Meta-Plaquette Expansion for the Triplet Excitation Spectrum in CaV${}_4$O${}_9$

We study antiferromagnetic, $S=1\mathrm{/}2$ Heisenberg models with nearest and second neighbor interactions on the one-fifth depleted square lattice which describes the spin degrees of freedom in the spin-gap system CaV${}_4$O${}_9$. The meta-plaquette expansion for the triplet excitation spectrum is extended to fifth order, and the results are compared with experimental data on CaV${}_4$O${}_9$. We attempt to locate the phase boundary between magnetically ordered and gapped phases.