Ground-state energies of the nonlinear sigma model and the Heisenberg spin chains

String order in half-integer-spin antiferromagnetic Heisenberg chains

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.