  Using linear flavor-wave theory (LFWT) and auxiliary field quantum Monte
Carlo (QMC), we investigate the properties of the SU(4) Heisenberg model on the
anisotropic square lattice in the fully antisymmetric six-dimensional
irreducible representation, a model that describes interacting fermions with
four flavors at half-filling. Thanks to the calculations on very large systems,
we have been able to convincingly demonstrate that QMC results are consistent
with a small but finite antiferromagnetic moment at the isotropic point, in
qualitative agreement with LFWT obtained earlier [F. H. Kim {\it{et al.}},
Phys. Rev. B {\bf 96}, 205142 (2017)], and in quantitative agreement with
results obtained previously on the Hubbard model [D. Wang {\it{et al.}}, Phys.
Rev. Lett. {\bf 112}, 156403 (2014)] after extrapolation to infinite $U/t$. The
presence of a long-range antiferromagnetic order has been further confirmed by
showing that a phase transition takes place into a valence-bond solid (VBS)
phase not too far from the isotropic point when reducing the coupling constant
along one direction on the way to decoupled chains.
