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arXiv:2607.07835

Super-Logarithmic Entanglement Scaling in a Monitored Superconducting Chain

Rui-Jing Guo, Zhi-Yuan Wei

quant-phcond-mat.stat-mechcond-mat.supr-conSubmitted Jul 8, 2026• Announced Jul 10, 2026• 0 citations
v1 · Jul 8, 2026v2 · Aug 23, 2026
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Abstract

We develop a Keldysh-replica non-linear sigma model (NLSM) for the entanglement dynamics of a monitored one-dimensional spinful $s$-wave BCS chain in the rare-measurement regime, $γ\ll J,Δ$. Although the clean spinful $s$-wave BCS Hamiltonian belongs to symmetry class CI, spin-resolved measurements and projection to a conserved $f$-sector reduce the effective problem to class C. Starting from the corresponding parent symplectic saddle, we show that measurement backaction and the pairing amplitude impose complementary mass constraints that gap out different fluctuation channels. Their interplay dynamically projects the surviving massless modes onto an $\textrm{SO(R)}$ target manifold in replica space. A one-loop renormalization group analysis of this $\textrm{SO(R)}$ NLSM shows that, in the replica limit $R\to1$, the beta function becomes negative, producing a weak-anti-localization flow. This flow yields a super-logarithmic steady-state entanglement scaling $S(L)\sim \ln^2 L$ in the rare-measurement regime. Our field-theoretic result explains the numerical evidence reported in the companion Letter [arXiv:2604.04375] and shows that a topologically trivial monitored $s$-wave superconductor can realize an $\textrm{SO(R)}$ weak-anti-localizing critical phase without relying on a Wess-Zumino-Witten term.

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Absence of a measurement-induced phase transition in a one-dimensional free fermionic system under phase-dependent quantum state diffusion

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This thesis studies the effects of continuous measurements of local occupation numbers on a one-dimensional free fermionic system with periodic boundary conditions and a fixed number of particles. The specific model that is studied here, is a phase-dependent quantum state diffusion of the free fermionic system. The system is studied by both analytical and numerical means.\\ Analytically, this thesis uses the framework of replica Keldysh field theory in order to obtain a field theoretical description, in terms of a functional integral formulation, that describes the conditional dynamics of the quantum system, which in turn is described by a stochastic Lindblad master equation. From the replica Keldysh field theory, a Gaussian theory that incorporates linear fluctuations around the saddle point of the replica Keldysh action is derived, yielding analytical insights into the behavior of density-density correlations and the entanglement entropy of the system at different system sizes. In addition to this, the effects of non-linear fluctuations around the Gaussian saddle point, in particular the effects of massless Goldstone modes that are described by a non-linear sigma model, are taken into account. The resulting RG flow equation of the NLSM coupling shows that the system enters a strong-measurement phase at exponentially large system sizes $l_*$, resulting in the absence of a measurement-induced phase transition. A one-loop renormalization group analysis leads to a logarithmic correction of the Gaussian theory at large system sizes. Due to this correction, the Fourier space density-density correlation function is expected to display local maxima at momenta $q_{C,\mathrm{max}}$ that scale as $q_{C,\mathrm{max}}\sim \gamma^2$, where $\gamma$ is the measurement rate. \\ As for the entanglement entropy, the Gaussian theory predicts an unbounded logarithmic scaling of the entanglement entropy with the subsystem size $\ell$, resulting in a constant scale-dependent effective central charge $c_\ell \sim \mathrm{const.}$ at large system sizes $\ell\rightarrow \infty$. However, due to the RG correction to the Gaussian theory a local maximum of the central charge at intermediate length scales is formed, where the position $l_{c,\mathrm{max}}$ is expected to scale as $l_{c,\mathrm{max}}\sim \gamma^{-3/2}$.\\ The analytical predictions from the replica Keldysh field theory are backed up by numerical data of a system with $L=1000$ lattice sites and $N=L/2$ particles, obtained by means of a quantum trajectory simulation for different unraveling phases $\varphi\in[0,\pi/2]$ and a variety of different measurement rates $\gamma$. The numerical data clearly shows the predicted scalings $q_{C,\mathrm{max}}\sim \gamma^2$ and $l_{c,\mathrm{max}}\sim \gamma^{-3/2}$ for unraveling phases $\varphi\ne \pi/2$, however modifying the scaling of the central charge maximum to $l_{c,\mathrm{max}}\sim \gamma^{-1}$ for smaller values of $\gamma$. The modified scaling can be explained by an emergent local maximum in the Gaussian prediction for the central charge for the corresponding values of $\varphi$. Conclusively, the thesis comes to the conclusion that there is no measurement-induced phase transition in the studied system. However, for the unraveling $\varphi = \pi/2$ the system is always found to be in the rare-measurement phase, characterized by volume-law entanglement.

