Valence-bond crystal and lattice distortions in a pyrochlore antiferromagnet with orbital degeneracy

We discuss the ground state properties of a spin- 1 2 magnetic ion with threefold t2g orbital degeneracy on a highly frustrated pyrochlore lattice, like Ti3+ ion in B-spinel Mg Ti2 O4. We formulate an effective spin-orbital Hamiltonian and study its low energy sector by constructing several exact eigenstates in the limit of vanishing Hund's coupling. We find that orbital degrees of freedom modulate the spin-exchange energies, release the infinite spin degeneracy of pyrochlore structure, and drive the system to a nonmagnetic spin-singlet manifold. The latter is a collection of spin-singlet dimers and is, however, highly degenerate with respect to dimer orientations. This "orientational" degeneracy is then lifted by a magneto-elastic interaction that optimizes the previous energy gain by distorting the bonds in suitable directions and leading to a tetragonal phase. In this way a valence bond crystal state is formed, through the condensation of dimers along helical chains running around the tetragonal c axis, as actually observed in Mg Ti2 O4. The orbitally ordered pattern in the dimerized phase is predicted to be of ferro type along the helices and of antiferro type between them. Finally, through analytical considerations as well as numerical ab initio simulations, we predict a possible experimental tool for the observation of such an orbital ordering, through resonant x-ray scattering.