  We study the magnetic ordering of Vanadium spinels by Quantum Monte Carlo
simulations of a three-band Hubbard model. Vanadium spinels, AV$_2$O$_4$,
exhibit a unique "up-up-down-down" spin ordering at low temperatures. While
this magnetic ordering was originally measured in 1973, its origin has remained
unclear for many years due to the lack of unbiased approaches for solving the
relevant model. A three-band Hubbard model on the spinel lattice (corner
sharing tetrahedra) is a minimal Hamiltonian for describing the $t_{2g}$
electrons of the V$^{2+}$ ions. One of the main difficulties is that this
family of compounds belongs to the elusive intermediate-coupling regime ($U
\gtrsim t$) for which there is no small parameter that can justify a
perturbative expansion. We present a controlled Quantum Monte-Carlo approach to
the three-band Hubbard model relevant for this materials that reproduces the
up-up-down-down spin ordering. The method is free of the sign problem that is
usually the main limiting factor for simulating fermionic systems in dimension
higher than one.
