Origin of Spin Gap in ${\mathrm{CaV}}_4{O}_9$: Effects of Frustration and Lattice Distortions

Spin gap behavior of ${\mathrm{CaV}}_4{\mathrm{O}}_9$ is studied by including the effects of frustrating magnetic interactions and lattice distortions. The spectrum of triplet excitations is calculated for a Heisenberg model on the $1/5$-depleted square lattice. In the spin gap phase, the location of the minima of the spectrum in the Brillouin zone is found to depend nontrivially on the exchange parameters. Experimental consequences of the temperature-dependent lattice distortion including its effect on the uniform susceptibility and the spin gap are explored.