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We investigate a generalized multi-orbital tight-binding model on atriangular lattice, a system prevalent in a wide range of two-dimensionalmaterials, and particularly relevant for simulating transition metaldichalcogenide monolayers. We show that the interplay between spin-orbitcoupling and different symmetry-breaking mechanisms leads to the emergence offour distinct topological phases [Eck, P., \textit{et al.}, Phys. Rev. B, 107(11), 115130 (2023)]. Remarkably, this interplay also triggers the orbital Halleffect with distinguished characteristics. Furthermore, by employing theLandauer-B\"uttiker formula, we establish that in the orbital Hall insulatingphase, the orbital angular momentum is carried by edge states present innanoribbons with specific terminations. We also show that, as expected, they donot have topological protection against the disorder of the edge statesbelonging to a first-order topological insulator.

Orbital Hall effect and topology on a two-dimensional triangular lattice: from bulk to edge