SO($N$) singlet-projection model on the pyrochlore lattice

We present an extensive quantum Monte Carlo study of a nearest-neighbor, singlet-projection model on the pyrochlore lattice that exhibits SO($N$) symmetry and is sign-problem-free. We find that in contrast to the previously studied two-dimensional variations of this model that harbor critical points between their ground state phases, the non-bipartite pyrochlore lattice in three spatial dimensions appears to exhibit a first-order transition between a magnetically-ordered phase and some, as yet uncharacterized, paramagnetic phase. We also observe that the magnetically-ordered phase survives to a relatively large value of $N=8$, and that it is gone for $N=9$.