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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. Theinterplay between the staggered potential and exchange field offers thepossibility of attaining different topological phases. By analyzing the Berrycurvatures and computing the Chern numbers and Hall conductivities, wedemonstrate that the system is time-reversal-symmetry-broken quantum spin Hallinsulator when $m_b<\lambda_{so}$, where $m_b$ is the exchange field operatingon 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 atvalley $K(K^{\prime})$ is topologically trivial (non-trivial) with Chern number$C=1$ and valley Chern number $C_v=-1$, indicating that the system isvalley-polarized quantum anomalous Hall insulator. We show that the topology ofeach valley is swapped (the Chern number becomes $C=-1$) by reversing the signof the exchange field. The system transitions to a valley-polarized metal andquantum valley Hall phase as V increases. Along the phase boundaries, weobserve a single-Dirac cone semimetal states. These findings shed more light onthe possibility of realizing and controlling topological phases in spintronicsand valleytronics devices.

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