Ultrasound Velocity Measurements in the Orbital-Degenerate Frustrated Spinel MgV$_2$O$_4$

Magnesium vanadate spinel MgV$_2$O$_4$ is a geometrically frustrated magnet with $t_{2g}$-orbital degeneracy of V$^{3+}$ (3$d^2$), which undergoes a cubic-to-tetragonal structural transition at $T_s$ = 65 K and an antiferromagnetic (AF) transition at $T_N$ = 42 K. For MgV$_2$O$_4$, it is considered that the occurrence of $t_{2g}$-orbital order at $T_s$ causes the release of frustration by the AF ordering at $T_N$ lower than $T_s$. We performed ultrasound velocity measurements in high-purity single crystal of MgV$_2$O$_4$. Temperature dependence of the tetragonal shear modulus ($C_{11}-C_{12}$)/2 exhibits huge Curie-type softening in the cubic paramagnetic (PM) phase ($T>T_s$), which should be a precursor to the cubic-to-tetragonal lattice distortion at $T_s$. The trigonal shear modulus $C_{44}(T)$ exhibits softening with an upturn curvature in the cubic PM phase, indicating a coupling of the lattice to magnetic excitations. These softenings suggest the coexistence of the dynamical Jahn-Teller effect and the dynamical magnetic state in the cubic PM phase.