Using both exact diagonalizations and diagonalizations in a subset of short-range valence-bond singlets, we address the nature of the groundstate of the Heisenberg spin-
1∕2 antiferromagnet on the square lattice with competing next-nearest and next-next-nearest neighbor antiferromagnetic couplings (${J}_{1}\ensuremath{-}{J}_{2}\ensuremath{-}{J}_{3}$ model). A detailed comparison of the two approaches reveals a region along the line
(J2+J3)∕J1=1∕2, where the description in terms of nearest-neighbor singlet coverings is excellent, therefore providing evidence for a magnetically disordered region. Furthermore, a careful analysis of dimer-dimer correlation functions, dimer structure factors and plaquette-plaquette correlation functions provides striking evidence for the presence of a plaquette valence bond crystal order in part of the magnetically disordered region.
We use series expansion methods to investigate the phase diagram of a frustrated spin-
21 Heisenberg antiferromagnet on the simple cubic lattice, with interactions extending to third neighbors. Our results do not support a recent prediction of a rather large paramagnetic phase at zero temperature, but do show behavior which may be indicative of a small disordered region near the point of maximum frustration.