  Electronic state and the Verwey transition in magnetite (Fe_3O_4) are studied
using a spinless three-band Hubbard model for 3d electrons on the B sites with
the Hartree-Fock approximation and the exact diagonalisation method.
Complex-orbital, e.g., 1/sqrt(2)[|zx> + i |yz>], ordered (COO) states having
noncollinear orbital moments ~ 0.4 mu_B on the B sites are obtained with the
cubic lattice structure of the high-temperature phase. The COO state is a novel
form of magnetic ordering within the orbital degree of freedom. It arises from
the formation of Hund's second rule states of spinless pseudo-d molecular
orbitals in the Fe_4 tetrahedral units of the B sites and ferromagnetic
alignment of their fictitious orbital moments. A COO state with longer
periodicity is obtained with pseudo-orthorhombic Pmca and Pmc2_1 structures for
the low-temperature phase. The state spontaneously lowers the crystal symmetry
to the monoclinic and explains experimentally observed rhombohedral cell
deformation and Jahn-Teller like distortion. From these findings, we consider
that at the Verwey transition temperature, the COO state remaining to be
short-range order impeded by dynamical lattice distortion in high temperature
is developed into that with long-range order coupled with the monoclinic
lattice distortion.
