Effects of magnetic fields and orbital angular momentum on excitonic condensation in two-orbital Hubbard model

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.