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arXiv:2309.12391 · Physical review. B./Physical review. B

Entanglement phases, localization, and multifractality of monitored free fermions in two dimensions

K. Chahine, Michael Buchhold

Published Aug 23, 2024• 37 citations
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Abstract

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.

DOI
10.1103/physrevb.110.054313

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We investigate the entanglement structure and wave function characteristics of continuously monitored free fermions with U(1)\mathrm{U}(1)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 SU(R)\mathrm{SU}(R)SU(R)-symmetric field theory of symmetry class AIII in (2+12+12+1) space-time dimensions, or (ii) for bipartite lattices, third quantization leading to a non-Hermitian SU(2R)\mathrm{SU}(2R)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 LlogLLlogLLlogL entanglement growth and multifractal dimension Dq=2{D}_{q}=2Dq​=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 DDD dimensions and quantum statistical mechanics in D+1D+1D+1 dimensions.
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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.
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