  We study the properties of the Heisenberg antiferromagnet with spatially
anisotropic nearest-neighbour exchange couplings on the kagome net, i.e. with
coupling J in one lattice direction and couplings J' along the other two
directions. For J/J' > 1, this model is believed to describe the magnetic
properties of the mineral volborthite. In the classical limit, it exhibits two
kinds of ground states: a ferrimagnetic state for J/J' < 1/2 and a large
manifold of canted spin states for J/J' > 1/2. To include quantum effects
self-consistently, we investigate the Sp(N) symmetric generalisation of the
original SU(2) symmetric model in the large-N limit. In addition to the
dependence on the anisotropy, the Sp(N) symmetric model depends on a parameter
kappa that measures the importance of quantum effects. Our numerical
calculations reveal that in the kappa-J/J' plane, the system shows a rich phase
diagram containing a ferrimagnetic phase, an incommensurate phase, and a
decoupled chain phase, the latter two with short- and long-range order. We
corroborate these results by showing that the boundaries between the various
phases and several other features of the Sp(N) phase diagram can be determined
by analytical calculations. Finally, the application of a block-spin
perturbation expansion to the trimerised version of the original spin-1/2 model
leads us to suggest that in the limit of strong anisotropy, J/J' >> 1, the
ground state of the original model is a collinearly ordered antiferromagnet,
which is separated from the incommensurate state by a quantum phase transition.
