Rational Approximations of Quasi-Periodic Problems via Projected Green's Functions

Abstract

We introduce the projected Green's function technique to study quasi-periodic systems such as the Andre-Aubry-Harper (AAH) model and beyond. In particular, we use projected Green's functions to construct a "rational approximate" sequence of transfer matrix equations consistent with quasi-periodic topology, where convergence of these sequences corresponds to the existence of extended eigenfunctions. We motivate this framework by applying it to a few well studied cases such as the almost-Mathieu operator (AAH model), as well as more generic non-dual models that challenge standard routines. The technique is flexible and can be used to extract both analytic and numerical results, e.g. we analytically extract a modified phase diagram for Liouville irrationals. As a numerical tool, it does not require the fixing of boundary conditions and circumvents a primary failing of numerical techniques in quasi-periodic systems, extrapolation from finite size. Instead, it uses finite size scaling to define convergence bounds on the full irrational limit.

(main) 6 pages 4 figures, (supplemental) 14 pages, 2 figures

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• Theory of polarization textures in crystal supercells
• Localization via Quasi-Periodic Bulk-Bulk Correspondence
• Quasiperiodic disorder induced critical phases in a periodically driven dimerized $p$-wave Kitaev chain
• Hidden self-duality and exact mobility edges in quasiperiodic network models
• From topological phase to Anderson localization in a two-dimensional quasiperiodic system
• Exact non-Hermitian mobility edges and robust flat bands in two-dimensional Lieb lattices with imaginary quasiperiodic potentials
• Emergent multi-loop nested point gap in a non-Hermitian quasiperiodic lattice
• Localization and mobility edges in non-Hermitian continuous quasiperiodic systems
• Quasiperiodicity protects quantized transport in disordered systems without gaps
• Generic non-Hermitian mobility edges in a class of duality-breaking quasicrystals