Quantum kagomé antiferromagnet in a magnetic field: Low-lying nonmagnetic excitations versus valence-bond crystal order

We study the ground state properties of a quantum antiferromagnet on the kagom\'e lattice in the presence of a magnetic field, paying particular attention to the stability of the plateau at magnetization $1∕3$ of saturation and the nature of its ground state. We discuss fluctuations around classical ground states and argue that quantum and classical calculations at the harmonic level do not lead to the same result in contrast to the zero-field case. For spin $S=1∕2$ we find a magnetic gap below which an exponential number of nonmagnetic excitations are present. Moreover, such non-magnetic excitations also have a (much smaller) gap above the threefold degenerate ground state. We provide evidence that the ground state has long-range order of valence-bond crystal type with nine spins in the unit cell.