Hybrid-order topological phase and transition in 1<i>H</i> transition metal compounds

Inspired by recent experimental observations of hybrid topological states [Hossain et al. Nature 628, 527 (2024)], we predict hybrid-order topological insulators in 1H transition metal compounds (TMCs), where both second-order and first-order topological (FOT) states coexist near the Fermi level. Initially, 1H-TMCs exhibit a second-order topological phase due to the d orbital bandgap. Upon coupling of p and d orbitals through the crystal field effect, first-order topological characteristics emerge. This hybrid-order topological phase transition can be tuned via crystal field effects. Combined with first-principles calculations, we illustrate the phase transition with WTe2 and NbSe2. The WTe2 exhibits hybrid-order under ambient conditions, while NbSe2 transitions to hybrid-order under pressure. Additionally, the first-order topological bandgap in the HyOTI demonstrates a strong spin Hall effect. Our findings reveal a hybrid-order topological phase in two-dimensional electron materials and underscore spintronic applications.