  We investigate the magnetic-field effects on a two-orbital Hubbard model that
describes multiple spin states. Cobalt oxides have been investigated as
materials possessing spin-state degrees of freedom due to the interplay between
the Hund coupling interaction and crystalline field effect. In the competing
region, quantum hybridizations between distinct spin states are expected to
emerge, corresponding to excitonic condensation. Applied magnetic fields could
also induce such a competition. To understand magnetic-field effects on
excitonic condensation in multi-orbital systems, it is crucial to account for
contributions from both spin and orbital degrees of freedom to magnetic
properties. Here, we study field-induced phenomena in the two-orbital Hubbard
model by focusing on the role of the orbital angular momentum. We
comprehensively analyze this model on a square lattice employing the
Hartree-Fock approximation. Omitting contributions from the orbital moment, we
find that an applied magnetic field gives rise to two excitonic phases, besides
the spin-state ordered phase, between the nonmagnetic low-spin and
spin-polarized high-spin phases. One of these excitonic phases manifests a
staggered-type spin-state order, interpreted as an excitonic supersolid state.
Conversely, the other phase is not accompanied by it and exhibits only a spin
polarization due to the applied magnetic field. When spin-orbit coupling is
present, this phase displays a ferrimagnetic spin alignment attributed to spin
anisotropy. Our analysis also reveals that incorporating the contribution of
the orbital magnetic moment to the Zeeman term significantly alters the overall
structure of the phase diagram. Notably, the orbital magnetization destabilizes
the excitonic phase in contrast to scenarios without this contribution. We also
discuss the relevance of our findings to real materials, such as cobalt oxides.
