The extension of the linear flavor-wave theory (LFWT) to fully antisymmetric irreducible representations (irreps) of SU(N ) is presented in order to investigate the color order of SU(N ) antiferromagnetic Heisenberg models in several two-dimensional geometries. The square, triangular and honeycomb lattices are considered with m fermionic particles per site. We present two different methods: the first method is the generalization of the multiboson spin-wave approach to SU(N ) which consists of associating a Schwinger boson to each state on a site. The second method adopts the Read and Sachdev bosons which are an extension of the Schwinger bosons that introduces one boson for each color and each line of the Young tableau. The two methods yield the same dispersing modes, a good indication that they properly capture the semi-classical fluctuations, but the first one leads to spurious flat modes of finite frequency not present in the second one. Both methods lead to the same physical conclusions otherwise: long-range Néel-type order is likely for the square lattice for SU(4) with two particles per site, but quantum fluctuations probably destroy order for more than two particles per site, with N = 2m. By contrast, quantum fluctuations always lead to corrections larger than the classical order parameter for the tripartite triangular lattice (with N = 3m) or the bipartite honeycomb lattice (with N = 2m) for more than one particle per site, m > 1, making the presence of color very unlikely except maybe for m = 2 on the honeycomb lattice, for which the correction is only marginally larger than the classical order parameter.

It was recently argued that SU(3) chains in the p-box symmetric irreducible representation (irrep) exhibit a "Haldane gap" when p is a multiple of 3 and are otherwise gapless [Nucl. Phys. B 924, 508 (2017)]. We extend this argument to the self-conjugate irreps of SU(3) with p columns of length 2 and p columns of length 1 in the Young tableau (p = 1 corresponding to the adjoint irrep), arguing that they are always gapped but have spontaneously broken parity symmetry for p odd but not even.

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 et al., Phys. Rev. B 96, 205142 (2017)], and in quantitative agreement with results obtained previously on the Hubbard model [D. Wang et al., Phys. Rev. Lett. 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.