  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 $\Delta$ in the range $-1 \leq \Delta < \infty$ of interest,
including both the easy-plane $(-1 < \Delta < 1)$ and easy-axis $(\Delta > 1)$
regimes, where $\Delta \rightarrow \infty$ 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 ($\Delta = 1$), where
the model undergoes a phase transition from a gapped state (for $\Delta > 1$)
to a gapless state (for $\Delta \leq 1$), and compare our results there with
those from spin-wave theory (SWT). Interestingly, the nature of the criticality
at $\Delta=1$ 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.
