# Stiefel-Whitney classes and topological phases in band theory

## TL;DR

This paper reviews recent advances in understanding topological phases in systems with space-time inversion symmetry, focusing on Stiefel-Whitney classes as invariants for real wave functions in various topological materials.

## Contribution

It introduces the use of Stiefel-Whitney numbers to classify topological phases in systems with anti-unitary symmetries like $I_{ST}$, expanding the topological classification beyond Berry curvature-based invariants.

## Key findings

- Stiefel-Whitney numbers classify topological phases with real wave functions.
- First Stiefel-Whitney number relates to 1D topological insulators and Dirac semimetals.
- Second Stiefel-Whitney number characterizes 3D nodal line semimetals with $Z_2$ monopole charge.

## Abstract

In this article, we review the recent progress in the study of topological phases in systems with space-time inversion symmetry $I_{\text{ST}}$. $I_{\text{ST}}$ is an anti-unitary symmetry which is local in momentum space and satisfies $I_{\text{ST}}^2=1$ such as $PT$ or $C_{2}T$ symmetry where $P$, $T$, $C_2$ indicate inversion, time-reversal, and two-fold rotation symmetries, respectively. Under $I_{\text{ST}}$, the Hamiltonian and the Bloch wave function can be constrained to be real-valued, which makes the Berry curvature and the Chern number to vanish. In this class of systems, gapped band structures of real wave functions can be topologically distinguished by Stiefel-Whitney numbers instead. The first and second Stiefel-Whitney numbers $w_1$ and $w_2$, respectively, are the corresponding invariants in 1D and 2D, which are equivalent to the quantized Berry phase and the $Z_2$ monopole charge, respectively. We first describe the topological phases characterized by the first Stiefel-Whitney number, including 1D topological insulators with quantized charge polarization, 2D Dirac semimetals, and 3D nodal line semimetals. Next we review how the second Stiefel-Whitney class characterizes the 3D nodal line semimetals carrying a $Z_{2}$ monopole charge. In particular, we explain how the second Stiefel-Whitney number $w_2$, the $Z_{2}$ monopole charge, and the linking number between nodal lines are related. Finally, we review the properties of 2D and 3D topological insulators characterized by the nontrivial second Stiefel Whitney class.