  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.

Mesoscale and Nanoscale Physics