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Electronic state and the Verwey transition in magnetite (Fe_3O_4) are studiedusing a spinless three-band Hubbard model for 3d electrons on the B sites withthe Hartree-Fock approximation and the exact diagonalisation method.Complex-orbital, e.g., 1/sqrt(2)[|zx> + i |yz>], ordered (COO) states havingnoncollinear orbital moments ~ 0.4 mu_B on the B sites are obtained with thecubic lattice structure of the high-temperature phase. The COO state is a novelform of magnetic ordering within the orbital degree of freedom. It arises fromthe formation of Hund's second rule states of spinless pseudo-d molecularorbitals in the Fe_4 tetrahedral units of the B sites and ferromagneticalignment of their fictitious orbital moments. A COO state with longerperiodicity is obtained with pseudo-orthorhombic Pmca and Pmc2_1 structures forthe low-temperature phase. The state spontaneously lowers the crystal symmetryto the monoclinic and explains experimentally observed rhombohedral celldeformation and Jahn-Teller like distortion. From these findings, we considerthat at the Verwey transition temperature, the COO state remaining to beshort-range order impeded by dynamical lattice distortion in high temperatureis developed into that with long-range order coupled with the monocliniclattice distortion.

Complex-Orbital Order in Fe_3O_4 and Mechanism of the Verwey Transition