  The spin-1/2 Heisenberg antiferromagnet on the frustrated diamond-decorated
square lattice is known to feature various zero-field ground-state phases,
consisting of extended monomer-dimer and dimer-tetramer ground states as well
as a ferrimagnetic regime. Using a combination of analytical arguments, density
matrix renormalization group (DMRG), exact diagonalization, as well as
sign-problem-free quantum Monte Carlo (QMC) calculations, we investigate the
properties of this system and the related Lieb lattice in the presence of a
finite magnetic field, addressing both the ground-state phase diagram as well
as several thermodynamic properties. In addition to the zero-field ground
states, we find at high magnetic field a spin-canted phase with a continuously
rising magnetization for increasing magnetic field strength, as well as the
fully polarized paramagnetic phase. At intermediate field strength, we identify
a first-order quantum phase transition line between the ferrimagnetic and the
monomer-dimer regime. This first-order line extends to finite temperatures,
terminating in a line of critical points that belong to the universality class
of the two-dimensional Ising model.
