Manipulation of Dirac points in graphene-like crystals

TL;DR

This paper reviews how Dirac points in 2D crystals like graphene can move and merge, classifying scenarios by topological invariants, and analyzing their Landau level spectra.

Contribution

It provides a topological classification of Dirac point merging scenarios and calculates their Landau level spectra using semiclassical quantization.

Findings

Different merging scenarios are characterized by distinct topological invariants.

Landau level spectra depend on the number of Dirac points enclosed.

Zero energy Landau levels are topologically protected in various scenarios.

Abstract

We review different scenarios for the motion and merging of Dirac points in two dimensional crystals. These different types of merging can be classified according to a winding number (a topological Berry phase) attached to each Dirac point. For each scenario, we calculate the Landau level spectrum and show that it can be quantitatively described by a semiclassical quantization rule for the constant energy areas. This quantization depends on how many Dirac points are enclosed by these areas. We also emphasize that different scenarios are characterized by different numbers of topologically protected zero energy Landau levels