From Quasiperiodicity to a Complete Coloring of the Kohmoto Butterfly

2025/09/28 by Ram Band, Band, Ram, Siegfried Beckus +1 · 1 citation
Agricultural and Biological Sciences · Biochemistry, Genetics and Molecular Biology · #FOS: Physical sciences #Mathematical Physics (math-ph) #Plant and Fungal Species Descriptions #Plant and animal studies #Plant biochemistry and biosynthesis #Quantum Physics (quant-ph)

paper · pdf · doi:10.48550/arxiv.2509.24025

openalex publication_date 2025/09/28 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

Abstract

The spectra of the Kohmoto model give rise to a fractal phase diagram, known as the Kohmoto butterfly. The butterfly encapsulates the spectra of all periodic Kohmoto Hamiltonians, whose index invariants are sought after. Topological methods - such as Chern numbers - are ill defined due to the discontinuous potential, and hence fail to provide index invariants. This Letter overcomes that obstacle and provides a complete classification of the Kohmoto model indices. Our approach encodes the Kohmoto butterfly as a spectral tree graph, reflecting the quasiperiodic nature via the periodic spectra. This yields a complete coloring of the phase diagram and a new perspective on other spectral butterflies.

Citations

• Augmentation and Bulk Edge Correspondence for one dimensional aperiodic tight binding operators
• A review of a work by Raymond: Sturmian Hamiltonians with a large coupling constant -- periodic approximations and gap labels
• Local topology for periodic Hamiltonians and fuzzy tori
• The Dry Ten Martini Problem for Sturmian Hamiltonians
• MFO Report: The dry ten Martini problem for Sturmian dynamical systems
• Gap Labelling for Discrete One-Dimensional Ergodic Schrödinger Operators
• Topology in quasicrystals
• The Fibonacci Hamiltonian

Cited by

• Augmentation and Bulk Edge Correspondence for one dimensional aperiodic tight binding operators

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