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.