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arXiv:2307.01638

From single-particle to many-body mobility edges and the fate of overlapped spectra in coupled disorder models

Xiaoshui Lin, Ming Gong, Guang‐Can Guo

Published Jul 4, 2023• 0 citations
arXivHTMLPDF

Abstract

Mobility edge (ME) has played an essential role in disordered models. However, while this concept has been well established in disordered single-particle models, its existence in disordered many-body models is still under controversy. Here, a general approach based on coupling between extended and localized states in their overlapped spectra for ME is presented. We show that in the one-dimensional (1d) disordered single-particle models, all states are localized by direct coupling between them. However, in $d \ge 2$ disordered single-particle and 1d disordered many-body models, the resonant hybridization between these states in their overlapped spectra makes all states be extended, while these in the un-overlapped spectra are unchanged, leading to tunable MEs. We propose several models, including two disordered many-body spin models, to verify this mechanism. Our results establish a unified mechanism for MEs and demonstrate its universality in single-particle and many-body models, which opens an intriguing avenue for the realization and verification of MEs in many-body localization.

DOI
10.48550/arxiv.2307.01638

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Mobility edge (ME) has played an essential role in disordered models. However, while this concept has been well established in disordered single-particle models, its existence in disordered many-body models is still under controversy. Here, a general approach based on coupling between extended and localized states in their overlapped spectra for ME is presented. We show that in the one-dimensional (1d) disordered single-particle models, all states are localized by direct coupling between them. However, in d≥2d \ge 2d≥2 disordered single-particle and 1d disordered many-body models, the resonant hybridization between these states in their overlapped spectra makes all states be extended, while these in the un-overlapped spectra are unchanged, leading to tunable MEs. We propose several models, including two disordered many-body spin models, to verify this mechanism. Our results establish a unified mechanism for MEs and demonstrate its universality in single-particle and many-body models, which opens an intriguing avenue for the realization and verification of MEs in many-body localization.
We construct a solvable spin chain model of many-body localization (MBL) with a tunable mobility edge. This simple model not only demonstrates analytically the existence of mobility edges in interacting one-dimensional (1D) disordered systems, but also allows us to study their physics. By establishing a connection between MBL and a quantum central limit theorem (QCLT), we show that many-body localization-delocalization transitions can be visualized as tuning a mobility edge in the energy spectrum. Since the effective disorder strength for individual eigenstates depends on energy density, we identify "energy-resolved disorder strength" as a physical mechanism for the appearance of mobility edges, and support the universality of this mechanism by arguing its presence in a large class of models including the random-field Heisenberg chain. We also construct models with multiple mobility edges. All our constructions can be made translationally invariant.
The disorder systems host three types of fundamental quantum states, known as the extended, localized, and critical states, of which the critical states remain being much less explored. Here we propose a class of exactly solvable models which host a novel type of exact mobility edges (MEs) separating localized states from robust critical states, and propose experimental realization. Here the robustness refers to the stability against both single-particle perturbation and interactions in the few-body regime. The exactly solvable one-dimensional models are featured by a quasiperiodic mosaic type of both hopping terms and on-site potentials. The analytic results enable us to unambiguously obtain the critical states which otherwise require arduous numerical verification including the careful finite size scalings. The critical states and new MEs are shown to be robust, illustrating a generic mechanism unveiled here that the critical states are protected by zeros of quasiperiodic hopping terms in the thermodynamic limit. Further, we propose a novel experimental scheme to realize the exactly solvable model and the new MEs in an incommensurate Rydberg Raman superarray. This Letter may pave a way to precisely explore the critical states and new ME physics with experimental feasibility.
