Spin-Orbit-Free Topological Insulators

TL;DR

This paper reviews spin-orbit-free topological insulators, focusing on crystalline symmetries and novel invariants like halved-mirror chirality and bent Chern numbers, expanding understanding of topological phases without spin-orbit coupling.

Contribution

It introduces new topological invariants and describes phases of insulators without spin-orbit coupling, broadening the scope of topological materials beyond traditional models.

Findings

Identification of topological phases with robust surface modes

Introduction of halved-mirror chirality and bent Chern number invariants

Description of Weyl semimetallic phase bridging gapped phases

Abstract

In this lecture for the Nobel symposium, we review previous research on a class of translational-invariant insulators without spin-orbit coupling. These may be realized in intrinsically spinless systems such as photonic crystals and ultra-cold atoms. Some of these insulators have no time-reversal symmetry as well, i.e., the relevant symmetries are purely crystalline. Nevertheless, topological phases exist which are distinguished by their robust surface modes. To describe these phases, we introduce the notions of (a) a halved-mirror chirality: an integer invariant which characterizes half-mirror planes in the 3D Brillouin zone, and (b) a bent Chern number: the traditional Thouless-Kohmoto-Nightingale-Nijs invariant generalized to bent 2D manifolds. Like other well-known topological phases, their band topology is unveiled by the crystalline analog of Berry phases, i.e., parallel transport across certain non-contractible loops in the Brillouin zone. We also identify certain topological phases without any robust surface modes - they are uniquely distinguished by parallel transport along bent loops, whose shapes are determined by the symmetry group. Finally, we describe the Weyl semimetallic phase that intermediates two distinct, gapped phases.