Long-range spin-pairing order and spin defects in quantum spin-1/2 ladders

For w-legged antiferromagnetic spin-1/2 Heisenberg ladders, a long-range sp in pairing order can be identified which enables the separation of the spac e spanned by finite-range (covalent) valence-bond configurations into w + 1 subspaces. Since every subspace has an equivalent counter subspace connect ed by translational symmetry, twofold degeneracy, breaking translational sy mmetry is found except for the subspace where the ground state of w = even belongs to. In terms of energy ordering, (non)degeneracy and the discontinu ities introduced in the long-range spin pairing order by topological spin d efects, the differences between even and odd ladders are explained in a gen eral and systematic way.