The Spin-Half {\it XXZ} Antiferromagnet on the Square Lattice Revisited: A High-Order Coupled Cluster Treatment

We use the coupled cluster method (CCM) to study the ground-state properties and lowest-lying triplet excited state of the spin-half {\it XXZ} antiferromagnet on the square lattice. The CCM is applied to it to high orders of approximation by using an efficient computer code that has been written by us and which has been implemented to run on massively parallelized computer platforms. We are able therefore to present precise data for the basic quantities of this model over a wide range of values for the anisotropy parameter $Δ$ in the range $-1 \leq Δ< \infty$ of interest, including both the easy-plane $(-1 < Δ< 1)$ and easy-axis $(Δ> 1)$ regimes, where $Δ\rightarrow \infty$ represents the Ising limit. We present results for the ground-state energy, the sublattice magnetization, the zero-field transverse magnetic susceptibility, the spin stiffness, and the triplet spin gap. Our results provide a useful yardstick against which other approximate methods and/or experimental studies of relevant antiferromagnetic square-lattice compounds may now compare their own results. We also focus particular attention on the behaviour of these parameters for the easy-axis system in the vicinity of the isotropic Heisenberg point ($Δ= 1$), where the model undergoes a phase transition from a gapped state (for $Δ> 1$) to a gapless state (for $Δ\leq 1$), and compare our results there with those from spin-wave theory (SWT). Interestingly, the nature of the criticality at $Δ=1$ for the present model with spins of spin quantum number $s=\frac{1}{2}$ that is revealed by our CCM results seems to differ qualitatively from that predicted by SWT, which becomes exact only for its near-classical large-$s$ counterpart.