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We develop an analytical approach to the study of one-dimensional free fermions subject to random projective measurements of local site occupation numbers, based on the Keldysh path-integral formalism and replica trick.In the limit of rare measurements, γ=J ≪ 1 (where γ is measurement rate per site and J is hopping constant in the tight-binding model), we derive a nonlinear sigma model (NLSM) as an effective field theory of the problem.Its replica-symmetric sector is described by a Uð2Þ=Uð1Þ × Uð1Þ ≃ S 2 sigma model with diffusive behavior, and the replica-asymmetric sector is a two-dimensional NLSM defined on SUðRÞ manifold with the replica limit R → 1.On the Gaussian level, valid in the limit γ=J → 0, this model predicts a logarithmic behavior for the second cumulant of number of particles in a subsystem and for the entanglement entropy.However, the one-loop renormalization group analysis allows us to demonstrate that this logarithmic growth saturates at a finite value ∼ðJ=γÞ 2 even for rare measurements, which corresponds to the area-law phase.This implies the absence of a measurement-induced entanglement phase transition for free fermions.The crossover between logarithmic growth and saturation, however, happens at exponentially large scale, ln l corr ∼ J=γ.This makes this crossover very sharp as a function of the measurement frequency γ=J, which can be easily confused with a transition from the logarithmic to area law in finite-size numerical calculations.We have performed a careful numerical analysis, which supports our analytical predictions.

