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arXiv:2208.10714 · Physical review. B./Physical review. B

Diagnostics of nonergodic extended states and many body localization proximity effect through real-space and Fock-space excitations

Nilanjan Roy, Jagannath Sutradhar, Sumilan Banerjee

Published Mar 28, 2023• 18 citations
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Abstract

We provide real-space and Fock-space (FS) characterizations of ergodic, nonergodic extended (NEE) and many-body localized (MBL) phases in an interacting quasiperiodic system, namely, the generalized Aubry-Andr\'e-Harper model, which possesses a mobility edge in the noninteracting limit. We show that a mobility edge in the single-particle (SP) excitations survives even in the presence of interaction in the NEE phase. In contrast, all single-particle excitations get localized in the MBL phase due to the MBL proximity effect. We give complementary insights into the distinction of the NEE states from the ergodic and MBL states by computing local FS self-energies and decay length associated, respectively, with the local and the nonlocal FS propagators. Based on a finite-size scaling analysis of the typical local self-energy across the NEE to ergodic transition, we show that MBL and NEE states exhibit qualitatively similar multifractal character. However, we find that the NEE and MBL states can be distinguished in terms of the distribution of local self-energy and the decay of the nonlocal propagator in the FS, whereas the typical local FS self-energy cannot tell them apart.

