Determination of boundary scattering, magnon-magnon scattering, and the Haldane gap in Heisenberg spin chains

Low-lying magnon dispersion in a $S=1$ Heisenberg antiferromagnetic (AF) chain is analyzed using the non-Abelian density-matrix-renormalization-group (DMRG) method. The scattering length $a_{\mathrm{b}}$ of the boundary coupling and the intermagnon scattering length $a$ are determined. The scattering length $a_{\mathrm{b}}$ is found to exhibit a characteristic diverging behavior at the crossover point. In contrast, the Haldane gap $\Delta$, the magnon velocity $v$, and $a$ remain constant at the crossover. Our method allowed estimation of the gap of the $S=2$ AF chain to be $\Delta =0.0891623\left(9\right)$ using a chain length longer than the correlation length $\xi$.

Figure 1

Distribution of local magnetization $\langle S_j^z\rangle$ for a single-magnon state with $S_{\mathrm{tot}}=1$ for various values of $J_{\mathrm{end}}$.

Figure 2

Single-magnon energy with $J_{\mathrm{end}}$ under the condition $m_s=512(m_{s^{\mathrm{z}}}\sim 2500)$. The dotted line represents $v^2{\sin }^2[\pi /(L-2a_{\mathrm{b}})]$.

Figure 3

Boundary-scattering length and intermagnon scattering length as a function of $J_{\mathrm{end}}$. The dotted line represents $J_{\mathrm{end}}=0.50865$.