenstate properties of the entanglement Hamiltonian.<n>Together, these results establish the entanglement Hamiltonian as a powerful framework for diagnosing metallic, localized, and multifractal regimes in monitored quantum dynamics.
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• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 non-trivial 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 $\langle \tilde r\rangle$ and the Kullback-Leibler divergence $KL_1$, 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.

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