Coexistence of a long-range and short-range ordered state in MnV$_2$O$_4$

We report $^{51}$V zero-field NMR of manganese vanadate spinel, MnV$_2$O$_4$ together with both ac and dc magnetization measurements. The zero-field NMR spectrum consists of multiple lines distributed from 240 MHz to 320 MHz. Its temperature dependence reveals that the ground state is formed by the delicate balance between long-range ferrimagnetic order and second short-range order, which causes reentrant-spin-glass-like behavior. The unusual ground state is ascribed to the competition between Mn$^{2+}$ and V$^{3+}$ exchange couplings and the orbital ordering of the V site induced by the structural phase transition.