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.

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

References in corpus (34)

• Localization transitions in non-Hermitian quantum mechanics
• Topological States and Adiabatic Pumping in Quasicrystals
• Non-Hermitian Boundary Modes
• The space group classification of topological band insulators
• Topological phase transition in non-Hermitian quasicrystals
• Many-Body Localization in a Quasiperiodic System
• Nearest neighbor tight binding models with an exact mobility edge in one dimension
• Single particle Green's functions and interacting topological insulators
• Dirac Electrons in a Dodecagonal Graphene Quasicrystal
• Edge states and the bulk-boundary correspondence in Dirac Hamiltonians
• Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model
• Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry
• Impurity Bound States and Greens Function Zeroes as Local Signatures of Topology
• Localization in one dimensional lattices with non-nearest-neighbor hopping: Generalized Anderson and Aubry-André models
• Localization and adiabatic pumping in a generalized Aubry-André-Harper model
• Unified bulk-boundary correspondence for band insulators
• The translational side of topological band insulators
• Sharp Phase transitions for the almost Mathieu operator
• Topological equivalence of crystal and quasicrystal band structures
• Of Bulk and Boundaries: Generalized Transfer Matrices for Tight-Binding Models
• Mixed spectra and partially extended states in a two-dimensional quasiperiodic model
• Virtual Topological Insulators with Real Quantized Physics
• Non-Commutative Chern Numbers for Generic Aperiodic Discrete Systems
• The Bulk-Edge Correspondence for Disordered Chiral Chains
• Universality and Quantum Criticality in Quasiperiodic Spin Chains
• The non-commutative n-th Chern number
• Fractal Weyl Law for Open Chaotic Maps
• Quasiperiodicity, band topology, and moiré graphene
• Criterion for the occurrence of many body localization in the presence of a single particle mobility edge
• Holder continuity of absolutely continuous spectral measures for one-frequency Schrodinger operators
• Semiclassical Dynamics, Berry Curvature and Spiral Holonomy in Optical Quasicrystals
• Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum
• Localization transitions and mobility edges in quasiperiodic ladder
• On the spectrum of critical almost Mathieu operators in the rational case

Cited by in corpus (14)

• Critical phase dualities in 1D exactly-solvable quasiperiodic models
• Bulk-Bulk Correspondence in Disordered Non-Hermitian Systems
• Renormalization-Group Theory of 1D quasiperiodic lattice models with commensurate approximants
• Incommensurability enabled quasi-fractal order in 1D narrow-band moiré systems
• 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