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arXiv:2010.09251 · Physical review. B./Physical review. B

Population imbalance for a family of one-dimensional incommensurate models with mobility edges

Sayantan Roy, Subroto Mukerjee, Manas Kulkarni

Published May 10, 2021• 5 citations
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Abstract

In this paper, we look at four generalizations of the one-dimensional Aubry-Andr\'e-Harper (AAH) model which possess mobility edges. We map out a phase diagram in terms of population imbalance and look at the system size dependence of the steady-state imbalance. We find nonmonotonic behavior of imbalance with the system parameters, which contradicts the idea that the relaxation of an initial imbalance is fixed only by the ratio of the number of extended states to the number of localized states. We propose that there exist dimensionless parameters, which depend on the fraction of single-particle localized states, single-particle extended states, and the mean participation ratio of these states. These ingredients fully control the imbalance in the long time limit and we present numerical evidence of this claim. Among the four models considered, three of them have interesting duality relations and their locations of mobility edges are known. One of the models (next-nearest-neighbor hopping) has no known duality but a mobility edge exists and the model has been experimentally realized. Our findings are an important step forward to understanding nonequilibrium phenomena in a family of interesting models with incommensurate potentials.

DOI
10.1103/physrevb.103.184203

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•Jan 1, 2017•21 citations
In this paper, we look at four generalizations of the one-dimensional Aubry-Andr'e-Harper (AAH) model which possess mobility edges. We map out a phase diagram in terms of population imbalance and look at the system size dependence of the steady-state imbalance. We find nonmonotonic behavior of imbalance with the system parameters, which contradicts the idea that the relaxation of an initial imbalance is fixed only by the ratio of the number of extended states to the number of localized states. We propose that there exist dimensionless parameters, which depend on the fraction of single-particle localized states, single-particle extended states, and the mean participation ratio of these states. These ingredients fully control the imbalance in the long time limit and we present numerical evidence of this claim. Among the four models considered, three of them have interesting duality relations and their locations of mobility edges are known. One of the models (next-nearest-neighbor hopping) has no known duality but a mobility edge exists and the model has been experimentally realized. Our findings are an important step forward to understanding nonequilibrium phenomena in a family of interesting models with incommensurate potentials.
We investigate and map out the nonequilibrium phase diagram of a generalization of the well known Aubry-Andr\'e-Harper (AAH) model. This generalized AAH (GAAH) model is known to have a single-particle mobility edge which also has an additional self-dual property akin to that of the critical point of the AAH model. By calculating the population imbalance, we get hints of a rich phase diagram. We also find a fascinating connection between single particle wave functions near the mobility edge of the GAAH model and the wave functions of the critical AAH model. By placing this model far from equilibrium with the aid of two baths, we investigate the open system transport via system size scaling of nonequilibrium steady state (NESS) current, calculated by fully exact nonequilibrium Green's function (NEGF) formalism. The critical point of the AAH model now generalizes to a `critical' line separating regions of ballistic and localized transport. Like the critical point of the AAH model, current scales subdiffusively with system size on the `critical' line ($I\ensuremath{\sim}{N}^{\ensuremath{-}2\ifmmode\pm\else\textpm\fi{}0.1}$). However, remarkably, the scaling exponent on this line is distinctly different from that obtained for the critical AAH model (where $I\ensuremath{\sim}{N}^{\ensuremath{-}1.4\ifmmode\pm\else\textpm\fi{}0.05}$). All these results can be understood from the above-mentioned connection between states near the mobility edge of the GAAH model and those of the critical AAH model. A very interesting high temperature nonequilibrium phase diagram of the GAAH model emerges from our calculations.
Due to the topological nature of the Aubry-Andr\'e-Harper (AAH) model, interesting edge states have been found existing in one-dimensional periodic and quasiperiodic lattices. In this article, we investigate continuous-time quantum walks of identical particles initially located on either edge of commensurate AAH lattices in detail. It is shown that the quantum walker is delocalized among the whole lattice until the strength of periodic modulation is strong enough. The inverse participation ratios (IPRs) for all of the eigenstates are calculated. It is found that the localization properties of the quantum walker is mainly determined by the IPRs of the topologically protected edge states. More interestingly, the edge states are shown to have an evident ``repulsion'' effect on quantum walkers initiated from the lattice sites inside the bulk. Furthermore, we examine the role of nearest-neighbor interaction on the quantum walks of two identical fermions. Clear enhancement of the repulsion effect by strong interaction has been shown.
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 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.
We uncover a systematic structure in the single-particle phase diagram of the quasiperiodic Aubry-Andr\'e-Harper (AAH) model with power-law hoppings ($\ensuremath{\sim}\frac{1}{{r}^{\ensuremath{\sigma}}}$) when the quasiperiodicity parameter is chosen to be a member of the metallic mean family of irrational Diophantine numbers. In addition to the fully delocalized and localized phases, we find a coexistence of multifractal (localized) states with the delocalized states for $\ensuremath{\sigma}<1$ ($\ensuremath{\sigma}>1$). The fraction of delocalized eigenstates in these phases can be obtained from a general sequence, which is a manifestation of a mathematical property of the metallic mean family. The entanglement entropy of the noninteracting many-body ground states respects the area law if the Fermi level belongs in the localized regime while logarithmically violating it if the Fermi level belongs in the delocalized or multifractal regimes. The prefactor of the logarithmically violating term is significantly larger in the delocalized phase in comparison to that in the multifractal phase. Entanglement entropy shows the area law even in the delocalized regime for special filling fractions, which are related to the metallic means.
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.