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

TL;DR

This paper introduces a novel projected Green's function method for analyzing quasi-periodic systems, enabling better numerical and analytical insights into their spectral properties and phase diagrams without boundary condition issues.

Contribution

The paper presents a new rational approximation technique using projected Green's functions for quasi-periodic systems, improving analysis of their spectral and phase properties.

Findings

Successfully applied to the AAH model and Liouville irrationals.

Provides a boundary-condition-free numerical approach.

Analytically derives a modified phase diagram.

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.