Dirac points annihilation and its obstruction characterized by Euler number and quaternionic charges in kagome lattice

We investigate the topological phenomenon of Dirac point annihilation and its obstruction in three-band, real symmetric Hamiltonians with time-reversal symmetry, and their relation to the Euler number, a well-known topological invariant. For this purpose, we study the example of the kagome lattice using a simple tight-binding model. By tuning the parameters of the lattice continuously, we illustrate situations where two Dirac points are able to annihilate, and others, where this annihilation is topologically obstructed. For a system with no gaps between the three bands, like in the kagome lattice, the Euler number of two bands is ill-defined on the whole Brillouin zone, which requires the introduction of the so-called ``patch" Euler number on a subregion without additional degeneracies coming from the third band. A non-zero patch Euler number means that the annihilation of the Dirac points is impossible. We also illustrate another point of view, using homotopy theory, associating the Dirac points with quaternionic charges. We prove that the non-abelian braiding of the Dirac points in k-space conjugates their quaternionic charge and explains the possible obstruction to the annihilation of Dirac points. Finally, we show that the proposed deformation of the kagome lattice can be achieved in realistic photonic systems.