Quantization of Hall Conductance in Double Exchange Systems: Topology and Lattice Gauge Field

We study quantization conditions of the Hall conductivity for a two dimensional system described by a double exchange Hamiltonian with and without an external magnetic field. This is obtained by an extension of the topological arguments familiar from the theory of the integer quantum Hall effect. The quantization conditions are related to spontaneous breaking of spin O(3), time-reversal, and spin chiral symmetries. Extension to systems with higher dimensions is briefly discussed.