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

• URL: http://arxiv.org/abs/2109.13933v1
• Date: Tue, 28 Sep 2021 18:00:00 GMT
• Title: Rational Approximations of Quasi-Periodic Problems via Projected Green's Functions
• Authors: Dan S. Borgnia, Ashvin Vishwanath, Robert-Jan Slager
• Abstract summary: We introduce the projected Green's function technique to study quasi-periodic systems. The technique is flexible and can be used to extract both analytic and numerical results.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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