Topological Lévy crystal insulator

Abstract

We investigate the band topological aspects of a Lévy crystal. This is done by first generalizing the two-dimensional Hamiltonian in fractional quantum mechanics into a two-band Hamiltonian model with fractional dispersion α that reduces to the Dirac Hamiltonian for α = 1. We then calculate the Berry connection and the Berry curvature. We found that there is a redistribution of the Berry curvature in the momentum space as α is varied across α = 1, showing that the fractional dispersion changes the quantum band geometry of the system. Despite the redistribution of the Berry curvature, we found that the change in Chern number across the topological transition remains quantized to an integer value even for fractional α.