Effect of anisotropy on the ground-state magnetic ordering of the spin-half quantum $J_1^{XXZ}$--$J_2^{XXZ}$ model on the square lattice
We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $Δ$ on the $z$-aligned Néel and (collinear) stripe states, as well as on the $xy$-planar-aligned Néel and collinear stripe states, are examined. All four of these quasiclassical states are chosen in turn as model states on top of which we systematically include the quantum correlations using a coupled cluster method analysis carried out to very high orders. We find strong evidence for two {\it quantum triple points} (QTP's) at ($Δ^{c} = -0.10 \pm 0.15, J_{2}^{c}/J_{1} = 0.505 \pm 0.015$) and ($Δ^{c} = 2.05 \pm 0.15, J_{2}^{c}/J_{1} = 0.530 \pm 0.015$), between which an intermediate magnetically-disordered phase emerges to separate the quasiclassical Néel and stripe collinear phases. Above the upper QTP ($Δ\gtrsim 2.0$) we find a direct first-order phase transition between the Néel and stripe phases, exactly as for the classical case. The $z$-aligned and $xy$-planar-aligned phases meet precisely at $Δ= 1$, also as for the classical case. For all values of the anisotropy parameter between those of the two QTP's there exists a narrow range of values of $J_{2}/J_{1}$, $α^{c_1}(Δ)<J_{2}/J_{1} <α^{c_2}(Δ)$, centered near the point of maximum classical frustration, $J_{2}/J_{1} = {1/2}$, for which the intermediate phase exists. This range is widest precisely at the isotropic point, $Δ= 1$, where $α^{c_1}(1) = 0.44 \pm 0.01$ and $α^{c_2}(1) = 0.59 \pm 0.01$. The two QTP's are characterized by values $Δ= Δ^{c}$ at which $α^{c_1}(Δ^{c})=α^{c_2}(Δ^{c})$.