Euler band topology in spin-orbit coupled magnetic systems

TL;DR

This paper explores the topological properties of magnetic Euler bands in spin-orbit coupled systems, revealing new phases like magnetic Euler insulators and their phase transitions mediated by topological semimetals, with candidate materials proposed.

Contribution

It introduces the concept of magnetic Euler insulators in spin-orbit coupled magnetic systems and identifies candidate materials through first-principles calculations.

Findings

Magnetic Euler bands can emerge in quantum spin Hall insulators with in-plane Zeeman fields.

Topological phase transitions involve Dirac nodes with non-Abelian charges and braiding.

ZrTe$_5$ bilayers under in-plane ferromagnetism are potential magnetic Stiefel-Whitney insulators.

Abstract

The Euler class characterizes the topology of two real bands isolated from other bands in two-dimensions. Despite various intriguing topological properties predicted up to now, the candidate real materials hosting electronic Euler bands are extremely rare. Here, we show that in a quantum spin Hall insulator with two-fold rotation $C_{2z}$ about the $z$-axis, a pair of bands with nontrivial $\mathbb{Z}_2$ invariant turn into magnetic Euler bands under in-plane Zeeman field or in-plane ferromagnetic ordering. The resulting magnetic insulator generally carries a nontrivial second Stiefel-Whitney invariant. In particular, when the topmost pair of occupied bands carry a nonzero Euler number, the corresponding magnetic insulator can be called a magnetic Euler insulator. Moreover, the topological phase transition between a trivial magnetic insulator and a magnetic Stiefel-Whitney or Euler insulator is mediated by a stable topological semimetal phase in which Dirac nodes carrying non-Abelian topological charges exhibit braiding processes across the transition. Using the first-principles calculations, we propose various candidate materials hosting magnetic Euler bands. We especially show that ZrTe$_5$ bilayers under in-plane ferromagnetism are a candidate system for magnetic Stiefel-Whitney insulators in which the non-Abelian braiding-induced topological phase transitions can occur under pressure.