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

Abstract

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