Topological Map of the Hofstadter Butterfly and Van Hove Singularities

TL;DR

This paper provides a detailed topological map of the Hofstadter butterfly, revealing how Van Hove singularities relate to topological invariants and induce quantum phase transitions under perturbations.

Contribution

It introduces simple rules for determining Chern numbers across the Hofstadter butterfly and links Van Hove singularities to topological collapses and phase transitions.

Findings

Identified critical points where topological patterns annihilate.

Mapped Van Hove singularities as topological anomalies.

Demonstrated perturbation-induced quantum phase transitions.

Abstract

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. Our study reveals the existence of a set of critical points, each corresponding to a macroscopic annihilation of orderly patterns of both the positive and the negative Cherns that appears as a fine structure in the butterfly. Such topological collapses are identified with the Van Hove singularities that exists at every band center in the butterfly landscape. We thus associate a topological character to the Van Hove anomalies. Finally, we show that this fine structure is amplified under perturbation, inducing quantum phase transitions to higher Chern states in the system.