Localization in one-dimensional lattices with non-nearest-neighbor hopping: Generalized Anderson and Aubry-André models

Abstract

We study the quantum localization phenomena of noninteracting particles in one-dimensional lattices based on tight-binding models with various forms of hopping terms beyond the nearest neighbor, which are generalizations of the famous Aubry-André and noninteracting Anderson model. For the case with deterministic disordered potential induced by a secondary incommensurate lattice (i.e. the Aubry-André model), we identify a class of self-dual models, for which the boundary between localized and extended eigenstates are determined analytically by employing a generalized Aubry-André transformation. We also numerically investigate the localization properties of nondual models with next-nearest-neighbor hopping, Gaussian, and power-law decay hopping terms. We find that even for these non-dual models, the numerically obtained mobility edges can be well approximated by the analytically obtained condition for localization transition in the self dual models, as long as the decay of the hopping rate with respect to distance is sufficiently fast. For the disordered potential with genuinely random character, we examine scenarios with next-nearestneighbor hopping, exponential, Gaussian, and power-law decay hopping terms numerically. We find that the higher-order hopping terms can remove the symmetry in the localization length about the energy band center compared to the Anderson model. Furthermore, our results demonstrate that for the power-law decay case, there exists a critical exponent below which mobility edges can be found. Our theoretical results could, in principle, be directly tested in shallow atomic optical lattice systems enabling non-nearest-neighbor hopping.

Engineering a Flux-Dependent Mobility Edge in Disordered Zigzag Chains

“…The high ground state (GS) degeneracy of the quartic band in this system and of flatbands in similar multirange hopping models has attracted great interest [37][38][39]. Such systems promise interesting localization properties under disorder [13], and the inherent high single-particle degeneracy allows for the study of emergent physics driven by interactions [37][38][39][40]. For all other flux values (ϕ ≠ AEπ=2) the dispersion of the bands at low and high energies is asymmetric, and this system permits the localization properties of the extremal energy eigenstates to be tuned through modification of the effective mass at low and high energies.…”

“…This fine-tuning results in a metal-insulator transition that occurs at the same critical disorder value (in units of the tunneling energy) for all energy eigenstates, and thus the absence of a mobility edge. By moving away from this fine-tuned scenario in any number of ways-by introducing longer-range hopping [13], by modifying the pseudodisorder correlations [14], or by adding nonlinear interactions [11,12,[31][32][33]]-a SPME can be introduced into the AA model.…”

“…While this form of correlated pseudodisorder allows for a localization transition in 1D, the fine-tuning of the cosine-distributed site energies and the cosine nearest-neighbor (NN) band dispersion results in an energy-independent metal-insulator transition, and thus the absence of a SPME. By deviating from this fine-tuned condition, either by modifying the band dispersion [13] or by modifying the form of the pseudodisorder [14], one can, in principle, controllably introduce a SPME in such a system.…”

Signatures of the single-particle mobility edge in the ground-state properties of Tonks-Girardeau and noninteracting Fermi gases in a bichromatic potential

“…In the continuum, outside the range of validity of the TB approximation, it is known that the sharp delocalizedto-localized transition of the lowest energy band transforms into a mobility edge, whose position in energy changes with V 2 [20,21]. Our aim is to study in detail how this crossover from the discrete to the continuum behavior occurs, and to show that signatures of this transition are displayed in the many-body properties of both non-interacting fermions and strongly-interacting bosons.…”

Constructing a Weyl semimetal by stacking one-dimensional topological phases

“…The central goal of our work is to understand TS in 3D from the point of view of 1D models manifesting topological phases. In this work, we show that the π-flux state of the generalized Aubry-AndreHarper (AAH) model [13][14][15] in 1D can be used as a theoretically unifying framework to understand semimetallic phases in both two and three dimensions. The simplest AAH model is a 1D tight-binding model with onsite cosine modulation with a phase parameter corresponding to the momentum in the second dimension 16 .…”