From quasiperiodicity to a complete coloring of the Kohmoto butterfly

¹ Department of Mathematics, Technion-Israel Institute of Technology - Haifa, Israel
² Institute of Mathematics, University of Potsdam - Potsdam, Germany

Received: 18 December 2025
Accepted: 27 March 2026

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 are ill defined due to the discontinuous periodic potentials, and hence fail to provide index invariants. This letter overcomes that obstacle and provides a complete classification of the Kohmoto model indices —suggesting new physical invariants instead of Chern 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.