Winding vector: how to annihilate two Dirac points with the same charge

TL;DR

This paper explains how two Dirac points with the same charge can merge or emerge through different scenarios by analyzing their winding numbers and their evolution on the Bloch sphere within a simple lattice model.

Contribution

It clarifies the apparent contradiction in Dirac point merging by analyzing winding number evolution in a two-band lattice model.

Findings

Dirac points with same charge can emerge and merge under different distortions.

Winding number evolution on the Bloch sphere explains the transition between scenarios.

The analysis is demonstrated within the Mielke lattice model with a flat band.

Abstract

The merging or emergence of a pair of Dirac points may be classified according to whether the winding numbers which characterize them are opposite ($+-$ scenario) or identical ($++$ scenario). From the touching point between two parabolic bands (one of them can be flat), two Dirac points with the {\it same} winding number emerge under appropriate distortion (interaction, etc), following the $++$ scenario. Under further distortion, these Dirac points merge following the $+-$ scenario, that is corresponding to {\it opposite} winding numbers. This apparent contradiction is solved by the fact that the winding number is actually defined around a unit vector on the Bloch sphere and that this vector rotates during the motion of the Dirac points. This is shown here within the simplest two-band lattice model (Mielke) exhibiting a flat band. We argue on several examples that the evolution between the two scenarios is general.