Convergent expansions for properties of the Heisenberg model for   CaV$_4$O$_9$
  We have carried out a wide range of calculations for the $S=1/2$ Heisenberg model with nearest- and second-neighbor interactions on a two-dimensional lattice which describes the geometry of the vanadium ions in the spin-gap system CaV$_4$O$_9$. The methods used were convergent high-order perturbation expansions (``Ising'' and ``Plaquette'' expansions at $T=0$, as well as high-temperature expansions) for quantities such as the uniform susceptibility, sublattice magnetization, and triplet elementary excitation spectrum. Comparison with the data for CaV$_4$O$_9$ indicates that its magnetic properties are well described by nearest-neighbor exchange of about 200K in conjunction with second-neighbor exchange of about 100K. 
M.P. Gelfand (1), Z. Weihong (2), R.R.P. Singh (3), J. Oitmaa (2), and   C.J. Hamer (2) ((1) Colorado State U, (2) U New Sou...