Orbital ordering phenomena in d- and f-electron systems

In recent decades, novel magnetism of d- and f-electron compounds has been discussed very intensively both in experimental and theoretical research fields of condensed matter physics. It has been recognized that these material groups are in the same category of strongly correlated electron systems, while the low-energy physics of d- and f-electron compounds has been separately investigated in rather different manners. One of the common features of both d and f-electron systems is certainly the existence of active orbital degree of freedom, but in f-electron materials, due to the strong spin-orbit interaction in rare-earth and actinide ions, the physics seems to be quite different from that of d-electron systems. In general, when the number of internal degrees of freedom and relevant interactions is increased, it is possible to obtain a rich phase diagram including large varieties of magnetic phases by using several kinds of theoretical techniques. However, we should not be simply satisfied with the reproduction of a rich phase diagram. It is believed that the more essential point is to seek a simple principle penetrating complicated phenomena in common with d- and f-electron materials, which opens the door to a new stage in orbital physics. In this sense, it is considered to be an important task of this paper to explain common features of magnetism in d- and f-electron systems from a microscopic viewpoint, using a key concept of orbital ordering, in addition to the review of the complex phase diagram of each material group. As a typical d-electron complex material exhibiting orbital order, we first focus on perovskite manganites, in which remarkable colossal magneto-resistance effect has been intensively studied. The manganites provide us with a good platform to understand that a simple mechanism works for the formation of complex spin, charge and orbital ordering. We also explain intriguing striped charge ordering on the orbital-ordered background in nickelates and the effect of orbital ordering to resolve spin frustration in geometrically frustrated e(g) electron systems. Note that orbital ordering phenomena are also found in t(2g) electron systems. Here we review recent advances in the understanding of orbital ordering phenomenon in Ca2RuO4. Next we discuss another spin-charge-orbital complex system such as the f-electron compound. After detailed explanation of the construction of microscopic models on the basis of a j-j coupling scheme, we introduce a d-electron-like scenario to understand novel magnetism in some actinide compounds with the HoCoGa5-type tetragonal crystal structure. Finally, we show that complicated multipole order can be understood from the spin-orbital model on the basis of the j-j coupling scheme. As a typical material with multipole order, we pick up NpO2 which has been believed to exhibit peculiar octupole order. Throughout this review, it is emphasized that the same orbital physics works both in d- and f-electron complex materials in spite of the difference between d and f orbitals.