Quantum 120° model on pyrochlore lattice: Orbital ordering in MnV ₂O₄

We present an analytical model of orbital ordering in vanadium spinel MnV₂O₄. The model is based on recent first-principles calculation indicating a strong trigonal distortion at the vanadium sites of this compound. At the single-ion level, the trigonal crystal field leaves a doubly degenerate atomic ground state and breaks the approximate rotational symmetry of t2g orbitals. We find that the effective interaction between the low-energy doublets is described by a quantum antiferromagnetic 120° model on the pyrochlore lattice. We obtain the classical ground state and show its stability against quantum fluctuations. The corresponding orbital order consisting of two inequivalent orbital chains is consistent with the experimentally observed tetragonal symmetry. A periodic modulation of electron density function along orbital chains is shown to arise from the staggering of local trigonal axes. In the presence of orbital order, single-ion spin anisotropy arising from relativistic spin-orbit interaction stabilizes the experimentally observed orthogonal magnetic structure.