  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.
