  We consider doping of non-magnetic impurities in the spin-1/2, 1/5-depleted
square lattice. This structure, whose undoped phase diagram offers both
magnetically ordered and spin-liquid ground states, is realized physically in
CaV_4O_9. Doping into the ordered phase results in a progressive loss of order,
which becomes complete at the percolation threshold. By contrast, non-magnetic
impurities introduced in the spin liquids create a phase of weak but
long-ranged antiferromagnetic order coexisting with the gapped state. The
latter may be viewed as a true order-by-disorder phenomenon. We study the phase
diagram of the doped system by computing the static susceptibility and
staggered magnetization using a stochastic series-expansion quantum Monte Carlo
technique.
