Spin-Peierls, valence-bond solid, and Néel ground states of low-dimensional quantum antiferromagnets

We examine large-N limits of the nearest-neighbor SU(N) quantum antiferromagnets on bipartite lattices in d=1,2. In d=2 the model displays a transition from a Néel to a disordered phase. The properties of the disordered phase close to the phase boundary are crucially dependent upon the nature of ‘‘hedgehog’’-like instanton tunneling events. We calculate the Berry phases of the instantons and show that, at scales larger than the spin-correlation length, the system can be described by a Coulomb plasma of instantons with complex fugacities. The properties of the Coulomb plasma vary periodically with the ‘‘spin’’ ${\mathit{n}}_{\mathit{c}}$ of the states at each site, with periodicity given by the coordination number Z of the lattice [${\mathit{n}}_{\mathit{c}}$=2S for SU(2)]. For ${\mathit{n}}_{\mathit{c}}$≠0 (mod Z) the disordered phase has a broken lattice symmetry with spin-Peierls order, while for ${\mathit{n}}_{\mathit{c}}$=0 (mod Z), the ground state is a valence-bond solid state with no broken symmetry. Related topological effects for the d=1 chain lead to spin-Peierls order for odd ${\mathit{n}}_{\mathit{c}}$. These results are for a class of models which have, at sites of the A sublattice, representations of SU(N) described by a Young tableau with a single row, and the conjugate on the B sublattice. Similar results are also obtained for representations with m rows, using U(m) gauge theory.