Origin of spin-gap in CaV$_4$O$_9$: effect of frustration and lattice   distortion
  We study the origin of spin-gap in recently discovered material CaV$_4$O$_9$. We analyze the spin-$1/2$ Heisenberg model on the $1/5$ depleted square lattice with nearest neighbor (nn) and next nearest neighbor (nnn) interactions, in terms of the singlet and triplet states of the 4-spin plaquettes and 2-spin dimers. Phase diagram of the model is obtained within a linear ``spin-wave"-like approximation, and is shown to agree well with the earlier results of QMC simulations for nn interactions. We further propose that the special lattice structure of CaV$_4$O$_9$ naturally leads to lattice distortions, which enhances the spin-gap via a spin-Peierls mechanism. 
O. A. Starykh, M. E. Zhitomirsky, D. I. Khomskii, R. R. P. Singh, and   K. Ueda