Charge and Orbital Order in Half-Doped Manganites

An explanation is given for the charge order, orbital order, and insulating state observed in half-doped manganese oxides, such as ${\mathrm{Nd}}_{1/2}{\mathrm{Sr}}_{1/2}{\mathrm{MnO}}_3$. The competition between the kinetic energy of the electrons and the magnetic exchange energy drives the formation of effectively one-dimensional ferromagnetic zigzag chains. Because of a topological phase factor in the hopping, the chains are intrinsically insulating and orbital ordered. Most surprisingly, the strong Coulomb interaction between electrons on the same Mn ion leads to the experimentally observed charge ordering. For doping less than $1/2$ the system is unstable towards phase separation into a ferromagnetic metallic and charge-ordered insulating phase.