Determination of Boundary Scattering, Intermagnon Scattering, and the Haldane Gap in Heisenberg Chains

Low-lying magnon dispersion of the S=1 Heisenberg anti-ferromagnetic (AF) chain is analyzed using the non-Abelian DMRG method. The scattering length $a_{\rm b}$ of the boundary coupling and the inter-magnon scattering length $a$ are determined. A characteristic diverging behavior of $a_{\rm b}$ is shown at a crossover point. While the Haldane gap $\Delta$, the magnon velocity $v$, and $a$ remain constants at the crossover. Our method allows estimation of the gap of the S=2 AF chain to be $\Delta = 0.0891623(9)$ using a chain length longer than $\xi$.