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

TL;DR

This paper introduces a method to identify the 3D winding number of Kramers-degenerate bands in topological insulators using symmetry indicators and Wilson loops, aiding the classification of topological crystalline phases.

Contribution

It proposes a novel approach to determine the 3D winding number from ab initio band structures and symmetry indicators, enhancing topological classification tools.

Findings

Identifies the 3D winding number from ab initio data.

Distinguishes trivial and non-trivial topological phases.

Applies method to Bi and tight-binding models.

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 ($\theta=2 s \pi$, with $s \neq 0$) topological crystalline insulators. In Part II, we demonstrate $\mathbb{Z}$-classification of $\theta$ by computing induced electric charge (Witten effect) on magnetic Dirac monopoles.