Tunable topological phases in monolayer Pt$_2$HgSe$_3$ with exchange fields

We investigate topological phases of monolayer jacutingaite (Pt$_2$HgSe$_3$) that arise when considering the competing effects of spin-orbit coupling (SOC), magnetic exchange interactions, and staggered sublattice potential V. The interplay between the staggered potential and exchange field offers the possibility of attaining different topological phases. By analyzing the Berry curvatures and computing the Chern numbers and Hall conductivities, we demonstrate that the system is time-reversal-symmetry-broken quantum spin Hall insulator when $m_b<\lambda_{so}$, where $m_b$ is the exchange field operating on the bottom Hg sublattice and $\lambda_{so}$ is the intrinsic SOC. For $m_b > \lambda_{so}$ and in the presence of Rashba SOC, we find that the band gap at valley $K(K^{\prime})$ is topologically trivial (non-trivial) with Chern number $C=1$ and valley Chern number $C_v=-1$, indicating that the system is valley-polarized quantum anomalous Hall insulator. We show that the topology of each valley is swapped (the Chern number becomes $C=-1$) by reversing the sign of the exchange field. The system transitions to a valley-polarized metal and quantum valley Hall phase as V increases. Along the phase boundaries, we observe a single-Dirac cone semimetal states. These findings shed more light on the possibility of realizing and controlling topological phases in spintronics and valleytronics devices.