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arXiv:2203.13213 · Physical review. B./Physical review. B

Localization of pairs in one-dimensional quasicrystals with power-law hopping

G. A. Domínguez‐Castro, R. Paredes

Published Oct 26, 2022• 3 citations
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Abstract

Pair localization in one-dimensional quasicrystals with nearest-neighbor hopping is independent of whether short-range interactions are repulsive or attractive. We numerically demonstrate that this symmetry is broken when the hopping follows a power law $1/{r}^{\ensuremath{\alpha}}$. In particular, for repulsively bound states, we find that the critical quasiperiodicity that signals the transition to localization is always bounded by the standard Aubry-Andr\'e critical point, whereas attractively bound dimers get localized at larger quasiperiodic modulations as the range of the hopping increases. Extensive numerical calculations establish the contrasting nature of the pair energy gap for repulsive and attractive interactions, as well as the behavior of the algebraic localization of the pairs as a function of quasiperiodicity, interaction strength, and power-law hops. The results here discussed are of direct relevance to the study of the quantum dynamics of systems with power-law couplings.

DOI
10.1103/physrevb.106.134208

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Pair localization in one-dimensional quasicrystals with nearest-neighbor hopping is independent of whether short-range interactions are repulsive or attractive. We numerically demonstrate that this symmetry is broken when the hopping follows a power law 1/r\ensuremathα1/{r}^{\ensuremath{\alpha}}1/r\ensuremathα. In particular, for repulsively bound states, we find that the critical quasiperiodicity that signals the transition to localization is always bounded by the standard Aubry-Andr'e critical point, whereas attractively bound dimers get localized at larger quasiperiodic modulations as the range of the hopping increases. Extensive numerical calculations establish the contrasting nature of the pair energy gap for repulsive and attractive interactions, as well as the behavior of the algebraic localization of the pairs as a function of quasiperiodicity, interaction strength, and power-law hops. The results here discussed are of direct relevance to the study of the quantum dynamics of systems with power-law couplings.
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.
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 investigate the formation of bound states made of two interacting atoms moving in a one dimensional (1D) quasiperiodic optical lattice. We derive the quantum phase diagram for Anderson localization of both attractively and repulsively bound pairs. We calculate the pair binding energy and show analytically that its behavior as a function of the interaction strength depends crucially on the nature-extended, multifractal, localized-of the single-particle atomic states. Experimental implications of our results are discussed.
Models with correlated disorder are rather common in physics. In some of them, like the Aubry-André (AA) model, the localization phase diagram can be found from the (self)duality with respect to the Fourier transform. In others, like the all-to-all translation-invariant Rosenzweig-Porter (TI RP) ensemble or the many-body localization (MBL), one needs to develop more sophisticated and usually phenomenological methods to obtain the localization transition. In addition, such models contain not only localization, but also the ergodicity-breaking transition in the Hilbert-space eigenstate structure, giving way to the nonergodic (i.e., MBL) phase of states with nontrivial fractal dimensions D q > 0 . In this work, we suggest a method to calculate both the above transitions and a lower bound to the fractal dimensions D 2 and D ∞ , relevant for the physical observables. To verify this method, we apply it to the class of long-range (self-)dual models, interpolating between a tight-binding ballistic model and TI RP via both power-law dependencies of the hopping terms and the on-site disorder correlations, and, thus, being out of the validity range of the previously developed methods. We show that the interplay of the correlated disorder and the power-law decaying hopping terms leads to the emergence of the two types of nonergodic phases in the entire range of parameters, even without having any quasiperiodicity of the AA potential. The analytical results of the above method are in full agreement with the extensive numerical calculations.
We propose a general framework for constructing self-dual one-dimensional quasiperiodic lattice models with arbitrary-range hoppings and multifractal behaviors. Our framework generates a broad spectrum of one-dimensional quasicrystals, ranging from the off-diagonal Aubry-Andr\'e-Harper models on one end, to those featuring long-range hoppings with varied quasiperiodic modulations on another. Focusing on models with off-diagonal quasiperiodic hoppings with power-law decay, we exploit the fact that, when the self-dual condition is satisfied, the system must be in the critical state with multifractal properties. This enables the engineering of models with competing extended, critical, and localized phases, with richly structured mobility edges separating them. As an outstanding example, we show that a limiting case of our family of self-dual quasicrystals can be implemented using Rydberg-atom arrays. Our work offers a systematic route toward critical phases from self-duality considerations, and would facilitate the experimental simulation of these exotic states.
The one-dimensional (1D) tight-binding model with random nearest-neighbor hopping is known to have a singularity of the density of states and of the localization length at the band center. We study numerically the effects of random long-range (power-law) hopping with an ensemble average magnitude $〈|{t}_{\mathrm{ij}}|〉\ensuremath{\propto}|i\ensuremath{-}j{|}^{\ensuremath{-}\ensuremath{\sigma}}$ in the 1D chain, while maintaining the particle-hole symmetry present in the nearest-neighbor model. We find, in agreement with results of real-space renormalization-group techniques applied to the random XY spin chain with power-law interactions, that there is a change of behavior when the power-law exponent $\ensuremath{\sigma}$ becomes smaller than 2.