Localization in one-dimensional incommensurate lattices beyond the Aubry-André model

Abstract

Localization properties of particles in one-dimensional incommensurate lattices without interaction are investigated with models beyond the tight-binding Aubry-André (AA) model. Based on a tight-binding t1 − t2 model with finite next-nearest-neighbor hopping t2, we find the localization properties qualitatively different from those of the AA model, signaled by the appearance of mobility edges. We then further go beyond the tight-binding assumption and directly study the system based on the more fundamental single-particle Schrödinger equation. With this approach, we also observe the presence of mobility edges and localization properties dependent on incommensuration.PACS numbers: 37.10.Jk, The physics of quantum transport in random disordered potentials has been a subject of substantial interest for condensed matter physicists for decades. The extended Bloch waves in a periodic lattice could undergo a quantum interference induced transition into localized states due to random disorder by a mechanism commonly referred to as Anderson localization [1]. Matter waves can also be localized in deterministic potentials that exhibit some similarities to random disorder [2,3,4,5]. Quasi-periodic potentials, such as incommensurate lattices (the superposition of two or more lattices with incommensurate periods), are notable examples and have been extensively studied with the Aubry-André model [2]. Such potentials have been shown to exhibit interesting quantum transport phenomena in themselves. Incommensurate potentials, for example, are theorized to have fractal spectrums [6]. However, it remains challenging to study these phenomena in solid state experiments, as it is difficult to systematically control the disorder in solid state systems. In contrast to the solid state systems, ultracold atoms loaded in optical lattices offer remarkable controllability over the system parameters, making it an attractive platform for the study of the localization of matter waves. Recently, Anderson localization of noninteracting Bose-Einstein condensates (BEC) has been observed in a one-dimensional matter waveguide with a random potential introduced with laser speckles [7]. Similar experiments have also been done in quasi-periodic optical lattices [8,9].Localization of noninteracting particles in one dimensional incommensurate lattices is often studied with the Aubry-André model (AA) with nearest neighbor (nn) hopping, where one of the lattices is assumed to be relatively weak and can be treated as a perturbation . Within the framework of the AA model, there is a duality point, at which a sharp transition from all eigenstates being extended to all being localized occurs. However, in ultracold atom experiments, one can tune the depth of each lattice in a controllable way and bring the system out of the tight-binding regime. To explore the physics of localization for shallow lattices, it is of interest to go beyond the AA model and the tight-binding assumption [10].In this work, we first study the tight-binding t 1 − t 2 model, which extends...

One-dimensional repulsive Fermi gas in a tunable periodic potential

“…Away from this regime multi-band processes come into play, and the effect of the independent tuning of the OL intensity and the interaction strength can be captured only via multi-band or continuous-space models. Recent theoretical and experimental studies have addressed the regime of shallow OLs and strong interactions, investigating intriguing phenomena such as Mott and pinning bosonic localization transitions [14][15][16][17][18], Anderson localization [19][20][21], Bose-Glass phases [22], and itinerant ferromagnetism [23,24].…”

Universal scheme to generate metal–insulator transition in disordered systems

“…A band of extended states will emerge in the binary alloy when the site energies are long-range correlated [23]. In addition, other theoretical models have been suggested to produce conducting states in low-dimensional disordered systems [24][25][26][27][28][29][30][31][32][33], and some of them have been corroborated in GaAs-AlGaAs superlattices [34] and Bose-Einstein condensate (BEC) [14]. Nevertheless, we notice that all electronic states become localized in these correlated disordered systems when the disorder degree is extremely large.…”

“…3(b) we obtain β=−1], due to the Anderson localization effects, although they possess either the diagonal disorder (dashed lines) or the off-diagonal disorder (dotted lines) and are more ordered than the former case. the disorder degree is very large [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33], we can see from Fig. 3(c) that ξ L is independent of W for the former ladder and all states are always extended in the gray energy region [ Fig.…”

Spatially-antisymmetric localization of matter wave in a bichromatic optical lattice

“…Although, the present localization is very similar to Anderson localization in a fully disordered potential, the bichromatic OL potential is quasi periodic and hence deterministic in nature. The localization considered here is well described by the 1D discrete AubryAndré model of quasi-periodic confinement [36,37]. * yong shan@163.com † adhikari@ift.unesp.br; URL: www.ift.unesp.br/users/adhikari If the bichromatic OL potential V (x) has the symmetry V (x) = V (−x), the localized states φ(x) of the non-interacting BEC has the symmetry φ(x) = ±φ(−x).…”