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arXiv:2311.00019 · Physical review. B./Physical review. B

Stable many-body localization under random continuous measurements in the no-click limit

Giuseppe De Tomasi, Ivan M. Khaymovich

Published May 10, 2024• 13 citations
arXivHTMLPDF

Abstract

In this work, we investigate the localization properties of a paradigmatic model, coupled to a monitoring environment and possessing a many-body localized (MBL) phase. We focus on the postselected no-click limit with quench random rates, i.e., random gains and losses. In this limit, the system is modeled by adding an imaginary random potential, rendering non-Hermiticity in the system. Numerically, we provide evidence that the system is localized for any finite amount of disorder. To analytically understand our results, we extend the quantum random energy model (QREM) to the non-Hermitian scenario. The Hermitian QREM has been used previously as a benchmark model for MBL. The QREM exhibits a size-dependent MBL transition, where the critical value scales as Wc∼LlnL with system size and presenting many-body mobility edges. We reveal that the non-Hermitian QREM with random gain-loss offers a significantly stronger form of localization, evident in the nature of the many-body mobility edges and the value for the transition, which scales as Wc∼ln1/2L with the system size. Published by the American Physical Society 2024

DOI
10.1103/physrevb.109.174205

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In this work, we investigate the localization properties of a paradigmatic model, coupled to a monitoring environment and possessing a many-body localized (MBL) phase. We focus on the postselected no-click limit with quench random rates, i.e., random gains and losses. In this limit, the system is modeled by adding an imaginary random potential, rendering non-Hermiticity in the system. Numerically, we provide evidence that the system is localized for any finite amount of disorder. To analytically understand our results, we extend the quantum random energy model (QREM) to the non-Hermitian scenario. The Hermitian QREM has been used previously as a benchmark model for MBL. The QREM exhibits a size-dependent MBL transition, where the critical value scales as Wc∼LlnL with system size and presenting many-body mobility edges. We reveal that the non-Hermitian QREM with random gain-loss offers a significantly stronger form of localization, evident in the nature of the many-body mobility edges and the value for the transition, which scales as Wc∼ln1/2L with the system size. Published by the American Physical Society 2024
The many-body localization (MBL) transition is a quantum phase transition involving highly excited eigenstates of a disordered quantum many-body Hamiltonian, which evolve from “extended/ergodic" (exhibiting extensive entanglement entropies and fluctuations) to “localized" (exhibiting area-law scaling of entanglement and fluctuations). The MBL transition can be driven by the strength of disorder in a given spectral range, or by the energy density at fixed disorder – if the system possesses a many-body mobility edge. Here we propose to explore the latter mechanism by using “quantum-quench spectroscopy", namely via quantum quenches of variable width which prepare the state of the system in a superposition of eigenstates of the Hamiltonian within a controllable spectral region. Studying numerically a chain of interacting spinless fermions in a quasi-periodic potential, we argue that this system has a many-body mobility edge; and we show that its existence translates into a clear dynamical transition in the time evolution immediately following a quench in the strength of the quasi-periodic potential, as well as a transition in the scaling properties of the quasi-stationary state at long times. Our results suggest a practical scheme for the experimental observation of many-body mobility edges using cold-atom setups.
Non-Hermitian many-body localization (NH MBL) has emerged as a possible scenario for stable localization in open systems, as suggested by spectral indicators identifying a putative transition for finite system sizes. In this work, we shift the focus to dynamical probes, specifically the steady-state spin current, to investigate transport properties in a disordered, non-Hermitian XXZ spin chain. Through exact diagonalization for small systems and tensor-network methods for larger chains, we demonstrate that the steady-state current remains finite and decays exponentially with disorder strength, showing no evidence of a transition up to disorder values far beyond the previously claimed critical point. Our results reveal a stark discrepancy between spectral indicators, which suggest localization, and transport behavior, which indicates delocalization. This highlights the importance of dynamical observables in characterizing NH MBL and suggests that traditional spectral measures may not fully capture the physics of non-Hermitian systems. Additionally, we observe a noncommutativity of limits in system size and time, further complicating the interpretation of finite-size studies. These findings challenge the existence of NH MBL in the studied model and underscore the need for alternative approaches to understanding localization in non-Hermitian settings.
