Fate of critical states and possible emergence of extended states at finite quasiperiodic strength in ABF ladders

Determine, for the full non-Abelian Aubry–André–Harper Hamiltonian H = ABF + W describing the one-dimensional two-band all-bands-flat ladder with quasiperiodic onsite fields W1 and W2, the behavior of the spectrum at moderate quasiperiodic potential strengths λ1, λ2: specifically, ascertain what happens to the critical states identified in the weak-perturbation regime and whether any extended (delocalized) eigenstates emerge, and characterize their localization and transport properties if present.

Background

The thesis analyzes one-dimensional two-band all-bands-flat (ABF) ladders subjected to quasiperiodic perturbations composed of two incommensurate onsite potentials W1 and W2. In the weak-perturbation limit, the ABF models can be projected onto effective one-dimensional quasiperiodic Jacobi operators that map to the extended Harper model or the off-diagonal Harper model, leading to spectra that are either fully critical or localized depending on parameters. Transport in the critical regime is subdiffusive or nearly diffusive, depending on the mapping.

At finite perturbation strengths, the projected-model description breaks down and the original ABF Hamiltonian takes a non-Abelian Aubry–André–Harper form with quasiperiodic blocks and fixed hopping blocks. While strong quasiperiodic potentials are expected to localize all states, the behavior at moderate strengths is explicitly left open in the text, including whether extended states might appear and the fate of previously critical states.