Hidden dualities in 1D quasiperiodic lattice models

TL;DR

This paper uncovers hidden dualities in 1D quasiperiodic lattice models that relate extended and localized phases, providing a new method to analyze mobility edges and eigenstate properties near phase transitions.

Contribution

It introduces a novel approach to identify and construct spectral and eigenstate dualities in 1D quasiperiodic systems using commensurate approximants and auxiliary Fermi surfaces.

Findings

Hidden dualities generalize Aubry-André model duality

Universal form of auxiliary Fermi surface at criticality

Accurate method for mobility edge and duality determination

Abstract

We find that quasiperiodicity-induced transitions between extended and localized phases in generic 1D systems are associated with hidden dualities that generalize the well-known duality of the Aubry-Andr\'e model. These spectral and eigenstate dualities are locally defined near the transition and can, in many cases, be explicitly constructed by considering relatively small commensurate approximants. The construction relies on auxiliary 2D Fermi surfaces obtained as functions of the phase-twisting boundary conditions and of the phase-shifting real-space structure. We show that, around the critical point of the limiting quasiperiodic system, the auxiliary Fermi surface of a high-enough-order approximant converges to a universal form. This allows us to devise a highly-accurate method to obtain mobility edges and duality transformations for generic 1D quasiperiodic systems through their commensurate approximants. To illustrate the power of this approach, we consider several previously studied systems, including generalized Aubry-Andr\'e models and coupled Moir\'e chains. Our findings bring a new perspective to examine quasiperiodicity-induced extended-to-localized transitions in 1D, provide a working criterion for the appearance of mobility edges, and an explicit way to understand the properties of eigenstates close to and at the transition.