Which theory can best explain different types of orbital ordering in manganites?

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Tapan Chatterji

The orbital ordering in manganites is best explained by the Kugel-Khomskii spin-orbital model. The Hamiltonian consists of three terms: (1) spin part J, Si.Sj, (2) the orbital part J2(TiTj), and the mixed term J3(SiSj)(TiTj). Here Ti and Tj are pseudospins and Si and Sj are ordinary spins.

You can read the chapter 6 (Orbital effects in manganites by J. van den Drink, G. Khaliullin and D. Khomskii) of the book"Colosal Magnetoresistive Manganites", ed. Tapan Chatterji, Kluwer Academic Publishers, Dordrecht (2004). Also you should consult the recent text book (I cannot remember the title) by D. Khomskii.