Anisotropic pyrochlores and the global phase diagram of the checkerboard antiferromagnet

We study the phase diagram of two models of spin-$1∕2$ antiferromagnets composed of corner-sharing tetrahedra, the basis of the pyrochlore structure. Primarily, we focus on the Heisenberg antiferromaget on the checkerboard lattice (also called the planar pyrochlore and crossed-chain model). This model has an anisotropic limit, when the dimensionless ratio of two exchange constants J_×∕J⪡1, in which it consists of one-dimensional spin chains coupled weakly together in a frustrated fashion. Using recently developed techniques combining renormalization group ideas and one-dimensional bosonization and current algebra methods, we show that in this limit the model enters a crossed-dimer state with twofold spontaneous symmetry breaking but no magnetic order. We complement this result by an approximate ``quadrumer triplet boson'' calculation, which qualitatively captures the physics of the ``plaquette valence-bond solid'' state believed to obtain for J_×∕J≈1. Using these known points in parameter space, the instabilities pointed to by the quadrumer boson calculation, and the simple limit J_×∕J⪢1, we construct a few candidate global phase diagrams for the model, and discuss the nature of the quantum phase transitions contained therein. Finally, we apply our quasi-one-dimensional techniques to an anisotropic limit of the three-dimensional pyrochlore antiferromagnet, an approximate model for magnetism in GeCu₂O₄. A crossed-dimer state is predicted here as well.