  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.

Mesoscale and Nanoscale Physics;Statistical Mechanics;Strongly Correlated Electrons