  We show that, when $N$ is a multiple of 6 ($N=6m$, $m$ integer), the \SU{N}
Heisenberg model on the honeycomb lattice with $m$ particles per site has a
clear tendency toward chiral order as soon as $m\geq 2$. This conclusion has
been reached by a systematic variational Monte Carlo investigation of
Gutzwiller projected wave-functions as a function of $m$ between the case of
one particle per site ($m=1$), for which the ground state has recently been
shown to be in a plaquette singlet state, and the $m\rightarrow \infty$ limit,
where a mean-field approach has established that the ground state has chiral
order. This demonstrates that the chiral phase can indeed be stabilized for not
too large values of $m$, opening the way to its experimental realisations in
other lattices.

Quantum Gases