Nearest Neighbor Tight Binding Models with an Exact Mobility Edge in One Dimension

Abstract

We investigate localization properties in a family of deterministic (i.e. no disorder) nearest neighbor tight binding models with quasiperiodic onsite modulation. We prove that this family is self-dual under a generalized duality transformation. The self-dual condition for this general model turns out to be a simple closed form function of the model parameters and energy. We introduce the typical density of states as an order parameter for localization in quasiperiodic systems. By direct calculations of the inverse participation ratio and the typical density of states we numerically verify that this self-dual line indeed defines a mobility edge in energy separating localized and extended states. Our model is a first example of a nearest neighbor tight binding model manifesting a mobility edge protected by a duality symmetry. We propose a realistic experimental scheme to realize our results in atomic optical lattices and photonic waveguides.Anderson localization [1] is a universal and extensively studied property of a quantum particle or a wave in a disordered medium. An interesting consequence of Anderson localization is a quantum phase transition between extended and localized states as a function of the disorder strength. In three dimensional systems with random (uncorrelated) disorder, a localization transition occurs as the strength of disorder crosses a critical value forming a sharp energy dependent mobility edge at the phase boundary separating localized/extended states below/above the mobility edge. Scaling theory [2] has shown the absence of this critical behavior in one and two dimensions where all states, at least in the absence of interaction, are known to be localized in a disordered system, pushing the mobility edge effectively to infinite energy. However, in one dimension this picture changes for a quasiperiodic system with two incommensurate (but deterministic) lattice potentials, which in a loose qualitative sense might be construed to be a highly correlated disorder, albeit perfectly well-defined with no randomness whatsoever. Aubry and Andre [3] showed that a 1D tight binding model with an onsite cosine modulation incommensurate with the underlying lattice has a self-dual symmetry and manifests an energy independent localization transition as a function of the modulation strength, i.e., all states are either localized or extended depending on the relative strength of the incommensurate modulation potential with respect to the lattice potential. [21] to show the existence of a many body localization transition in the quasiperiodic AAH model in the presence of weak interactions. The duality-driven and energy independent localization transition in the AAH model does not manifest a mobility edge, which is a hallmark of the disorder tuned localization transition in 3D. The 1D localization transition in the AAH model, defined by the self-duality point, thus has no analog in disorder-driven Anderson localization, and does not give any insight into the physics of 3D mobility edges. Recent works have ...

Anderson localization transitions with and without random potentials

“…Furthermore, if there is a single localization transition between these two phases it must be at exactly v c = 1, where the model is invariant under the duality. This is indeed the case for the AA model (although there are variations that experience a richer transition [12][13][14][15][16][17][18][19][20][21][22] , so self-duality alone does not imply a single transition that occurs simultaneously at all energies). Now let's consider adding a weak site-random potential to this model.…”

Engineering a Flux-Dependent Mobility Edge in Disordered Zigzag Chains

“…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.…”

Dynamical observation of mobility edges in one-dimensional incommensurate optical lattices

“…One can obtain a one-dimensional model displaying the mobility edge when the so-called self-dual symmetry is broken, such as a system with a shallow one-dimensional quasi-periodic potential [47][48][49][50][51]. Another class of systems with the mobility edge by introducing a long-range hopping term [31] or a special form of the on-site incommensurate potential [52] present the energy-dependent self-duality in the compactly analytic form. Recently, great attention has been paid to the properties of the intermediate phase characterized by the mobility edge in the quasi-periodic lattices, such as many-body localization in the presence of a single particle mobility edge [53][54][55][56][57][58][59][60][61] and the existence of Bose glass phase in finite temperature [62,63].…”