Validity of the single-band model

There is an ongoing debate as to whether the generalized two-band Hubbard model is equivalent to the single-band {ital t}-{ital J} model. Starting from a two-band model, by constructing the singlet and triplet bands and considering their hybridization, we are able to reproduce very well the exact solution in a ferromagnetic background. Hence the singlet-band model is a good approximation for low-energy states, although the spin expectation is not exactly zero. The oxygen-oxygen direct hopping is also considered to argue that the single-band model might still be valid even in this case.