Part I: Staggered index and 3D winding number of Kramers-degenerate bands

Abstract

For three-dimensional (3D) crystalline insulators, preserving space-inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetries, the third homotopy class of two-fold, Kramers-degenerate bands is described by a 3D winding number $n_{3,j}\in \mathbb{Z}$, where $j$ is the band index. It governs space group symmetry-protected, instanton or tunneling configurations of $SU(2)$ Berry connection, and the quantization of magneto-electric coefficient ${\theta }_j=n_{3,j}\pi$. We show that $|n_{3,j}|$ for realistic, \emph{ab initio} band structures can be identified from a staggered symmetry-indicator ${\kappa }_{AF,j}\in \mathbb{Z}$ and the gauge-invariant spectrum of $SU(2)$ Wilson loops. The procedure is elucidated for $4$-band and $8$-band tight-binding models and \emph{ab initio} band structure of Bi, which is a ${\mathbb{Z}}_2$-trivial, higher-order, topological crystalline insulator. When the tunneling is protected by $C_{nh}$ and $D_{nh}$ point groups, the proposed method can also identify the signed winding number $n_{3,j}$. Our analysis distinguishes between magneto-electrically trivial ($\theta =0$) and non-trivial ($\theta =2s\pi$, with $s\ne 0$) topological crystalline insulators. In Part II, we demonstrate $\mathbb{Z}$-classification of $\theta$ by computing induced electric charge (Witten effect) on magnetic Dirac monopoles.

Comments: 18 pages, 12 figures; substantial revision of staggered symmetry indicator and Wilson loop calculations for tight-binding models and ab initio band structure of bismuth. Real space response is studied in arXiv:2206.10636[ https://doi.org/10.48550/arXiv.2206.10636]