  The zero-temperature phase diagrams of a two-dimensional frustrated quantum
antiferromagnetic system, namely the Union Jack model, are studied using the
coupled cluster method (CCM) for the two cases when the lattice spins have spin
quantum number $s=1$ and $s=3/2$. The system is defined on a square lattice and
the spins interact via isotropic Heisenberg interactions such that all
nearest-neighbour (NN) exchange bonds are present with identical strength
$J_{1}>0$, and only half of the next-nearest-neighbour (NNN) exchange bonds are
present with identical strength $J_{2} \equiv \kappa J_{1} > 0$. The bonds are
arranged such that on the $2 \times 2$ unit cell they form the pattern of the
Union Jack flag. Clearly, the NN bonds by themselves (viz., with $J_{2}=0$)
produce an antiferromagnetic N\'{e}el-ordered phase, but as the relative
strength $\kappa$ of the frustrating NNN bonds is increased a phase transition
occurs in the classical case ($s \rightarrow \infty$) at $\kappa^{\rm
cl}_{c}=0.5$ to a canted ferrimagnetic phase. In the quantum cases considered
here we also find strong evidence for a corresponding phase transition between
a N\'{e}el-ordered phase and a quantum canted ferrimagnetic phase at a critical
coupling $\kappa_{c_{1}}=0.580 \pm 0.015$ for $s=1$ and $\kappa_{c_{1}}=0.545
\pm 0.015$ for $s=3/2$. In both cases the ground-state energy $E$ and its first
derivative $dE/d\kappa$ seem continuous, thus providing a typical scenario of a
second-order phase transition at $\kappa=\kappa_{c_{1}}$, although the order
parameter for the transition (viz., the average ground-state on-site
magnetization) does not go to zero there on either side of the transition.
