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arXiv:2411.06332 · Physical review. B./Physical review. B

Quantum feedback induced entanglement relaxation and dynamical phase transition in monitored free fermion chains with a Wannier-Stark ladder

Xuyang Huang, Han-Ze Li, Yu‐Jun Zhao, Shuo Liu, Jianxin Zhong

Published May 5, 2025• 5 citations

Abstract

In recent years, measurement induced entanglement transitions (MIETs) have attracted significant attention. However, the dynamical transition associated with the feedback induced skin effect, which exhibits a wealth of intriguing phenomena, has not been fully understood. In this work, we investigate a dynamical phase transition in a tilted free-fermion chain under measurement-feedback protocols, emphasizing the particle density and entanglement entropy dynamics. We reveal a feedback induced skin effect, enhanced by the Wannier-Stark ladder potential, that creates localization at one boundary and generates an effective pseudoedge under periodic conditions. The observables show a two-stage evolution: a rapid initial logarithmic growth followed by decay into an area-law steady state. Using a rescaling analysis, we pinpoint the critical behavior and offer an intuitive physical picture that links it to the feedback-driven suppression of quantum jump fluctuations. The resulting entanglement dynamics appear to be governed by a system-size-dependent delay, followed by a size-independent relaxation process. This behavior is consistent with the ballistic propagation of free fermions toward a domain-wall-like steady state and does not exhibit any signatures of nontrivial criticality. This work provides an effective supplement to the dynamical transition. It provides valuable references for linking the dynamical understanding of the role feedback plays in MIETs.

DOI: 10.1103/physrevb.111.184302

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In recent years, measurement induced entanglement transitions (MIETs) have attracted significant attention. However, the dynamical transition associated with the feedback induced skin effect, which exhibits a wealth of intriguing phenomena, has not been fully understood. In this work, we investigate a dynamical phase transition in a tilted free-fermion chain under measurement-feedback protocols, emphasizing the particle density and entanglement entropy dynamics. We reveal a feedback induced skin effect, enhanced by the Wannier-Stark ladder potential, that creates localization at one boundary and generates an effective pseudoedge under periodic conditions. The observables show a two-stage evolution: a rapid initial logarithmic growth followed by decay into an area-law steady state. Using a rescaling analysis, we pinpoint the critical behavior and offer an intuitive physical picture that links it to the feedback-driven suppression of quantum jump fluctuations. The resulting entanglement dynamics appear to be governed by a system-size-dependent delay, followed by a size-independent relaxation process. This behavior is consistent with the ballistic propagation of free fermions toward a domain-wall-like steady state and does not exhibit any signatures of nontrivial criticality. This work provides an effective supplement to the dynamical transition. It provides valuable references for linking the dynamical understanding of the role feedback plays in MIETs.

The traditional dynamical phase transition refers to the appearance of singularities in an observable with respect to a control parameter for a late-time state or singularities in the rate function of the Loschmidt echo with respect to time. Here, we study the many-body dynamics in a continuously monitored free fermion system with conditional feedback under open boundary conditions. We surprisingly find a novel dynamical transition from a logarithmic scaling of the entanglement entropy to an area-law scaling as time evolves. The transition, which is noticeably different from the conventional dynamical phase transition, arises from the competition between the bulk dynamics and boundary skin effects. In addition, we find that while quasidisorder or disorder cannot drive a transition for the steady state, a transition occurs for the maximum entanglement entropy during the time evolution, which agrees well with the entanglement transition for the steady state of the dynamics under periodic boundary conditions.

The Wannier-Stark localization dynamically alters the entanglement behavior of non-Hermitian free fermions. Utilizing the single-particle correlation matrix technique, we analyze the effective Hamiltonian of these fermions with a Wannier-Stark ladder. Under open boundary conditions, we observe steady-state half-chain entanglement entropy (EE) and identify two distinct area-law regions and an algebraic scaling region. Finite-sized scaling analysis reveals critical scaling behavior of the half-chain EE. Notably, the system demonstrates unique entanglement characteristics under periodic boundary conditions, which diverge from the

(1+1)d

conformal field theory predictions for Anderson localization. Our findings highlight entanglement phases emerging from the interplay between the non-Hermitian skin effect and disorder-free localization.

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.

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.

We study the entanglement dynamics of a one-dimensional chain of monitored non-interacting complex fermions with random power-law hopping characterized by a decay exponent

α

. For

α\lesssim 1

, in stark contrast with the case of hopping to nearest neighbors, the scaling of the entanglement entropy (EE) of the steady state with system size

L

is faster than logarithmic for any monitoring or disorder strength and it tends towards a linear (volume-law) scaling for sufficiently small

α\lesssim 1/2

. For

α> 3/2

, the EE is in the area-law phase, namely, no scaling with

L

, for any monitoring strength. For

1 < α\lesssim 3/2

, we identify an

α

-dependent measurement-induced phase transition (MIPT) at a critical value of the monitoring strength separating the mentioned area-law and sub-volume-law phases. At this critical point, the EE scales logarithmically with system size, and the density-density correlation function, closely related to the EE, exhibits multifractal features. These results highlight the importance of superdiffusive classical hopping in the entanglement dynamic of quantum many-body systems and also help differentiate its role with respect to conventional sources of entanglement such as genuine quantum non-locality.

A quantum many-body system subject to unitary evolution and repeated local measurements with an increasing rate undergoes a measurement-induced entanglement transition from extensive (or subextensive) to area law entropy scaling. We find that certain open boundary systems under ``generalized monitoring,'' consisting of ``projective monitoring'' and conditional feedback, display an anomalous late-time particle concentration on the edge, reminiscent of the ``skin effect'' in non-Hermitian systems. Such feedback-induced skin effect will suppress the entanglement generation, rendering the system short-range entangled without measurement-induced entanglement transition. While initially emerged in noninteracting models, such skin effect can also occur in chaotic interacting systems and Floquet quantum circuits subjected to random generalized measurements. Since the dynamics of the skin effect do not require postselection and can be observed at the particle number level, the phenomenon is experimentally relevant and accessible in noisy intermediate-scale quantum platforms, such as trapped ions.