Building on recent progress in the study of Anderson and many-body localization via the renormalization group (RG), we examine the scaling theory of localization in the quantum Random Energy Model (QREM). The QREM is known to undergo a localization-delocalization transition at finite energy density, while remaining fully ergodic at the center of the spectrum. At zero energy density, we show that RG trajectories consistently flow toward the ergodic phase, and are characterized by an unconventional scaling of the fractal dimension near the ergodic fixed point. When the disorder amplitude is rescaled, as suggested by the forward scattering approximation approach, a localization transition emerges also at the center of the spectrum, with properties analogous to the Anderson transition on expander graphs. At finite energy density, a localization transition takes place without disorder rescaling, and yet it exhibits a scaling behavior analogous to the one observed on expander graphs. The universality class of the model remains unchanged under the rescaling of the disorder, reflecting the independence of the RG from microscopic details. Our findings demonstrate the robustness of the scaling behavior of random graphs and offer new insights into the many-body localization transition.
Many-body-localization (MBL) transitions are studied in a family of single-spin-flip spin- 1 2 models, including the one-dimensional (1D) chain with nearest-neighbor interactions, the quantum dot (QD) model with all-to-all pair interactions, and the quantum random energy model (QREM). We investigate the generalized imbalance that characterizes propagation in Fock space out of an initial basis state and, at the same time, can be efficiently probed by real-space measurements. For all models considered, the average imbalance and its quantum and mesoscopic fluctuations provide excellent indicators for the position of the MBL transition W c ( n ) , where n is the number of spins. Combining these findings with earlier results on level statistics, we determine phase diagrams of the MBL transitions in the n − W plane. Our results provide evidence for a direct transition between the ergodic and MBL phases for each of the models, without any intermediate phase. For QREM and QD model, W c ( n ) grows as a power law of n (with logarithmic corrections), in agreement with analytical predictions W c QREM ( n ) ∼ n 1 / 2 ln n and W c QD ( n ) ≳ n 3 / 4 ln 1 / 2 n . This growth is in stark contrast to the 1D model, where W c ( n ) is essentially independent of n , consistent with the analytic expectation W c 1 D ( n → ∞ ) = const . We also determine the scaling of the transition width Δ W ( n ) / W c ( n ) and estimate the system size n needed to study the asymptotic scaling behavior. Although the corresponding values of n are larger than those accessible to exact simulations on a classical computer, they are within the reach of quantum simulators. Our results indicate feasibility of experimental studies of n − W phase diagrams and scaling properties of MBL transitions in models of 1D and QD type and in their extensions to other spatial geometry or distance-dependent interactions.
The explorations of non-Hermiticity have been devoted to investigate the disorder-induced many-body localization (MBL). However, the sensitivity of the spatial boundary conditions and the interplay of the non-Hermitian skin effect with many-body phenomena are not yet clear. For a MBL system in the presence of nonreciprocal tunnelings and random disorder potential, we identify two different complex-real spectral transitions, one is present for both open and periodic boundaries while the other is present only for open boundaries of coupled non-Hermitian chains. The later is driven due to the interchain coupling at weak disorder where the level statistics of the real eigenenergy phase follows a Gaussian orthogonal ensemble. We further characterize wave functions through the (biorthogonal) inverse participation ratio and fractal dimension, which reveal the suppression of skin effect in the non-Hermitian MBL phase. Finally, we demonstrate that the quench dynamics of the local particle density, spin imbalance, and entanglement entropy also signify the hallmark of the boundary effects and nonergodic character of many-body localization.
We present the non-Hermitian many-body localization (MBL) induced by imaginary disorder. By numerically calculating the entanglement entropy and complex level spacing, we identify a clear thermal-MBL transition: the entanglement entropy from the volume law to area law and the level spacing from the repulsive distribution to the uncorrelated Poisson distribution. In the complex energy plane, there exists a mobility loop separating different phases, which is similar to the mobility edge for the Hermitian MBL. Surprisingly, we find that the confluence of imaginary disorder and the localization length provides tunneling channels for dynamically jumping of the particles from the initial to the final state in both the real and Hilbert spaces. Our work provides an example of non-Hermitian many-body phases induced by imaginary disorder, which offers new possibilities for studying strongly correlated non-Hermitian physics.