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arXiv:2109.09621 · Physical review. B./Physical review. B

Emergence of multiple localization transitions in a one-dimensional quasiperiodic lattice

Ashirbad Padhan, Mrinal Kanti Giri, Suman Mondal, Tapan Mishra

Published Jun 3, 2022• 52 citations
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Abstract

Low-dimensional quasiperiodic systems exhibit localization transitions by turning all quantum states localized after a critical quasidisorder. While certain systems with modified or constrained quasiperiodic potential undergo multiple localization transitions in one dimension, we predict an emergence of multiple localization transitions without directly imposing any constraints on the quasiperiodic potential. By considering a one-dimensional system described by the Aubry-Andr\'e model, we show that an additional staggered on-site potential can drive the system through a series of localization transitions as a function of the staggered potential. Interestingly, we find that the number of localization transitions strongly depends on the strength of the quasiperiodic potential. Moreover, we obtain the signatures of these localization transitions in the expansion dynamics and propose an experimental scheme for their detection in the quantum gas experiment.

DOI
10.1103/physrevb.105.l220201

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Low-dimensional quasiperiodic systems exhibit localization transitions by turning all quantum states localized after a critical quasidisorder. While certain systems with modified or constrained quasiperiodic potential undergo multiple localization transitions in one dimension, we predict an emergence of multiple localization transitions without directly imposing any constraints on the quasiperiodic potential. By considering a one-dimensional system described by the Aubry-Andr'e model, we show that an additional staggered on-site potential can drive the system through a series of localization transitions as a function of the staggered potential. Interestingly, we find that the number of localization transitions strongly depends on the strength of the quasiperiodic potential. Moreover, we obtain the signatures of these localization transitions in the expansion dynamics and propose an experimental scheme for their detection in the quantum gas experiment.
We propose a one-dimensional generalized Aubry-Andr\'e-Harper (AAH) model with off-diagonal hopping and staggered on-site potential. We find that the localization transitions could be multiple reentrant with the increasing of staggered on-site potential. The multiple localization transitions are verified by the quantum static and dynamic measurements such as the inverse or normalized participation ratios, fractal dimension, and survival probability. Based on the finite-size scaling analysis, we also obtain an interesting intermediate phase where the extended, localized, and critical states are coexistent in certain regimes of model parameters. These results are quite different from those in the generalized AAH model with off-diagonal hopping, and can help us to find novel quantum phases and new localization phenomena in the disordered systems.
A reentrant localization transition was predicted recently in a one-dimensional quasiperiodic lattice with dimerized hopping between the nearest-neighbor sites [Roy et al., Phys. Rev. Lett. 126, 106803 (2021)]. It was shown that the interplay between the hopping dimerization and a staggered quasiperiodic disorder manifests two localization transitions through two intermediate phases, resulting in four critical points as a function of the quasiperiodic potential. In this paper, we study the critical nature of the states across the localization transitions by computing the mass exponents and the corresponding fractal dimensions of the states through a multifractal analysis. Moreover, we analyze the phenomenon of this reentrant localization transition by examining the spectral properties of the eigenstates. By performing a systematic finite-size scaling analysis for a fixed value of the hopping dimerization, we obtain accurate critical disorder strengths for different transitions and the associated critical exponents. Further, we complement the critical nature of the energy spectrum by computing the Hausdorff dimensions.
We study the combined effect of quasiperiodic disorder, driven, and interaction in the periodically kicked Aubry-Andr\'e model. In the noninteracting limit, by analyzing the quasienergy spectrum statistics, we verify the existence of a dynamical localization transition in the high-frequency region, whereas the spectrum statistics becomes intricate in the low-frequency region due to the emergence of the extended/localized-to-multifractal edges in the quasienergy spectrum, which separate the multifractal states from the extended (localized) states. When the interaction is introduced, we find the periodically kicked incommensurate potential can lead to a transition from the ergodic to many-body-localization phase in the high-frequency region. However, the many-body localization phase vanishes in the low-frequency region even for the strong quasiperiodic disorder. Our studies demonstrate that the periodically kicked Aubry-Andr\'e model displays rich dynamical phenomena and the driving frequency plays an important role in the formation of many-body localization in addition to the disorder strength.
Despite their potential similarity between the mosaic Aubry-André (MAA) and AA models, the MAA model allows mobility edges (MEs), whereas the AA model does not. Here we develop a new double quasiperiodic MAA (DMAA) model consisting of one primitive MAA with nonzero even-site potentials and the other modified one with both nonzero odd-site potentials and a tunable amplitude factor, to reveal how localization transitions evolve from MAA to AA models. Interplays and competitions among the extended, critical and localized states arising from superpositions of double quasi-periodic MAA potentials enable new twice and multiple localization-delocalization transitions besides the original single localization transition. Our numerical calculations on inverse participation ratio, normalized participation ratio, fractal dimension and real-space wavefunction distribution confirm such localization features. The continuum model simulations on the experimental polariton modes also yield consistent results and hence validate their experimental feasibility. The constructed DMAA model provides a new framework for studying the localization transition processes between two analogous quasiperiodic models and broadens the understanding of Anderson localization.
We study the localization properties of weakly interacting Bose gas in a quasiperiodic potential. The Hamiltonian of the non-interacting system reduces to the well known 'Aubry–André model', which shows the localization transition at a critical strength of the potential. In the presence of repulsive interaction we observe multi-site localization and obtain a phase diagram of the dilute Bose gas by computing the superfluid fraction and the inverse participation ratio. We construct a low-dimensional classical Hamiltonian map and show that the onset of localization is manifested by the chaotic phase space dynamics. The level spacing statistics also identify the transition to localized states resembling a Poisson distribution that are ubiquitous for both non-interacting and interacting systems. We also study the quantum fluctuations within the Bogoliubov approximation and compute the quasiparticle energy spectrum. Enhanced quantum fluctuation and multi-site localization phenomenon of non-condensate density are observed above the critical coupling of the potential. We briefly discuss the effect of the trapping potential on the localization of matter wave.
One-dimensional quasiperiodic systems with power-law hopping, 1/r^{a}, differ from both the standard Aubry-André (AA) model and from power-law systems with uncorrelated disorder. Whereas in the AA model all single-particle states undergo a transition from ergodic to localized at a critical quasidisorder strength, short-range power-law hops with a>1 can result in mobility edges. We find that there is no localization for long-range hops with a≤1, in contrast to the case of uncorrelated disorder. Systems with long-range hops rather present ergodic-to-multifractal edges and a phase transition from ergodic to multifractal (extended but nonergodic) states. Both mobility and ergodic-to-multifractal edges may be clearly revealed in experiments on expansion dynamics.