Dispersion of magnetic excitations in a spin-1 chain with easy-plane anisotropy

We study the dispersion of magnetic excitations in a Heisenberg antiferromagnetic chain of spin S=1 with exchange J and single-ion anisotropy D. Chains of length up to 18 sites are studied by a Lanczo¨s method. For the Haldane phase D<${\mathit{D}}_{\mathit{c}}$≊J we show that the gap at zero momentum and ${\mathit{S}}^{\mathit{z}}$=1 is given by the sum of the gaps in the sectors ${\mathit{S}}^{\mathit{z}}$=1 and ${\mathit{S}}^{\mathit{z}}$=0 at momentum π. The transition region near D=J is likely to be reduced to an isolated point. Large-D perturbation theory is smoothly recovered as soon as D>2J. We discuss the implications for experiments on Ni(${\mathrm{C}}_2$${\mathrm{H}}_8$${\mathrm{N}}_2$${)}_2$${\mathrm{NO}}_2$${\mathrm{ClO}}_4$ (NENP) as well as ${\mathrm{CsFeBr}}_3$.