Three-dimensional spinless Euler insulators with rotational symmetry

The Euler class is a Z-valued topological invariant that characterizes a pair of real bands in a two-dimensional Brillouin zone. One of the symmetries that permits its definition is C2zT, where C2z denotes a twofold rotation about the z axis and T denotes time-reversal symmetry. Here, we study three-dimensional spinless insulators characterized by the Euler class, focusing on the case where additional C4z or C6z rotational symmetry is present, and investigate the relationship between the Euler class of the occupied bands and their rotation eigenvalues. We first consider two-dimensional systems and clarify the transformation rules for the real Berry connection and curvature under point group operations, using the corresponding sewing matrices. Applying these rules to C4z and C6z operations, we obtain explicit formulas that relate the Euler class to the rotation eigenvalues at high-symmetry points. We then extend our analysis to three-dimensional systems, focusing on the difference in the Euler class between the two C2zT-invariant planes. We derive analytic expressions that relate the difference in the Euler class to two types of representation-protected invariants and analyze their phase transitions. We further construct tight-binding models and perform numerical calculations to support our analysis and elucidate the bulk-boundary correspondence.