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.
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.
Recent progress in the study of many-body localization in strongly disordered interacting spin chains has emphasized the importance of distinguishing finite time prethermal behavior from long time and large volume asymptotics. We reexamine a reported nonergodic extended (NEE) regime in the interacting quasiperiodic Ganeshan-Pixley-Das Sarma model from this perspective, and propose that this regime is a prethermal feature. Indeed, we argue that the NEE regime may be identified through a change in the functional form of spin-spin autocorrelation functions, demonstrating that the NEE regime is distinguishable within intermediate-time dynamics. This is in contrast with existing conjectures relating the NEE regime to the presence of an asymptotic mobility edge in the single-particle spectrum. Thus, we propose a mechanism for the formation of an NEE regime which does not rely on asymptotic properties of the spin chain. Namely, we propose that the NEE regime emerges due to regularly spaced deep wells in the disorder potential. The highly detuned sites suppress spin transport across the system, effectively cutting the chain and producing a separation of timescales between the spreading of different operators. To support this proposal, we show that the NEE phenomenology also occurs in random models with deep wells but with no mobility edges, and does not occur in quasiperiodic models with mobility edges but with no deep wells. Our results support the broad conclusion that there is not a sharp distinction between the dynamics of quasiperiodically and randomly disordered systems in the prethermal regime. More specifically, we find that generic interacting quasiperiodic models do not have stable intermediate dynamical phases arising from their single-particle mobility edges, and that NEE phenomenology in such models is transient.
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.
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.
The breaking of ergodicity in isolated quantum systems with a single-particle mobility edge is an intriguing subject that has not yet been fully understood. In particular, whether a nonergodic but metallic phase exists or not in the presence of a one-dimensional quasiperiodic potential is currently under active debate. In this Letter, we develop a neural-network-based approach to investigate the existence of this nonergodic metallic phase in a prototype model using many-body entanglement spectra as the sole diagnostic. We find that such a method identifies with high confidence the existence of a nonergodic metallic phase in the midspectrum at an intermediate quasiperiodic potential strength. Our neural-network-based approach shows how supervised machine learning can be applied not only in locating phase boundaries but also in providing a way to definitively examine the existence or not of a novel phase.
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, 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 U. 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.
Despite the exponentially large amount of information required in the quantum description of many-body systems, finite size numerical simulations have paved the way for recent progress in the understanding of quantum phases of matter both in and out-of-equilibrium. \n \nIn this spirit, using exact diagonalization for a small number of spins ~ 20, first, we investigate the fate of both ergodic quantum many-body systems that thermalize at long times and systems with quenched disorder that exhibit many-body localization (MBL) and fail to thermalize (due to an extensive set of local conserved degrees of freedom) under periodic driving. Ergodic systems always delocalize in energy space and heat up to infinite temperature, for both local driving, in which the time dependence of the Hamiltonian is restricted to a few contiguous degrees of freedom, and global driving. On the other hand, MBL systems remain localized at finite energy density for local driving, while in the case of global driving these systems can either heat up to infinite temperature or remain localized depending on the parameters of the drive. Underlying the latter is the emergence of an effective time independent MBL Hamiltonian describing the long time dynamics of the system. These numerical predictions have recently been verified experimentally in a one dimensional quasi-disordered optical lattice of interacting fermions wherein unitary time evolution can be experimentally probed. \n \nSubsequently, we study the stability of MBL systems with respect to another perturbation, in this case by thermal inclusions, which remains poorly understood. We consider a simple model to address this question: a two level system interacting with 𝑁 localized bits subject to random fields. In the large 𝑁 limit, the localized bits are thermalized for any nonzero ratio of interactions to disorder strength. Moreover, at finite 𝑁, in the transition region, the single-site eigenstate entanglement entropies exhibit bimodal distributions, so that localized bits are either "on" (strongly entangled) or "off" (weakly entangled) in eigenstates. The clusters of "on" bits vary significantly between eigenstates of the same sample, which provides evidence for a heterogenous discontinuous transition out of the localized phase in single site observables. These results also imply the MBL phase is unstable in systems with short-range interactions and quenched randomness in dimensions 𝑑 that are high but finite. \n \nThe second part of this thesis is motivated by the recent breakthroughs of artificial intelligence in domains such as computer vision involving highly dimensional data of complexity not unlike the one found in the description of many-body systems. Using Monte Carlo simulations to generate synthetic data, we find that kernel methods, a class of machine learning algorithms, can learn the mathematical \nform of physical discriminators associated with phase transitions, such as order parameters and Hamiltonian constraints, for a set of two dimensional spin models: the ferromagnetic Ising model, a conserved-order-parameter Ising model, and the Ising gauge theory. \n \nOur work shows that the combination of numerical simulations with the ability of machine learning algorithms for automatic feature detection hold promise to become standard tools in the exploration and characterization of different phases of matter in condensed matter and other many-body systems.