  We study the ground state properties of the double-exchange systems. The
phase factor of the hopping matrix elements arises from $t_{2g}$ spin texture
in two or more dimensions. A novel ``flux'' state is stabilized against the
canted antiferromagnetic and spiral spin states. In a certain range of hole
doping, the phase separation occurs between the ``flux'' state and
antiferromagnetic states. Constructing a trial state which provides the
rigorous upper bound on the ground state, we show that the metallic canted
antiferromagnetic state is not stable in the double exchange model.