We review the physics of many-body localization in models with incommensurate potentials. In particular, we consider one-dimensional quasiperiodic models with single-particle mobility edges. A conventional perspective suggests that delocalized states act as a thermalizing bath for the localized states in the presence of of interactions. However, contrary to this intuition there is evidence that such systems can display non-ergodicity. This is in part due to the fact that the delocalized states do not have any kind of protection due to symmetry or topology and are thus susceptible to localization. A study of such incommensurate models, in the non-interacting limit, shows that they admit extended, partially extended, and fully localized many-body states. Non-interacting incommensurate models cannot thermalize dynamically and remain localized upon the introduction of interactions. In particular, for a certain range of energy, the system can host a non-ergodic extended (i.e. metallic) phase in which the energy eigenstates violate the eigenstate thermalization hypothesis (ETH) but the entanglement entropy obeys volume-law scaling. The level statistics and entanglement growth also indicate the lack of ergodicity in these models. The phenomenon of localization and non-ergodicity in a system with interactions despite the presence of single-particle delocalized states is closely related to the so-called many-body proximity effect and can also be observed in models with disorder coupled to systems with delocalized degrees of freedom. Many-body localization in systems with incommensurate potentials (without single-particle mobility edges) have been realized experimentally, and we show how this can be modified to study the the effects of such mobility edges. Demonstrating the failure of thermalization in the presence of a single-particle mobility edge in the thermodynamic limit would indicate a more robust violation of the ETH.
We study many-body localization (MBL) in an interacting one-dimensional system with a deterministic aperiodic potential. Below the threshold potential h<hch<{h}_{c}h<hc​, the noninteracting system has single particle mobility edges (MEs) at $\ifmmode\pm\else\textpm\fi{}{E}_{c}$ while for h>hch>{h}_{c}h>hc​ all the single particle states are localized. We demonstrate that even in the presence of single particle MEs, interactions do not always delocalize the system and the interacting system can have MBL. Our numerical calculation of energy level spacing statistics, participation ratio in the Fock space, and Shannon entropy shows that for some regime of particle densities, even for h<hch<{h}_{c}h<hc​, many-body states at the top (with E>E2E>{E}_{2}E>E2​) and the bottom of the spectrum (with E<E1E<{E}_{1}E<E1​) remain localized, though states in the middle of the spectrum are delocalized. Variance of entanglement entropy (EE) also diverges at E1,2{E}_{1,2}E1,2​, indicating a transition from MBL to a delocalized regime, though transitions from volume to area law scaling for EE and from thermal to nonthermal behavior occur inside the MBL regime much below E1{E}_{1}E1​ and above E2{E}_{2}E2​.
Many-body localization provides a mechanism to avoid thermalization in isolated interacting quantum systems. The breakdown of thermalization may be complete, when all eigenstates in the many-body spectrum become localized, or partial, when the so-called many-body mobility edge separates localized and delocalized parts of the spectrum. Previously, De Roeck et al. [Phys. Rev. B 93, 014203 (2016)] suggested a possible instability of the many-body mobility edge in energy density. The local ergodic regions---so-called ``bubbles''---resonantly spread throughout the system, leading to delocalization. In order to study such instability mechanism, in this work we design a model featuring many-body mobility edge in particle density: the states at small particle density are localized, while increasing the density of particles leads to delocalization. Using numerical simulations with matrix product states, we demonstrate the stability of many-body localization with respect to small bubbles in large dilute systems for experimentally relevant timescales. In addition, we demonstrate that processes where the bubble spreads are favored over processes that lead to resonant tunneling, suggesting a possible mechanism behind the observed stability of many-body mobility edge. We conclude by proposing experiments to probe particle density mobility edge in the Bose-Hubbard model.
Disorder and interactions can lead to the breakdown of statistical mechanics in quantum systems by transitioning to a many-body localized (MBL) phase. The thermal-to-MBL transition occurs by either increasing the amount of disorder present in a system or by decreasing its energy until it crosses the so-called mobility edge. The authors construct a set of approximate integrals of motion that are quasi-universal across the energy spectrum and, thus, across the mobility edge, too. Interestingly, MBL eigenstates under the mobility edge are aware of the existence of a thermal phase at a higher energy. The phase diagram is probed close to the transition and in the MBL phase. One finds a universal distribution of the strength of the couplings of the integrals of motion in the strong disorder limit.