DOI
10.1103/physrevb.107.115155

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We provide real-space and Fock-space (FS) characterizations of ergodic, nonergodic extended (NEE) and many-body localized (MBL) phases in an interacting quasiperiodic system, namely, the generalized Aubry-Andr'e-Harper model, which possesses a mobility edge in the noninteracting limit. We show that a mobility edge in the single-particle (SP) excitations survives even in the presence of interaction in the NEE phase. In contrast, all single-particle excitations get localized in the MBL phase due to the MBL proximity effect. We give complementary insights into the distinction of the NEE states from the ergodic and MBL states by computing local FS self-energies and decay length associated, respectively, with the local and the nonlocal FS propagators. Based on a finite-size scaling analysis of the typical local self-energy across the NEE to ergodic transition, we show that MBL and NEE states exhibit qualitatively similar multifractal character. However, we find that the NEE and MBL states can be distinguished in terms of the distribution of local self-energy and the decay of the nonlocal propagator in the FS, whereas the typical local FS self-energy cannot tell them apart.
We study a quasiperiodic model in one dimension, namely the extended Aubry-Andr\'e-Harper (EAAH) chain, that realizes a critical phase comprising entirely single-particle critical states in the noninteracting limit. In the presence of short-range interactions, the noninteracting critical phase transforms to a many-body critical (MBC) phase, separated by lines of MBC-ergodic, MBC-many-body localized (MBL) and ergodic-MBL phase transitions that meet at a triple point. We elucidate the unusual characteristics of the MBC phase compared to the ergodic and MBL phases through the localization properties of the excitations in real space and Fock space (FS), and the eigenstate inverse participation ratio (IPR). We show that the MBC phase, like the MBL phase, is well described by a multifractal scaling of the IPR and a linear finite-size scaling ansatz near the transition to the ergodic and MBL phases. However, the MBC phase, at the same time, exhibits delocalization of all single-particle excitations and a system-size-dependent Fock-space localization length, analogous to the ergodic phase. Remarkably, we find evidence of unusual Widom lines on the phase diagram in the form of lines of pronounced peaks or dips in the FS localization properties inside the MBC and MBL phases. These Widom lines either emerge as a continuation of the precursor phase transition line terminating at the triple point, or originate from a phase boundary.
We study localization and many-body localization transition in one-dimensional systems in the presence of deterministic quasiperiodic potential. We use single-particle excitations obtained through single-particle Green's function in real space to characterize the localization to delocalization transition. A single parameter scaling analysis of the ratio of the typical to average value of the local density of states (LDOS) of single-particle excitations shows that the critical exponent with which the correlation length $\ensuremath{\xi}$ diverges at the transition point $\ensuremath{\xi}\ensuremath{\sim}|h\ensuremath{-}{h}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, coming from the localized side, satisfies the inequality $\ensuremath{\nu}\ensuremath{\ge}1$ for the noninteracting Aubry-Andre (AA) model. For the interacting system with AA potential, we study single-particle excitations produced in highly excited many-body eigenstates across the MBL transition and found that the critical exponent obtained from finite-size scaling of the ratio of the typical to average value of the LDOS satisfies $\ensuremath{\nu}\ensuremath{\ge}1$ here as well. This analysis of the local density of states shows that the localization and MBL transition in systems with quasiperiodic potential belong to a different universality class than the localization and MBL transition in systems with random disorder where $\ensuremath{\nu}\ensuremath{\ge}2$. In complete contrast to this, finite-size scaling of the level spacing ratio is known to support the same universality class for MBL transitions in systems with quasiperiodic as well as random disorder potentials. For the interacting systems with quasiperiodic potentials, though finite-size scaling of the level spacing ratio shows a transition at hclsr{h}_{c}^{\mathrm{lsr}}hclsr​ which is close to the transition point obtained from LDOS within numerical precision, the critical exponent obtained from finite-size scaling of level spacing ratio is $\ensuremath{\nu}\ensuremath{\sim}0.54$ in close similarity to the MBL systems with random disorder.
A generalization of the Aubry-Andr\'e model, the noninteracting Ganeshan-Pixley-Das Sarma model (GPD) introduced by Ganeshan et al. [Phys. Rev. Lett. 114, 146601 (2015)], is known analytically to possess a mobility edge, allowing both extended and localized eigenstates to coexist. This mobility edge has been hypothesized to survive in closed many-body interacting systems, giving rise to a new nonergodic metallic phase. In this work, coupling the interacting GPD model to a thermal bath, we provide direct numerical evidence for multiple qualitative behaviors in the parameter space of disorder strength and energy level. In particular, we look at the bath-induced saturation of entanglement entropy to classify three behaviors: thermalized, nonergodic extended, and localized. We also extract the localization length in the localized phase using the long-time dynamics of the entanglement entropy and the spin imbalance. Our work demonstrates the rich localization landscape of generalized Aubry-Andr\'e models containing mobility edges in contrast to the simple Aubry-Andr\'e model with no mobility edge.
Quantum thermalization of isolated systems undergoing unitary time evolution is a fundamental problem in quantum statistical mechanics. Its study has been revived recently in the context of many-body Anderson localization. Previous works have focused on localization of many-body systems with all the single-particle states being localized. As a significant step forward, this work studies localization aspects of noninteracting many-particle systems in the presence of a single-particle mobility edge. By systemically investigating entanglement entropy scaling and nonthermal fluctuations in various lattice models, the authors establish a nonergodic extended phase as a generic intermediate phase (between purely ergodic extended and nonergodic localized phases) for the many-body localization transition of noninteracting fermions. This work also sheds light on the interacting transition scenario as well.
We study many-body localization (MBL) in the quasiperiodic ${t}_{1}\text{\ensuremath{-}}{t}_{2}$ model, focusing on the role of next-nearest-neighbor (NNN) hopping t2{t}_{2}t2​, which introduces a single-particle mobility edge. The calculated phase diagram can be divided into three distinct regimes, depending on the strength of the short-range interaction UUU. For weak interactions ($U\ensuremath{\ll}{t}_{1}$), this model is always nonthermal. For intermediate interactions ($U\ensuremath{\sim}{t}_{1}$), the thermal-MBL phase transition in this model is qualitatively the same as that of the Aubry-Andre (AA) model, which is consistent with existing experimental observations. For strong interactions $(U\ensuremath{\gg}{t}_{1})$, the NNN hopping produces qualitatively new physics because it breaks down the Hilbert space fragmentation present in the AA model. The NNN hopping is thus irrelevant when the interaction is intermediate but relevant for strong interactions.
Many-body eigenstates that are neither thermal nor many-body localized (MBL) have been numerically found in certain interacting chains with moderate quasiperiodic potentials. The energy regime consisting of these nonergodic but extended (NEE) eigenstates has been extensively studied as a possible many-body mobility edge between the energy-resolved MBL and thermal phases. Recently, the NEE regime was proposed to be a prethermal phenomenon that generally occurs when different operators spread on sizably different timescales. Here, we numerically examine the mutual independence among the NEE, MBL, and thermal regimes through the lens of eigenstate entanglement spectra (ES). Given the complexity and rich information embedded in ES, we develop an unsupervised learning approach designed to quantify the mutual independence among general phases. Our method is first demonstrated on an illustrative toy example that uses RGB color data to represent phases, and then applied to the ES of an interacting generalized Aubry-Andre model from weak to strong potential strength. We find that while the MBL and thermal regimes are mutually independent, the NEE regime is dependent on the former two and smoothly emerges as the potential strength decreases. We attribute this numerical finding to the fact that the ES data in the NEE regime exhibit both an MBL-like fast decay and a thermal-like long tail.