  We consider a simplified model of the magnetic structure of the
two-dimensional compound $CaV_4O_9$ in terms of interacting square plaquettes
of spins with two distinct antiferromagnetic exchange constants. We analyze the
competition between two types of singlet ground states and the Neel ordered one
in terms of respectively, numerical cluster expansion and nonlinear spin wave
theory. The resulting phase diagram agrees well with known Quantum Monte Carlo
results and suggests a first order transition between ordered and singlet
ground states as a function of the exchange constants.

Condensed Matter