Orbital Dimer Model for Spin-Glass State in Y$_2$Mo$_2$O$_7$

The formation of a spin glass generally requires that magnetic interactions are both frustrated and disordered. Consequently, the origin of spin-glass behavior in Y$_2$Mo$_2$O$_7$ -- in which magnetic Mo$^{4+}$ ions occupy a frustrated pyrochlore lattice with minimal compositional disorder -- is a longstanding question. We use neutron and X-ray pair-distribution function (PDF) analysis to develop a disorder model that resolves apparent incompatibilities between previous PDF, EXAFS and NMR studies, and provides a new and physical explanation of the exchange disorder responsible for spin-glass formation. We show that Mo$^{4+}$ ions displace according to a local ``2-in/2-out" rule on Mo$_4$ tetrahedra, driven by orbital dimerization of Jahn-Teller active Mo$^{4+}$ ions. Long-range orbital order is prevented by the macroscopic degeneracy of dimer coverings permitted by the pyrochlore lattice. Cooperative O$^{2-}$ displacements yield a distribution of Mo--O--Mo angles, introducing disorder into magnetic interactions. Our presentation shows how frustration of atomic displacements can assume the role of compositional disorder in driving a spin-glass transition, and reveals a link between ice-like and spin-glass physics.