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We develop a Keldysh-replica non-linear sigma model (NLSM) for the entanglement dynamics of a monitored one-dimensional spinful sss-wave BCS chain in the rare-measurement regime, γ≪J,Δγ\ll J,Δγ≪J,Δ. Although the clean spinful sss-wave BCS Hamiltonian belongs to symmetry class CI, spin-resolved measurements and projection to a conserved fff-sector reduce the effective problem to class C. Starting from the corresponding parent symplectic saddle, we show that measurement backaction and the pairing amplitude impose complementary mass constraints that gap out different fluctuation channels. Their interplay dynamically projects the surviving massless modes onto an SO(R)\textrm{SO(R)}SO(R) target manifold in replica space. A one-loop renormalization group analysis of this SO(R)\textrm{SO(R)}SO(R) NLSM shows that, in the replica limit R→1R\to1R→1, the beta function becomes negative, producing a weak-anti-localization flow. This flow yields a super-logarithmic steady-state entanglement scaling S(L)∼ln⁡2LS(L)\sim \ln^2 LS(L)∼ln2L in the rare-measurement regime. Our field-theoretic result explains the numerical evidence reported in the companion Letter [arXiv:2604.04375] and shows that a topologically trivial monitored sss-wave superconductor can realize an SO(R)\textrm{SO(R)}SO(R) weak-anti-localizing critical phase without relying on a Wess-Zumino-Witten term.
A common view in monitored quantum dynamics is that local measurements suppress entanglement growth. We show that this intuition can fail in a one-dimensional spinful fermionic chain governed by a BCS Hamiltonian with pairing strength ΔΔΔ and subject to continuous, on-site, spin-resolved charge measurements at rate γγγ. Using free-fermion simulations and quasiparticle analysis, we show that pairing suppresses entanglement growth, while measurements suppress pairing. Their competition yields measurement-enhanced entanglement: for Δ>0Δ>0Δ>0, the steady-state entanglement Ss\mathcal{S}_sSs​ increases with γγγ over a finite interval 0<γ<γpeak0<γ<γ_{\mathrm{peak}}0<γ<γpeak​. This occurs because stronger measurements suppress pairing correlations, which would otherwise suppress entanglement growth. Using a nonlinear sigma-model calculation and free-fermion simulations, we provide evidence that for Δ>0Δ>0Δ>0 and small but finite γγγ, the steady-state entanglement scales as Ss(L)∼ln⁡2L\mathcal{S}_s(L)\sim \ln^2 LSs​(L)∼ln2L. This implies that, in this setting, measurement-enhanced entanglement does not persist in the thermodynamic limit.
Entanglement phase transitions driven by quantum measurements have emerged as a central paradigm in open quantum many-body physics. Such phase transitions are well established for systems with finite local Hilbert-space dimensions, such as qubits and fermions, while their realization in bosonic systems with unbounded local occupation numbers remains poorly understood. Even in the absence of interactions, number states of bosons are intrinsically non-Gaussian, preventing the use of standard correlation-matrix approaches. To address this problem, we develop a replica-free Keldysh field-theoretic framework that expresses the Renyi entropy of bosonic systems initialized in on-site Fock states in terms of permanents of matrices constructed from single-particle Green's functions. Applying this framework to a continuously monitored one-dimensional cross-stitch lattice conditioned on the no-click trajectory, we uncover a transition from volume-law to logarithmic entanglement scaling. We show that the transition is controlled by a restructuring of the non-Hermitian spectrum that changes the number of long-lived modes from extensive to finite. In the strongly monitored regime, bosons dynamically condense into a microscopic number of slowest-decaying modes, producing logarithmic entanglement scaling, whereas an extensive manifold of long-lived modes at weak monitoring gives rise to volume-law entanglement. Our results establish a distinct mechanism for measurement-induced entanglement transitions in free bosonic systems and provide a computationally efficient diagnostic of the measurement-induced bosonic condensation.
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.
This thesis studies the effects of continuous measurements of local occupation numbers on a one-dimensional free fermionic system with periodic boundary conditions and a fixed number of particles. The specific model that is studied here, is a phase-dependent quantum state diffusion of the free fermionic system. The system is studied by both analytical and numerical means.\\ Analytically, this thesis uses the framework of replica Keldysh field theory in order to obtain a field theoretical description, in terms of a functional integral formulation, that describes the conditional dynamics of the quantum system, which in turn is described by a stochastic Lindblad master equation. From the replica Keldysh field theory, a Gaussian theory that incorporates linear fluctuations around the saddle point of the replica Keldysh action is derived, yielding analytical insights into the behavior of density-density correlations and the entanglement entropy of the system at different system sizes. In addition to this, the effects of non-linear fluctuations around the Gaussian saddle point, in particular the effects of massless Goldstone modes that are described by a non-linear sigma model, are taken into account. The resulting RG flow equation of the NLSM coupling shows that the system enters a strong-measurement phase at exponentially large system sizes l∗l_*l∗​, resulting in the absence of a measurement-induced phase transition. A one-loop renormalization group analysis leads to a logarithmic correction of the Gaussian theory at large system sizes. Due to this correction, the Fourier space density-density correlation function is expected to display local maxima at momenta qC,maxq_{C,\mathrm{max}}qC,max​ that scale as qC,max∼γ2q_{C,\mathrm{max}}\sim \gamma^2qC,max​∼γ2, where γ\gammaγ is the measurement rate. \\ As for the entanglement entropy, the Gaussian theory predicts an unbounded logarithmic scaling of the entanglement entropy with the subsystem size ℓ\ellℓ, resulting in a constant scale-dependent effective central charge cℓ∼const.c_\ell \sim \mathrm{const.