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.
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.
Measurement-induced phase transitions are nonequilibrium transitions between phases characterized by distinct entanglement scaling behaviors, driven by the competition between unitary dynamics and measurements. Despite recent numerical efforts, how quenched disorder affects these transitions remains unclear. In this work, we study a
d-dimensional noninteracting fermionic system subject to both quenched disorder and continuous monitoring of the local particle density, and derive an effective field theory describing its long-time universal behaviors. We find that the system is governed by the same nonlinear sigma model as in the case of clean monitored fermions, with disorder entering only through a modification of model parameters. This result suggests that the presence or absence of a measurement-induced phase transition is unaffected by the introduction of disorder. In spatial dimension
d=1, the system exhibits only an area law phase and no transition, whereas in
d>1, a transition persists between an area
×log law phase and an area law phase, with the critical measurement rate suppressed by disorder. Numerical simulation confirms the absence of MIPT in
d=1, and the linear shift of the critical measurement rate by disorder scattering rate in higher dimensions predicted by analytical theory.
We analyze the quantum trajectory dynamics of free fermions subject to continuous monitoring. For weak monitoring, we identify a novel dynamical regime of subextensive entanglement growth, reminiscent of a critical phase with an emergent conformal invariance. For strong monitoring, however, the dynamics favors a transition into a quantum Zeno-like area-law regime. Close to the critical point, we observe logarithmic finite size corrections, indicating a Berezinskii-Kosterlitz-Thouless mechanism underlying the transition. This uncovers an unconventional entanglement transition in an elementary, physically realistic model for weak continuous measurements. In addition, we demonstrate that the measurement aspect in the dynamics is crucial for whether or not a phase transition takes place.
Measurement-induced phase transitions have largely been explored for projective or continuous measurements of Hermitian observables, assuming perfect detection without information loss. Yet such transitions also arise in more general settings, including quantum-jump processes with non-Hermitian jump operators, and under inefficient detection. A theoretical framework for treating these broader scenarios has been missing. Here we develop a comprehensive replica Keldysh field theory for general quantum-jump processes in both bosonic and fermionic systems. Our formalism provides a unified description of pure-state quantum trajectories under efficient detection and mixed-state dynamics emerging from inefficient monitoring, with deterministic Lindbladian evolution appearing as a limiting case. It thus establishes a direct connection between phase transitions in nonequilibrium steady states of driven open quantum matter and in measurement-induced dynamics. As an application, we study imbalanced and inefficient fermion counting in a one-dimensional lattice system: monitored gain and loss of fermions occurring at different rates, with a fraction of gain and loss jumps undetected. For imbalanced but efficient counting, we recover the qualitative picture of the balanced case: entanglement obeys an area law for any nonzero jump rate, with an extended quantum-critical regime emerging between two parametrically separated length scales. Inefficient detection introduces a finite correlation length beyond which entanglement, as quantified by the fermionic logarithmic negativity, obeys an area law, while the subsystem entropy shows volume-law scaling. Numerical simulations support our analytical findings. Our results offer a general and versatile theoretical foundation for studying measurement-induced phenomena across a wide class of monitored and open quantum systems.
The competition between unitary quantum dynamics and dissipative stochastic effects, as emerging from continuous-monitoring processes, can culminate in measurement-induced phase transitions. Here, a many-body system abruptly passes, when exceeding a critical measurement rate, from a highly entangled phase to a low-entanglement one. We consider a different perspective on entanglement phase transitions and explore whether these can emerge when the measurement process itself is modified, while keeping the measurement rate fixed. To illustrate this idea, we consider a noninteracting fermionic system and focus on diffusive detection processes. Through extensive numerical simulations, we show that, upon varying a suitable unraveling parameter---interpolating between measurements of different quadrature operators---the system displays a transition from a phase with area-law entanglement to one where entanglement scales logarithmically with the system size. Our findings may be relevant for tailoring quantum correlations in noisy quantum devices and for conceiving optimal classical simulation strategies.