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.
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
qC,max that scale as
qC,max∼γ2, where
γ 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, resulting in a constant scale-dependent effective central charge
cℓ∼const. at large system sizes
ℓ→∞. 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,max is expected to scale as
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=1000 lattice sites and
N=L/2 particles, obtained by means of a quantum trajectory simulation for different unraveling phases
φ∈[0,π/2] and a variety of different measurement rates
γ. The numerical data clearly shows the predicted scalings
qC,max∼γ2 and
lc,max∼γ−3/2 for unraveling phases
φ=π/2, however modifying the scaling of the central charge maximum to
lc,max∼γ−1 for smaller values of
γ. 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. 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 the system is always found to be in the rare-measurement phase, characterized by volume-law entanglement.
A theory of the measurement-induced entanglement phase transition for free-fermion models in
d>1 dimensions is developed. The critical point separates a gapless phase with ${\ensuremath{\ell}}^{d\ensuremath{-}1}\mathrm{ln}\ensuremath{\ell}$ scaling of the second cumulant of the particle number and of the entanglement entropy and an area-law phase with ${\ensuremath{\ell}}^{d\ensuremath{-}1}$ scaling, where $\ensuremath{\ell}$ is a size of the subsystem. The problem is mapped onto an
SU(R) replica nonlinear sigma model in
d+1 dimensions, with $R\ensuremath{\rightarrow}1$. Using renormalization-group analysis, we calculate critical indices in one-loop approximation justified for $d=1+\ensuremath{\epsilon}$ with $\ensuremath{\epsilon}\ensuremath{\ll}1$. Further, we carry out a numerical study of the transition for a
d=2 model on a square lattice, determine numerically the critical point, and estimate the critical index of the correlation length, $\ensuremath{\nu}\ensuremath{\approx}1.4$.
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.
We study a one-dimensional lattice system of free fermions subjected to a generalized measurement process: the system exchanges particles with its environment, but each fermion leaving or entering the system is counted. In contrast to the freezing of dynamics due to frequent measurements of lattice site occupation numbers, a high rate of fermion counts induces fast fluctuations in the state of the system. Still, through numerical simulations of quantum trajectories and an analytical approach based on replica Keldysh field theory, we find that instantaneous correlations and entanglement properties of free fermions subjected to fermion counting and local occupation measurements are strikingly similar. We explain this similarity through a generalized Zeno effect induced by fermion counting and a universal long-wavelength description in terms of a nonlinear sigma model. The physical requirements underlying this universal emergent behavior are conservation of the total number of particles in the system and its environment, and conservation of the purity of the state of the system by keeping a full record of all measurement outcomes. For both types of measurement processes, we present strong evidence against the existence of a critical phase with logarithmic entanglement and conformal invariance. Instead, we identify a finite critical range of length scales on which signatures of conformal invariance are observable. While area-law entanglement is established beyond a scale that is exponentially large in the measurement rate, the upper boundary of the critical range is only algebraically large and thus numerically accessible. Our finding that these properties do not rely on particle-number conservation has far-reaching implications for measurement-induced phenomena beyond noninteracting fermions, such as charge sharpening in random quantum circuits or generic interacting systems.
On finite system sizes, monitored one-dimensional free-fermion chains display a broad crossover in entanglement scaling as the measurement rate increases, from sub-extensive behavior at weak monitoring toward an area law at strong monitoring. Analytical field theory and recent large-scale simulations indicate that this apparent change is not a finite-rate transition in the thermodynamic limit. I test, using trajectory-resolved correlation-matrix simulations (chains up to
L=96 and up to 128 Born-rule trajectories per parameter point), whether two quantities built from the single-particle entanglement spectrum are useful finite-size diagnostics: the bipartite particle-number fluctuation
FA=∑kνk(1−νk) and a participation-style effective number of entangling modes,
MA=exp[−∑kwklnwk], with
wk∝νk(1−νk). Both quantities track the crossover. However, for trajectory- and time-averaged steady-state values,
M is nearly a deterministic function of
F: a pooled curve explains 99.7\% of its variance across all rates, so
M carries little independent information. Its main advantage is statistical: its trajectory-to-trajectory coefficient of variation is up to a factor of
∼2 smaller than that of the entropy or
F under strong monitoring. Subsystem-scaling comparisons at
L=48--
96 also show that a pure logarithmic law is not statistically preferred under weak monitoring; a small residual quasi-extensive component and a drifting logarithmic coefficient are instead consistent with a crossover rather than a critical phase. Thus,
F remains the natural experimentally motivated companion to the entropy, whereas
M is best viewed as a variance-reduced numerical summary of essentially the same information.
We study measurement-induced phases of free fermion systems with U ( 1 ) symmetry. Following a recent approach developed for Majorana chains, we derive a field theory description for the purity and bipartite entanglement at large space and timescales. We focus on a multiflavor one-dimensional chain with random complex hoppings and continuous monitoring of the local fermion density. By means of the replica trick, and using the number of flavors as a large parameter controlling our approximations, we derive an effective field theory made up of a SU(N) nonlinear sigma model ( NL σ M ) coupled to fluctuating hydrodynamics. Contrary to the case of noninteracting Majorana fermions, displaying no U ( 1 ) symmetry, we find that the bipartite entanglement entropy satisfies an area law for all monitoring rates but with a nontrivial scaling of entanglement when the correlation length is large. We provide numerical evidence supporting our claims. We briefly show how imposing a reality condition on the hoppings can change the NL σ M and also discuss higher-dimensional generalizations. Published by the American Physical Society 2024