Valence bond crystal in a pyrochlore antiferromagnet with orbital degeneracy

We discuss the ground state of a pyrochlore lattice of threefold-orbitally-degenerate $S=1/2$ magnetic ions. We derive an effective spin-orbital Hamiltonian and show that the orbital degrees of freedom can modulate the spin exchange, removing the infinite spin-degeneracy characteristic of pyrochlore structures. The resulting state is a collection of spin-singlet dimers, with a residual degeneracy due to their relative orientation. This latter is lifted by a magneto-elastic interaction, induced in the spin-singlet phase-space, that forces a tetragonal distortion. Such a theory provides an explanation for the helical spin-singlet pattern observed in the B-spinel MgTi$_2$O$_4$.