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

Low-lying magnon dispersion in a $S=1$ Heisenberg antiferromagnetic (AF) chain with boundary $S/2$ spins coupling antiferromagnetically ($J_{\rm end} > 0$) is analyzed by use of the non-Abelian DMRG method. The Haldane gap $\Delta$, the magnon velocity $v$, the inter-magnon scattering length $a$, and the scattering length $a_{\rm b}$ of the boundary coupling are evaluated. The length $a_{\rm b}$, which represents the contribution of boundary effects, depends on $J_{\rm end}$ drastically, while $\Delta$, $v$, and $a$ are constant irrespective of $J_{\rm end}$. Our method estimates the gap of the $S=2$ AF chain as $\Delta = 0.0891623(9)$ using a chain length up to 2048, which is longer than the correlation length.