Topological Phase Transitions from Harper to Fibonacci Crystals

TL;DR

This paper investigates topological phase transitions in Harper and Fibonacci chains by interpolating between them, revealing how band degeneracies and Chern number changes occur and relate to classical energy structures.

Contribution

It introduces an interpolating Hamiltonian connecting Harper and Fibonacci crystals, analyzing topological transitions and their spectral signatures across frequencies.

Findings

Topological phase transitions occur at band degeneracies.

Degeneracies align with classical separatrix energies.

Spectral features of Fibonacci crystals influence Hofstadter butterfly patterns.

Abstract

Topological properties of Harper and generalized Fibonacci chains are studied in crystalline cases, i.e., for rational values of the modulation frequency. The Harper and Fibonacci crystals at fixed frequency are connected by an interpolating one-parameter Hamiltonian. As the parameter is varied, one observes topological phase transitions, i.e., changes in the Chern integers of two bands due to the degeneracy of these bands at some parameter value. For small frequency, corresponding to a semiclassical regime, the degeneracies are shown to occur when the average energy of the two bands is approximately equal to the energy of the classical separatrix. Spectral and topological features of the Fibonacci crystal for small frequency leave a clear imprint on the corresponding Hofstadter butterfly for arbitrary frequency.