}cℓ​∼const. at large system sizes ℓ→∞\ell\rightarrow \inftyℓ→∞. However, due to the RG correction to the Gaussian theory a local maximum of the central charge at intermediate length scales is formed, where the position lc,maxl_{c,\mathrm{max}}lc,max​ is expected to scale as lc,max∼γ−3/2l_{c,\mathrm{max}}\sim \gamma^{-3/2}lc,max​∼γ−3/2.\\ The analytical predictions from the replica Keldysh field theory are backed up by numerical data of a system with L=1000L=1000L=1000 lattice sites and N=L/2N=L/2N=L/2 particles, obtained by means of a quantum trajectory simulation for different unraveling phases φ∈[0,π/2]\varphi\in[0,\pi/2]φ∈[0,π/2] and a variety of different measurement rates γ\gammaγ. The numerical data clearly shows the predicted scalings qC,max∼γ2q_{C,\mathrm{max}}\sim \gamma^2qC,max​∼γ2 and lc,max∼γ−3/2l_{c,\mathrm{max}}\sim \gamma^{-3/2}lc,max​∼γ−3/2 for unraveling phases φ≠π/2\varphi\ne \pi/2φ=π/2, however modifying the scaling of the central charge maximum to lc,max∼γ−1l_{c,\mathrm{max}}\sim \gamma^{-1}lc,max​∼γ−1 for smaller values of γ\gammaγ. The modified scaling can be explained by an emergent local maximum in the Gaussian prediction for the central charge for the corresponding values of φ\varphiφ. Conclusively, the thesis comes to the conclusion that there is no measurement-induced phase transition in the studied system. However, for the unraveling φ=π/2\varphi = \pi/2φ=π/2 the system is always found to be in the rare-measurement phase, characterized by volume-law entanglement.
We employ an nnn-replica Keldysh field theory to investigate the effects of measurements and decoherence on long distance behaviors of quantum critical states. We classify different measurements and decoherence based on their timescales and symmetry properties, and demonstrate that they can be described by nnn-replica Keldysh field theories with distinct physical and replica symmetries. Low energy effective theories for various scenarios are then derived using the symmetry and fundamental consistency conditions of the Keldysh formalism. We apply this framework to study the critical Ising model in both one and two spatial dimensions. In one dimension, we demonstrate that (1) measurements over a finite period of time along the transverse spin direction do not modify the asymptotic scaling of correlation functions and entanglement entropy, whereas (2) measurements along the longitudinal spin direction lead to an area law entangled phase. We also show that (3) decoherence noises over a finite time can be mapped to specific boundary conditions of a critical Ashkin-Teller model, and the entanglement characteristics of the resulting mixed state can be determined. For measurements and decoherence over an extensive time, we demonstrate that (4) the von Neumann entanglement entropy of a large subsystem can exhibit a (sub-)dominant logarithmic scaling in the stationary state for weak measurement (decoherence) performed in a basis that is symmetric under the Ising symmetry, but (5) reduces to an area law for measurements and decoherence in the longitudinal direction. Our results demonstrate that the Keldysh formalism is a useful tool for systematically studying the effects of measurements and decoherence on long-wavelength physics.
We develop an analytical approach to the study of one-dimensional free fermions subject to random projective measurements of local site occupation numbers, based on the Keldysh path-integral formalism and replica trick.In the limit of rare measurements, γ=J ≪ 1 (where γ is measurement rate per site and J is hopping constant in the tight-binding model), we derive a nonlinear sigma model (NLSM) as an effective field theory of the problem.Its replica-symmetric sector is described by a Uð2Þ=Uð1Þ × Uð1Þ ≃ S 2 sigma model with diffusive behavior, and the replica-asymmetric sector is a two-dimensional NLSM defined on SUðRÞ manifold with the replica limit R → 1.On the Gaussian level, valid in the limit γ=J → 0, this model predicts a logarithmic behavior for the second cumulant of number of particles in a subsystem and for the entanglement entropy.However, the one-loop renormalization group analysis allows us to demonstrate that this logarithmic growth saturates at a finite value ∼ðJ=γÞ 2 even for rare measurements, which corresponds to the area-law phase.This implies the absence of a measurement-induced entanglement phase transition for free fermions.The crossover between logarithmic growth and saturation, however, happens at exponentially large scale, ln l corr ∼ J=γ.This makes this crossover very sharp as a function of the measurement frequency γ=J, which can be easily confused with a transition from the logarithmic to area law in finite-size numerical calculations.We have performed a careful numerical analysis, which supports our analytical predictions.