  We consider bipartite SU($N$) spin Hamiltonians with a fundamental
representation on one sublattice and a conjugate to fundamental on the other
sublattice. By mapping these antiferromagnets to certain classical loop models
in one higher dimension, we provide a practical strategy to write down a large
family of SU($N$) symmetric spin Hamiltonians that satisfy Marshall's sign
condition. This family includes all previously known sign-free SU($N$) spin
models in this representation and in addition provides a large set of new
models that are Marshall positive and can hence be studied efficiently with
quantum Monte Carlo methods. As an application of our idea to the square
lattice, we show that in addition to Sandvik's $Q$-term, there is an
independent non-trivial four-spin $R$-term that is sign-free. Using numerical
simulations, we show how the $R$-term provides a new route to the study of
quantum criticality of N\'eel order.

Strongly Correlated Electrons