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Phonon dispersion, Raman spectra and evidence for spin-phonon coupling in MnV$_2$O$_4$ from first-principles

MnV$_2$O$_4$ in the spinel structure is known to exhibit coupled orbital and spin ordering, and its Raman spectra show interesting anomalies in its low-temperature phase. With a goal to explain this behavior involving coupled spins and phonons, we determine here the spin-phonon couplings in MnV$_2$O$_4$ from a theoretical analysis of its phonon spectra and their dependence on spin-ordering and electron correlations, obtained from first-principles density functional theoretical calculations. Using these in an analysis based on a Landau-like theory, we uncover the mechanism governing the Raman anomalies observed in its low-temperature phase.