Finite-size-scaling study of the spin-1 Heisenberg-Ising chain with uniaxial anisotropy

Finite-cell calculations (up to $N=12\mathrm{}$ spins) have been performed on the spin-1 Heisenberg-Ising chain with an uniaxial anisotropy, $\mathcal{H}=\Sigma i^{}[S_i^x{S}_{i+1\mathrm{}}^x+S_i^y{S}_{i+1\mathrm{}}^y+\lambda S_i^z{S}_{i+1\mathrm{}}^z+D{\left(S_i^z\right)}^2]$. From a scaling analysis of the gap between the ground state and the first excited state, a phase diagram has been drawn in the ($\lambda ,D$) plane and the transition lines between the "ferromagnetic," $X-Y$," "singlet-ground-state," and "antiferromagnetic" phases have been estimated for the infinite-$N$ system. One of the most important results is that a singlet-ground-state phase with a nonzero gap exists in an extended range of $\lambda$ and $D$ values including the Heisenberg point $\lambda =1\mathrm{}$, $D=0\mathrm{}$, in contrast with the spin-$\frac{1}{2}$ case. Moreover, for $\lambda \simeq 1$, the gap decreases with increasing positive anisotropy $D$, goes through a minimum, estimated to be zero, and then increases with $D$.