Three-dimensional spinless Euler insulators with rotational symmetry

The Euler class is a \ensuremath{\mathbb{Z}}-valued topological invariant characterizing two real bands protected by ${C}_{2z}$\ensuremath{\mathcal{T}} symmetry. The authors investigate here the relationship between the Euler class and rotation symmetry in spinless insulators with additional ${C}_{4z}$ or ${C}_{6z}$ symmetry. They derive general formulas relating the Euler class to rotation eigenvalues and two types of representation-protected invariants. They further construct tight-binding models and present numerical calculations of the Wilson loop spectra and topological surface states to support their analysis.