Spin-orbital exchange as a route to intertwined dipole-quadrupole orbital order in MnV$_2$O$_4$ under strong trigonal crystal field

Abstract

Orbitally degenerate systems provide a promising platform for realizing novel quantum phases driven by spin-orbital exchange interactions, as described by the Kugel-Khomskii model. Spinel vanadates, in which orbital degrees of freedom remain active, exhibit structural and magnetic transitions accompanied by orbital ordering, but the nature of the orbital state in MnV$_2$O$_4$ remains under debate. Here, we combine first-principles calculations with an effective spin-orbital model to address this problem. We show that a significant trigonal crystal field is present in high-temperature cubic phase and plays an essential role in determining the low-energy degrees of freedom. Based on the resulting parameters, we construct an effective Hamiltonian beyond the conventional dominant-hopping approximation and demonstrate that subdominant hopping processes strongly modify the spin-orbital exchange interactions. As a result, the system stabilizes a two-in/two-out magnetic configuration featuring spin canting and intertwined dipole-quadrupole orbital order.