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Green function formulations of weak field orbital magnetism of Bloch electrons are developed so as to take into account not only interband effects but also mutual interactions between electrons. The modified Bloch functions of Luttinger and Kohn are employed as basic functions to yield gauge- and phase-invariant expressions which, applied to the perfect crystal without many-body interactions, are equivalent to the exact formula. As examples of applications of the formulations for interacting Bloch electron systems, we first discuss the strongly correlated electron system represented by Hubbard's model Hamiltonian within his first approximation. The interband effect between two hands separated off by a Coulomb interaction is shown to have properties similar to Van-Vleck paramagnetism. Next we considered the effect of the correlation in the framework of a random phase approximation in the model of the periodic potential of the one-dimensional weak cosine type.
Hidetoshi Fukuyama (1971) studied this question.