A general multi-orbital Hubbard model, which includes on-site inter-orbital electron hoppings, is introduced and studied. It is shown that the on-site inter-orbital single electron hopping is one of the most basic interactions. Two electron spin-flip and pair-hoppings are shown to be correlation effects of higher order than the on-site inter-orbital single hopping. It is shown how the double and higher hopping interactions can be well-defined for arbitrary systems. The two-orbital Hubbard model is studied numerically to demonstrate the influence of the single electron hopping effect, leading to a change of the shape of the bands and a shrinking of the difference between the two bands. Inclusion of the on-site inter-orbital hopping suppresses the so-called orbital-selective Mott transition.PACS numbers: 71.10.Fd, 71.27.+a, 71.30.+h Strongly correlated electron systems have drawn great interest, as they exhibit many exotic phenomena, such as metal-insulator transitions, unusual forms of magnetism, superconductivity, and heavy-fermion behavior.1,2 These systems are therefore investigated theoretically in both model and ab initio calculations. One of the most essential tasks is to model the system appropriately, i.e., such that a realistic physical picture can be attained. In condensed matter theory the Hubbard model, which was originally proposed in the early sixties, 3-5 is one of the simplest, yet also the most important and most frequently studied lattice model to investigate strongly correlated electron systems. It sets up a competition between an inter-site quantum mechanical hopping term and an on-site Coulomb interaction term. As a consequence the model can describe various non-trivial phenomena. Due to its simplicity and because the model has captured the essence of strongly correlated electron systems, the Hubbard model has been widely used.
6-9In a realistic situation an atom in a correlated system will usually have several partial filled orbitals and should therefore be described with a multi-orbital (MO) Hubbard model. With the inclusion of orbital degrees of freedom, inter-orbital interactions have to be included for such a system. The inter-site hoppings should now be a sum of the inter-site intra-orbital hoppings and the inter-site inter-orbital hoppings, where the two kinds of hopping are defined according to the orbital indices of the start and the end orbitals for hopping electron are identical or not, respectively. Moreover, besides the remain competition of the inter-site hopping and on-site Coulomb interactions, one new kind of on-site interactions will join into this competition, i.e., the on-site inter-orbital hoppings. Acting as the inter-site hoppings, the on-site inter-orbital hoppings are also associated to the kinetic energy of electrons. All the inter-site hoppings, on-site inter-orbital hoppings and on-site Coulomb interactions (which also divides as on-site intra-orbital Coulomb interactions and on-site inter-orbital Coulomb interactions) constitute a competition for various inter...