  Above the saturation field, geometrically frustrated quantum antiferromagnets
have dispersionless low-energy branches of excitations corresponding to
localized spin-flip modes. Transition into a partially magnetized state occurs
via condensation of an infinite number of degrees of freedom. The ground state
below the phase transition is a magnon crystal, which breaks only translational
symmetry and preserves spin-rotations about the field direction. We give a
detailed review of recent works on physics of such phase transitions and
present further theoretical developments. Specifically, the low-energy degrees
of freedom of a spin-1/2 kagom\'e antiferromagnet are mapped to a hard hexagon
gas on a triangular lattice. Such a mapping allows to obtain a quantitative
description of the magnetothermodynamics of a quantum kagom\'e antiferromagnet
from the exact solution for a hard hexagon gas. In particular, we find the
exact critical behavior at the transition into a magnon crystal state, the
universal value of the entropy at the saturation field, and the position of
peaks in temperature- and field-dependence of the specific heat. Analogous
mapping is presented for the sawtooth chain, which is mapped onto a model of
classical hard dimers on a chain. The finite macroscopic entropies of
geometrically frustrated magnets at the saturation field lead to a large
magnetocaloric effect.
