We investigate analytically and numerically a one-dimensional periodically driven free-fermionic system subjected to monitoring of the local particle density. Based on the analytical approach that describes the long-wavelength physics of the time-dependent Hamiltonian in the field-theoretical language using the nonlinear sigma-model (NLSM), we reveal that driving does not alter the universality class of the problem. As a consequence, the system retains the area-law behavior in the thermodynamic limit, with an intermediate diffusive regime giving rise to logarithmic growth of entanglement entropy for a small monitoring rate. At the same time, driving leads to a renormalization of the bare coupling constant of the NLSM, which controls the space-time conductivity'' in the diffusive regime. We derive the analytic form of this renormalization, which becomes particularly strong in the case of a maximally symmetric'' drive and sufficiently short driving period. In addition, we employ the Wiener-Hopf method to investigate the ballistic-diffusive crossover. These analytical predictions are corroborated by numerical simulations of the von-Neumann entanglement entropy and the density correlation function. Our numerical results clearly demonstrate that, with an increase in the system size, there are successive crossovers from ballistic to diffusive behavior and ultimately to localization. Furthermore, in the diffusive regime, we observe weak-localization corrections that are in agreement with the analytical predictions of the NLSM. Overall, our results provide a unified analytical and numerical framework for understanding the effects of monitoring in time-modulated fermionic systems, paving a way for broader investigations of driven quantum matter.
It is well known that unitary evolution tends to increase entanglement, whereas continuous monitoring counteracts this growth by pinning the wavefunction trajectories to the eigenstates of the measurement operators. In this work, we investigate the fate of the measurement-induced phase transition in a periodically driven free-fermionic quantum system, where the hopping amplitude is modulated periodically in time using a square pulse. In the high-frequency limit, a renormalization group analysis of the non-Hermitian quantum sine-Gordon model [as proposed in {Phys. Rev. X 11, 041004 (2021)}] reveals that if the hopping amplitude is varied symmetrically around zero, the system always favors the area-law phase, where the steady-state entanglement entropy is independent of subsystem size. In contrast, asymmetry in the drive amplitudes tends to promote entanglement growth. Furthermore, numerical evidence for the system sizes accessible to us suggests that decreasing the drive frequency typically favors entanglement growth. For such driven systems, at least for reasonably small frequency regimes, as a function of measurement strength, we observe a potential signature of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition between a gapless critical phase, characterized by logarithmic growth of entanglement entropy with subsystem size, and a gapped area-law phase. However, it is almost impossible to rule out the possibility that the transition observed here is not an actual thermodynamic transition, but a finite-size crossover between logarithmic to area law entanglement phase. Even in that scenario, the critical length scale beyond which the area law phase prevails increases with the increasing time period of driving. On the other hand, for a symmetric drive, the system consistently exhibits an area-law phase, regardless of the driving frequency.
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.
We explore, both analytically and numerically, the quantum dynamics of a many-body free-fermion system subjected to local density measurements. We begin by extending the mapping to the nonlinear sigma-model (NLSM) field theory for the case of finite evolution time T and different classes of initial states, which lead to different NLSM boundary conditions. The analytical formalism is then used to study how quantum correlations gradually develop, with increasing T , from those determined by the initial state towards their steady-state form. The analytical results are confirmed by numerical simulations for several types of initial states. We further consider the long-time limit, when the system in d + 1 space-time dimensions becomes quasi-one-dimensional, and analyze the scaling of the “localization” time (which is simultaneously the purification time and the charge-sharpening time for this class of problems). The analytical predictions for scaling properties are fully confirmed by numerical simulations in a d = 2 model around the measurement-induced phase transition. We use this dynamical approach to determine numerically the measurement-induced transition point and the associated correlation-length critical exponent.
We investigate entanglement dynamics in continuously monitored open quantum systems featuring current-carrying nonequilibrium states. We focus on a prototypical one-dimensional model of boundary-driven noninteracting fermions with monitoring of the local density, whose average Lindblad dynamics features a well-studied ballistic to diffusive crossover in transport. Here we analyze the dynamics of the fermionic negativity, mutual information, and purity along different quantum trajectories. We show that monitoring this boundary-driven system enhances its entanglement negativity at long times, which otherwise decays to zero in the absence of measurements. This result is in contrast with the case of unitary evolution where monitoring suppresses entanglement production. For small values of $\ensuremath{\gamma}$, the stationary-state negativity shows a logarithmic scaling with system size, transitioning to an area-law scaling as $\ensuremath{\gamma}$ is increased beyond a critical value. Similar critical behavior is found in the mutual information, while the late-time purity shows no apparent signature of a transition, being
O(1) for all values of $\ensuremath{\gamma}$. Our work unveils the double role of weak monitoring in current-driven open quantum systems, simultaneously damping transport and enhancing entanglement.
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.
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.