  Site dilution of spin-gapped antiferromagnets leads to localized free
moments, which can order antiferromagnetically in two and higher dimensions.
Here we show how a weak magnetic field drives this order-by-disorder state into
a novel disordered-free-moment phase, characterized by the formation of local
singlets between neighboring moments and by localized moments aligned
antiparallel to the field. This disordered phase is characterized by the
absence of a gap, as it is the case in a Bose glass. The associated
field-driven quantum phase transition is consistent with the universality of a
superfluid-to-Bose-glass transition. The robustness of the
disordered-free-moment phase and its prominent features, in particular a series
of pseudo-plateaus in the magnetization curve, makes it accessible and relevant
to experiments.
