Non-Abelian Stokes theorem and quantized Berry flux

TL;DR

This paper develops a gauge-invariant method using a non-Abelian Stokes theorem to compute quantized $SU(2)$ Berry flux in degenerate bands, enabling topological classification of materials without detailed basis knowledge.

Contribution

It introduces a non-Abelian generalization of Stokes' theorem to define and compute $SU(2)$ Berry flux in degenerate bands, broadening topological analysis tools.

Findings

Method applied to classify band topology of Dirac materials

Enables calculation of spin-Chern numbers without basis details

Provides a unified framework for topological insulators and semimetals

Abstract

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