  We study Berry's connection potentials of many-body ground states of spin-one
bosons with antiferromagnetic interactions in adiabatically varying magnetic
fields. We find that Berry's connection potentials are generally determined by,
instead of usual singular monopoles, linearly positioned monosegments each of
which carries one unit of topological charge; in the absence of a magnetic
field gradient this distribution of monosegments becomes a linear chain of
monopoles. Consequently, Berry's phases consist of a series of step functions
of magnetic fields; a magnetic field gradient causes rounding of these
step-functions. We also calculate Berry's connection fields, profiles of
monosegments and show that the total topological charge is conserved in a
parameter space.
