In-plane Wilson loop for measurement of quantized non-Abelian Berry flux

Band topology of anomalous quantum Hall insulators can be precisely addressed by computing the Chern numbers of constituent nondegenerate bands, describing the presence of quantized, Abelian Berry flux through the two-dimensional Brillouin zone. Can Berry flux be captured for the SU (2) Berry connection of two-fold degenerate bands in spinful materials preserving space-inversion (P) and time-reversal (T) symmetries without detailed knowledge of underlying basis? We address this question by investigating the correspondence between a non-Abelian generalization of Stokes' theorem and the manifestly gauge-invariant eigenvalues of Wilson loops computed along in-plane contours which preserve the underlying crystalline symmetry. The importance of this correspondence is elucidated by performing natural number resolved classification of ab initio band structures of three-dimensional, Dirac materials. Our work underscores how identification of quantized Berry flux, both Abelian and non-Abelian, offers a unified framework for addressing first-order and higher-order topology of insulators and semimetals.