Measurement-induced entanglement phase transitions (MIETs) highlight how local measurements drive quantum systems between area-law and volume-law entangled states. This review surveys MIET in free fermion models, focusing on how unitary hopping competes with measurement-induced non-unitarity. We discuss controversies regarding the existence of MIET in one dimension, the impact of non-Hermitian skin effects, and potential experimental platforms. We conclude with open challenges, including feedback control and higher-dimensional extensions.
Measurement-induced entanglement phase transitions (MIETs) highlight how local measurements drive quantum systems between area-law and volume-law entangled states. This review surveys MIET in free fermion models, focusing on how unitary hopping competes with measurement-induced non-unitarity. We discuss controversies regarding the existence of MIET in one dimension, the impact of non-Hermitian skin effects, and potential experimental platforms. We conclude with open challenges, including feedback control and higher-dimensional extensions.
The competition between unitary time evolution and quantum measurements could induce phase transitions in the entanglement characteristics of quantum many-body dynamics. In this work, we reveal such entanglement transitions in the context of non-Hermitian Floquet systems. Focusing on noninteracting fermions in a representative bipartite lattice with balanced gain/loss and under time-periodic quenches, we uncover rich patterns of entanglement transitions due to the interplay between driving and non-Hermitian effects. Specially, we find that the monotonic increase of quenched hopping amplitude could flip the system between volume-law and area-law entangled Floquet phases, yielding alternated entanglement transitions. Meanwhile, the increase of gain/loss strength could trigger area-law to volume-law reentrant transitions in the scaling behavior of steady-state entanglement entropy, which are abnormal and highly unexpected in nondriven systems. Connections between entanglement transitions and parity-time-reversal (PT) transitions in Floquet spectra are further established. Our findings not only build a foundation for exploring entanglement phase transitions in Floquet non-Hermitian setups, but also provide efficient means to engineer and control such transitions by driving fields. Published by the American Physical Society 2024
Measurement-induced phase transitions in the presence of competition between projective measurement and random unitary evolution have attracted increasing attention due to the rich phenomenology of entanglement structures. However, in open quantum systems with free fermions, a generalized measurement with conditional feedback can induce the skin effect and render the system short-range entangled without any entanglement transition, meaning the system always remains in the ``area law'' entanglement phase. In this work, we demonstrate that power-law long-range hopping does not alter the absence of entanglement transition brought on by the measurement-induced skin effect for systems with open boundary conditions. In addition, for finite-size systems, we discover an algebraic scaling $S(L,L/4)\ensuremath{\sim}{L}^{3/2\ensuremath{-}p}$ when the power-law exponent p of long-range hopping is relatively small. For systems with periodic boundary conditions, we find that the measurement-induced skin effect disappears and observe entanglement phase transitions among ``algebraic law,'' ``logarithmic law,'' and ``area law'' phases.
Recently there has been an intense effort to understand measurement induced transitions, but we still lack a good understanding of non-Markovian effects on these phenomena. To that end, we consider two coupled chains of free fermions, one acting as the system of interest, and one as a bath. The bath chain is subject to Markovian measurements, resulting in an effective non-Markovian dissipative dynamics acting on the system chain which is still amenable to numerical studies in terms of quantum trajectories. Within this setting, we study the entanglement within the system chain, and use it to characterize the phase diagram depending on the ladder hopping parameters and on the measurement probability. For the case of pure state evolution, the system is in an area law phase when the internal hopping of the bath chain is small, while a non-area law phase appears when the dynamics of the bath is fast. The non-area law exhibits a logarithmic scaling of the entropy compatible with a conformal phase, but also displays linear corrections for the finite system sizes we can study. For the case of mixed state evolution, we instead observe regions with both area, and non-area scaling of the entanglement negativity. We quantify the non-Markovianity of the system chain dynamics and find that for the regimes of parameters we study, a stronger non-Markovianity is associated to a larger entanglement within the system.
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.
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, $\ensuremath{\nu}\ensuremath{\approx}1.4$.
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.