Euler band topology in spin-orbit coupled magnetic systems

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.