Fibonacci number systems and the localization criterion in the many-body Aubry-André model

2026/08/04 by Balázs Hetényi
Mathematics · Physics and Astronomy · #cond-mat.dis-nn #cond-mat.mes-hall #cond-mat.stat-mech #math-ph #math.MP

paper · pdf · doi:10.1088/1751-8121/ae91a5

arxiv created 2026/08/04 · arxiv updated 2026/08/05

Abstract

We introduce an extension of the Fibonacci number system, we call the fractional Fibonacci number system, which interpolates between the Fibonacci number system (used for natural numbers) and the irrational base-ϕ number system, which can be used to represent real numbers. The number system finds its use in interpreting the localization phase diagram of the many-body non-interacting Aubry-André model. For finite system sizes a localization criterion can be obtained if the particle density is written in the fractional Fibonacci number system. In the thermodynamic limit, the criterion remains, but in this case the particle density is expressed in base-ϕ. The nature of the thermodynamic limit is also discussed.

Citations

• Wigner distribution, Wigner entropy, and Anomalous Transport of a Generalized Aubry-André model
• Critical dynamics and its interferometry in the one-dimensional p-wave-paired Aubry-André-Harper model
• A numerical study of the localization transition of Aubry-André type models
• Enhanced sensing of Stark weak field under the influence of Aubry-André-Harper criticality
• Luttinger Liquid phase in the Aubry-André Hubbard chain
• Dry Ten Martini Problem in the non-critical case
• Localization of interacting fermions in the Aubry-Andre' model
• Kohn's localisation in disordered fermionic systems with and without interactions
• Nearest neighbor tight binding models with an exact mobility edge in one dimension
• Topological Zero-Energy Modes in Gapless Commensurate Aubry-André-Harper Models
• Moire bands in twisted double-layer graphene
• Localization in one dimensional lattices with non-nearest-neighbor hopping: Generalized Anderson and Aubry-André models
• Predicted mobility edges in one-dimensional incommensurate optical lattices: An exactly solvable model of Anderson localization
• Fifty years of Anderson localization
• Exponential localization in one-dimensional quasiperiodic optical lattices
• Anderson localization of a non-interacting Bose-Einstein condensate
• Direct observation of Anderson localization of matter-waves in a controlled disorder
• Anderson Transitions
• The Ten Martini Problem
• Metal-insulator transition for the almost Mathieu operator
• Electron Localization in the Insulating State
• The Quantum-Mechanical Position Operator in Extended Systems
• Quantized Hall Conductance in a Two-Dimensional Periodic Potential
• Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions