  We show that H-phase transition metal dichalcogenides (TMDs) monolayers such
as MoS$_2$ and WSe$_2$, are orbital Hall insulators. They present very large
orbital Hall conductivity plateaus in their semiconducting gap, where the spin
Hall conductivity vanishes. Our results open the possibility of using TMDs for
orbital current injection and orbital torque transfers that surpass their
spin-counterparts in spin-orbitronics devices. The orbital Hall effect (OHE) in
TMD monolayers occurs even in the absence of spin-orbit coupling. It can be
linked to exotic momentum-space Dresselhaus-like orbital textures, analogous to
the spin-momentum locking in 2D Dirac fermions that arise from a combination of
orbital attributes and lattice symmetry.
