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Band topology of anomalous quantum Hall insulators can be precisely addressedby computing Chern numbers of constituent non-degenerate bands that describequantized, Abelian Berry flux through two-dimensional Brillouin zone. Can Chernnumbers be defined for $SU(2)$ Berry connection of two-fold degenerate bands ofmaterials preserving space-inversion ($\mathcal{P}$) and time-reversal($\mathcal{T}$) symmetries or combined $\mathcal{PT}$ symmetry, withoutdetailed knowledge of underlying basis? We affirmatively answer this questionby employing a non-Abelian generalization of Stokes' theorem and describe amanifestly gauge-invariant method for computing magnitudes of quantized $SU(2)$Berry flux (spin-Chern number) from eigenvalues of Wilson loops. The power ofthis method is elucidated by performing $\mathbb{N}$-classification of \emph{abinitio} band structures of three-dimensional, Dirac materials. Our workoutlines a unified framework for addressing first-order and higher-ordertopology of insulators and semimetals, without relying on detailed symmetrydata.

Non-Abelian Stokes theorem and quantized Berry flux