  This paper presents a theoretical description of both the valley Zeeman
effect (g-factors) and Landau levels in two-dimensional H-phase transition
metal dichalcogenides (TMDs) using the Luttinger-Kohn approximation with
spin-orbit coupling. At the valley extrema in TMDs, energy bands split into
Landau levels with a Zeeman shift in the presence of a uniform out-of-plane
external magnetic field. The Landau level indices are symmetric in the $K$ and
$K'$ valleys. We develop a numerical approach to compute the single band
g-factors from first principles without the need for a sum over unoccupied
bands. Many-body effects are included perturbatively within the GW
approximation. Non-local exchange and correlation self-energy effects in the GW
calculations increase the magnitude of single band g-factors compared to those
obtained from density functional theory. Our first principles results give
spin- and valley-split Landau levels, in agreement with recent optical
experiments. The exciton g-factors deduced in this work are also in good
agreement with experiment for the bright and dark excitons in monolayer
WSe$_2$, as well as the lowest-energy bright excitons in MoSe$_2$-WSe$_2$
heterobilayers with different twist angles.

Materials Science;Mesoscale and Nanoscale Physics;Computational Physics