Rational approximations of quasiperiodicity via projected Green's functions

We introduce the projected Green's function technique to study quasiperiodic systems such as the Aubry-Andre-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 quasiperiodic 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 no-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 quasiperiodic systems-extrapolation from finite size. Instead, it uses finite-size scaling to define convergence bounds on the full irrational limit.