In this paper, we review theoretical investigations on the origin of the orbital selective phase where localized and itinerant electrons coexist in the d shell at intermediate strength of the on-site Coulomb interactions between electrons. In particular, the effect of spatial fluctuations on the phase diagram of the two-orbital Hubbard model with unequal bandwidths is discussed. And different band dispersions in different orbitals as well as different magnetically ordered states in different orbitals which are responsible for orbital selective phase transitions are emphasized. This is due to the fact that these two mechanisms are independent of the Hund's rule coupling, and are completely distinct from other well-known mechanisms like orbitals of unequal bandwidths and orbitals with different degeneracies. Moreover, crystal field splitting is not required in these two recently proposed mechanisms.
Spatial fluctuations of spin and orbital nature in the two-orbital Hubbard model