Spin/valley effects and finite-temperature effects on the Hall conductivity of jacutingaite

We theoretically investigate the spin-Hall and valley-Hall transport properties of monolayer jacutingaite (Pt2HgSe3) under an off-resonant circularly polarized light field and a perpendicular electric field. Using a low-energy massive Dirac model combined with linear resp Alipourzadeh onse theory, we analyze how spin-orbit coupling, inversion-symmetry breaking, optical driving, and finite temperature jointly influence the Hall conductivities. At zero temperature, the spin-Hall and valley-Hall conductivities exhibit step-like plateau features, reflecting the underlying topological character of the massive Dirac bands. The off-resonant circularly polarized light effectively renormalizes the Dirac mass and modifies the band structure, enabling controllable transitions between distinct Hall response regimes. While the spin-Hall effect arises intrinsically from strong spin-orbit coupling, the valley-Hall effect appears only when inversion symmetry is broken by a staggered sublattice potential, with characteristic critical points associated with gap closing. At finite temperatures, thermal broadening smooths these plateau-like features into continuous Fermi energy dependent responses and reduces the magnitude of both Hall effects. Nevertheless, the corresponding topological signatures remain robust at low and moderate temperatures, particularly near charge neutrality. We further show that circularly polarized light alone cannot generate a finite valley-Hall response at finite temperatures without inversion symmetry breaking. These results demonstrate that the combined action of optical driving, electric field control, and thermal effects provides an effective route to manipulate spin and valley transport in jacutingaite, highlighting its potential for tunable spintronic and valleytronic applications.