  A novel Bethe Ansatz scheme is proposed to calculate physical properties of
quantum integrable systems without $U(1)$ symmetry. As an example, the
anti-periodic XXZ spin chain, a typical correlated many-body system embedded in
a topological manifold, is examined. Conserved "momentum" and "charge"
operators are constructed despite the absence of translational invariance and
$U(1)$ symmetry. The ground state energy and elementary excitations are derived
exactly. It is found that two intrinsic fractional (one half) zero modes
accounting for the double degeneracy exist in the eigenstates. The elementary
excitations show quite a different picture from that of a periodic chain. This
method can be applied to other quantum integrable models either with or without
$U(1)$ symmetry.
