  Despite the recent success of identifying experimental signatures of the
orbital Hall effect (OHE), the research on the microscopic mechanisms behind
this unique phenomenon is still in its infancy. Here, using a gapped 2D Dirac
material as a model system of the OHE, we develop a microscopic theory of
orbital transport which captures extrinsic disorder effects non-perturbatively.
We show that it predicts several hitherto unknown effects, including (i) a
strong dependence of the OHE conductivity with the strength and symmetry of the
impurity scattering potential, and (ii) a smooth crossover from intrinsic to
extrinsic OHE as a function of the Fermi energy. In contrast to previous
(perturbative) studies, the OHE is found to exhibit bona fide diffusive
behavior in the dilute impurity limit, which we trace back to the dominance of
skew scattering-type processes. Our work unveils the crucial nature of
nonperturbative vertex corrections for a correct description of extrinsic
orbital Hall transport and confirms common short-range impurities as key
enablers of the OHE.
