  Motivated by the recent angle-resolved photoemission spectroscopy (ARPES) on
FeSe and iron pnictide families of iron-based superconductors, we have studied
the orbital nematic order and its interplay with antiferromagnetism within the
two-orbital Hubbard model. We used random phase approximation (RPA) to
calculate the dependence of the orbital and magnetic susceptibilities on the
strength of interactions and electron density (doping). To account for strong
electron correlations not captured by RPA, we further employed non-perturbative
variational cluster approximation (VCA) capable of capturing symmetry broken
magnetic and orbitally ordered phases. Both approaches show that the electron
and hole doping affect the two orders differently. While hole doping tends to
suppress both magnetism and orbital ordering, the electron doping suppresses
magnetism faster. Crucially, we find a realistic parameter regime for moderate
electron doping that stabilizes orbital nematicity in the absence of long-range
antiferromagnetic order. This is reminiscent of the non-magnetic orbital
nematic phase observed recently in FeSe and a number of iron pnictide materials
and raises the possibility that at least in some cases, the observed electronic
nematicity may be primarily due to orbital rather than magnetic fluctuations.
