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arXiv:2410.07334 · Physical review. B./Physical review. B

Measurement-induced transitions for interacting fermions

Igor Poboiko, Paul Pöpperl, I. V. Gornyi, A. D. Mirlin

Published Jan 8, 2025• 31 citations
arXivHTMLPDF

Abstract

The authors develop here a unifying field-theoretical framework to study effects of quantum measurements on interacting fermionic systems. The interaction reduces the problem's symmetry, dramatically affecting the phase diagram and scaling of observables. It leads to information-charge separation and stabilization of the volume-law phase for entanglement. In one-dimensional systems, it establishes measurement-induced transitions for entanglement and charge fluctuations. Analytical results are supported by numerical simulations utilizing matrix product states. This work paves the way to exploring measurement-induced phenomena in generic settings.

DOI
10.1103/physrevb.111.024204

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The authors develop here a unifying field-theoretical framework to study effects of quantum measurements on interacting fermionic systems. The interaction reduces the problem's symmetry, dramatically affecting the phase diagram and scaling of observables. It leads to information-charge separation and stabilization of the volume-law phase for entanglement. In one-dimensional systems, it establishes measurement-induced transitions for entanglement and charge fluctuations. Analytical results are supported by numerical simulations utilizing matrix product states. This work paves the way to exploring measurement-induced phenomena in generic settings.
Measurement-induced phase transitions (MIPTs) in monitored quantum dynamics are nonequilibrium phase transitions between quantum-chaotic (volume-law entangled) and entanglement-suppressed, area-law phases. Here we reveal how monitored dynamics are situated within the framework of general far-from-equilibrium, quantum condensed-matter physics. Measurement-induced heating effects scramble the distribution function in generic (interacting) monitored fermion systems, and this enables a simplified symmetry-based description of the dynamics. We demonstrate the equivalence of the Keldysh technique with the conventional Statistical-Mechanics Model for circuits, resulting from a doubled Hilbert-space (Choi-Jamio\l{}kowski) mapping. We illustrate this using the monitored dynamics of interacting fermions with a conserved charge, deriving a unified effective-field theory that captures all phases and phase transitions. The noninteracting counterpart in one-dimensional space only has an area-law phase, with no MIPT. This was explained via an effective nonlinear sigma model replica field theory possessing a very large symmetry. We show that other phases and phase transitions emerge when the replica symmetry is reduced by interactions. The reduced symmetry combines a replica permutation symmetry and charge-conservation within each replica. The former and its spontaneous breaking govern the MIPT, which can be recognized via a separatrix in the renormalization-group flow. The replica-resolved charge conservation dictates the ``charge-sharpening'' transition between two kinds of dynamics, where the global charge information is either hidden or reconstructible from the measurements. The field theory explains why the charge-sharpening transition should occur only in the volume-law phase. Our framework provides a template for other classes of MIPTs and situates these within the arena of nonequilibrium condensed-matter physics.
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 ddd-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=1d=1d=1, the system exhibits only an area law phase and no transition, whereas in d>1d>1d>1, a transition persists between an area×\times×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=1d=1d=1, and the linear shift of the critical measurement rate by disorder scattering rate in higher dimensions predicted by analytical theory.
At a generic volume- to area-law entanglement transition in a many-body system, quantum chaos is arrested. We argue that this tends to imply the vanishing of a certain "mass" term in the field theory of the measurement-induced phase transition (MIPT) for monitored, interacting fermions. To explore this idea, we consider the MIPT with no conserved quantities that describes 1D monitored, interacting Majorana fermions in class DIII. This is the most general problem of interacting fermions with weak fermion parity measurements. Without interactions, it is known that a noninteracting MIPT separates the area-law phase from a log-enhanced "thermal metal" phase at sufficiently weak monitoring. We conjecture that the MIPT with interactions is the same as the noninteracting one in this case; the volume-law phase arises through the dangerously irrelevant mass. The physical picture is that the mass represents a local Fermi's golden rule interparticle scattering rate density that is tantamount to the entangling rate density. The latter must vanish continuously at a continuous MIPT. On the other hand, the field theory capturing the MIPT for monitored fermions with additional continuous symmetries is expected to be different, because the interactions introduce additional terms associated to conserved Noether currents. We propose numerical tests of our conjecture. In addition, we analytically identify a candidate noninteracting critical point representing the MIPT, using a controlled εεε-expansion.
Interspersing unitary dynamics with local measurements results in measurement-induced phases and transitions in many-body quantum systems. When the evolution is driven by a local Hamiltonian, two types of transitions have been observed, characterized by an abrupt change in the system size scaling of entanglement entropy. The critical point separates the strongly monitored area-law phase from a volume law or a subextensive, typically logarithmiclike, one at low measurement rates. Identifying the key ingredients responsible for the entanglement scaling in the weakly monitored phase is the key purpose of this work. For this purpose, we consider prototypical one-dimensional spin chains with local monitoring featuring the presence/absence of U(1) symmetry, integrability, and interactions. Using exact numerical methods, the system sizes studied reveal that the presence of interaction is always correlated to a volume law weakly monitored phase. In contrast, noninteracting systems present subextensive scaling of entanglement. Other characteristics, namely integrability or U(1) symmetry, do not play a role in the character of the entanglement phase.
A new class of quantum entanglement transitions separating phases with different entanglement entropy scaling has been observed in recent numerical studies. Despite the numerical efforts, an analytical understanding of such transitions has remained elusive. Here, the authors propose a theory for the area-law to volume-law entanglement transition in many-body systems that undergo both random unitary evolutions and projective measurements. Using the replica method, the authors map analytically this entanglement transition to an ordering transition in a classical statistical mechanics model. They derive the general entanglement scaling properties at the transition and show a solvable limit where this transition can be mapped onto two-dimensional percolation.
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.