Geometry, Number Theory and the Butterfly Spectrum of Two-Dimensional Bloch Electrons

Abstract

We take a deeper dive into the geometry and the number theory that underlay the butterfly graphs of the Harper and the generalized Harper models of Bloch electrons in a magnetic field. Root of the number theoretical characteristics of the fractal spectrum is traced to a close relationship between the Farey tree -- the hierarchical tree that generates all rationals and the Wannier diagram -- a graph that labels all the gaps of the butterfly graph. The resulting Farey-Wannier hierarchical lattice of trapezoids provides geometrical representation of the nested pattern of butterflies in the butterfly graph. Some features of the energy spectrum such as absence of some of the Wannier trajectories in the butterfly graph fall outside the number theoretical framework, can be stated as a simple rule of "minimal violation of mirror symmetry". In a generalized Harper model, Farey-Wannier representation prevails as the lattice regroups to form some hexagonal unit cells creating new {\it species} of butterflies