Observing topological phase transition in ferromagnetic transition metal dichalcogenides

Two-dimensional transition metal dichalcogenides (TMDs) are considered a suitable platform to study topological properties such as the quantum anomalous Hall effect. However, this quantum transport property is usually found in systems with a small band gap. For large-band-gap TMDs, the nature of quantum transport is difficult to find. In this work, with the analysis of the $k\ifmmode\cdot\else\textperiodcentered\fi{}p$ model, we investigate the topological phases of ferromagnetic TMDs in the cases of a large gap and small gap. By analyzing the orbital Berry curvature, we reveal that the orbital Hall effect in the system possesses a nontrivial topological invariant ${C}_{L}=\ensuremath{-}1$ when $\mathrm{sgn}({\mathrm{\ensuremath{\Delta}}}_{+1}{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{-}1})>0$, where ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\tau}}$ represents the band gap at the $\ensuremath{\tau}$ valley (K or ${K}^{\ensuremath{'}}$ valley). Consequently, the large-gap ferromagnetic TMDs exhibit properties of an orbital Hall insulator. In the case of $\mathrm{sgn}({\mathrm{\ensuremath{\Delta}}}_{+1}{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{-}1})<0$, we demonstrate that the small-gap system transforms into a quantum anomalous Hall insulator with a Chern number $C=1$. Both topological phases are valley polarized and robust in the presence of an out-of-plane magnetic moment. Subsequently, we illustrate this transition from an orbital Hall insulator to a quantum anomalous Hall insulator in ${\mathrm{FeCl}}_{2}$, a representative ferromagnetic TMD, by tuning the Coulomb correction ${U}_{\mathrm{eff}}$ to manipulate the valley band gaps.