Magnons and solitons in a spin-1 antiferromagnetic Heisenberg-Ising ring

The excitation energies of magnons and solitons as well as the ground-state energy of the spin-1 antiferromagnetic Heisenberg-Ising ring are studied by the numerical method. The method of Vanden Broeck and Schwartz is used to estimate the infinite-lattice limits accurately. The result confirms Haldane’s prediction that a novel singlet phase with a nonzero excitation gap appears between the gapless XY phase and the doublet ground-state Néel phase. It extends over the range of anisotropy (${\Delta }_l$≤Δ≤${\Delta }_u$) around the isotropic Heisenberg point Δ=0.5. It is ascertained that at the upper transition point ${\Delta }_u$=0.540±0.001 (1) the longitudinal magnon gap closes, (2) the soliton existing in the Ising side softens, and (3) the second derivative of the ground-state energy diverges. Around the lower transition point of ${\Delta }_l$=0.38±0.02, the transverse magnon gap exhibits an exponential singularity accompanying with the slight anomaly of the second derivative of the ground-state energy.