We derive the extended string order parameter $O(S)$ for antiferromagnetic Heisenberg chains with half-integer spin $S$ in the valence-bond-solid picture. We obtain the analytic power-law scaling of $O(S)$ versus the chain length $L$ and show that $O(S)$ scales at an extremely slow pace that decreases rapidly with growing spin magnitude. Furthermore, accurate numerical calculations show that the power-law scaling sets in only when $L$ exceeds a characteristic length scale $l_S$ which increases very fast with growing $S$. Consequently, a pseudo-long-range string order exists in half-integer-spin Heisenberg chains. The implications of this result and its relationship to other topological features such as the end-chain states are discussed.

• Received 30 April 2003

DOI:https://doi.org/10.1103/PhysRevLett.91.087204