Effect of anisotropy on the ground-state magnetic ordering of the spin-one quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ model on the square lattice
We study the zero-temperature phase diagram of the $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ Heisenberg model for spin-1 particles on an infinite square lattice interacting via nearest-neighbour ($J_1 \equiv 1$) and next-nearest-neighbour ($J_2 > 0$) bonds. Both bonds have the same $XXZ$-type anisotropy in spin space. The effects on the quasiclassical Néel-ordered and collinear stripe-ordered states of varying the anisotropy parameter $Δ$ is investigated using the coupled cluster method carried out to high orders. By contrast with the spin-1/2 case studied previously, we predict no intermediate disordered phase between the Néel and collinear stripe phases, for any value of the frustration $J_2/J_1$, for either the $z$-aligned ($Δ> 1$) or $xy$-planar-aligned ($0 \leq Δ< 1$) states. The quantum phase transition is determined to be first-order for all values of $J_2/J_1$ and $Δ$. The position of the phase boundary $J_{2}^{c}(Δ)$ is determined accurately. It is observed to deviate most from its classical position $J_2^c = {1/2}$ (for all values of $Δ> 0$) at the Heisenberg isotropic point ($Δ= 1$), where $J_{2}^{c}(1) = 0.55 \pm 0.01$. By contrast, at the XY isotropic point ($Δ= 0$), we find $J_{2}^{c}(0) = 0.50 \pm 0.01$. In the Ising limit ($Δ\to \infty$) $J_2^c \to 0.5$ as expected.