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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 aclear tendency toward chiral order as soon as $m\geq 2$. This conclusion hasbeen reached by a systematic variational Monte Carlo investigation ofGutzwiller projected wave-functions as a function of $m$ between the case ofone particle per site ($m=1$), for which the ground state has recently beenshown 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 chiralorder. This demonstrates that the chiral phase can indeed be stabilized for nottoo large values of $m$, opening the way to its experimental realisations inother lattices.

Stabilization of the chiral phase of the SU($6m$) Heisenberg model on the honeycomb lattice with $m$ particles per site for $m$ larger than 1