Patch Euler class

We have implemented in Mathematica an algorithm that computes the patch Euler class of a pair of energy bands on a region (the 'patch') with a boundary. The code is downloadable from ResearchGate.

The patch Euler class is an invariant for systems with C₂T-symmetry (and also for spinless systems with PT-symmetry), which determines whether a pair (or a larger collection) of Dirac points are able to annihilate along a prescribed trajectory. The algorithm, as described in the supplementary material to

• A. Bouhon, Q.S. Wu, R.-J. Slager, H. Weng, O. V. Yazyev, and T. Bzdušek, Nat. Phys. 16, 1137–1143 (2020), DOI:10.1038/s41567-020-0967-9,

corresponds to a computationally efficient way to extract the most relevant bits of information from the non-Abelian topological invariant introduced previously in

• Q.S. Qu, A. A. Soluyanov, and T. Bzdušek, Science 365, 1273–1277 (2019), DOI:10.1126/science.aau8740.

The data above plot the Euler curvature computed from first principles near two distinct pairs of band nodes of strained ZrTe. Integral of Euler curvature constitutes one of the contribution towards the patch Euler class; the other is a boundary integral of Berry connection (not plotted). Upon collision, the pair of band nodes are preserved in the top scenario, while the pair annihilate in the second scanario – even though no difference is present in the Berry phases and Chern numbers of between the two cases.