Valence-Bond Crystal, and Lattice Distortions in a Pyrochlore Antiferromagnet with Orbital Degeneracy

TL;DR

This paper investigates how orbital degeneracy and lattice distortions in a frustrated pyrochlore antiferromagnet lead to a valence bond crystal state with specific orbital and spin arrangements, explaining experimental observations in MgTi₂O₄.

Contribution

It introduces an effective spin-orbital Hamiltonian for pyrochlore systems and demonstrates how orbital effects induce valence bond crystal formation and lattice distortions, a novel insight into such frustrated magnets.

Findings

Orbital degrees of freedom modulate spin-exchange energies.

System forms a valence bond crystal with dimer condensation.

Predicted orbital ordering pattern can be observed via resonant x-ray scattering.

Abstract

We discuss the ground state properties of a spin 1/2 magnetic ion with threefold $t_{2g}$ orbital degeneracy on a highly frustrated pyrochlore lattice, like Ti$^{3+}$ ion in B-spinel MgTi$_2$O$_4$. 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 non-magnetic spin-singlet manifold. The latter is a collection of spin-singlet dimers and is, however, highly degenerate with respect of 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 MgTi$_2$O$_4$. 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.