Topological map of the Hofstadter butterfly: Fine structure of Chern numbers and Van Hove singularities

• Use a higher dimensional approach to build a topological map of the Hofstadter butterfly.
• There is a fine structure of Chern numbers around each rational flux.
• Van Hove singularities are limiting points for topological sequences of the fine flux.

The Hofstadter butterfly is a quantum fractal with a highly complex nested set of gaps, where each gap represents a quantum Hall state whose quantized conductivity is characterized by topological invariants known as the Chern numbers. Here we obtain simple rules to determine the Chern numbers at all scales in the butterfly fractal and lay out a very detailed topological map of the butterfly by using a method used to describe quasicrystals: the cut and projection method. Our study reveals the existence of a set of critical points that separates orderly patterns of both positive and negative Cherns that appear as a fine structure in the butterfly. This fine structure can be understood as a small tilting of the projection subspace in the cut and projection method and by using a Chern meeting formula. Finally, we prove that the critical points are identified with the Van Hove singularities that exist at every band center in the butterfly landscape.