A variational investigation of the SU(N) Heisenberg model in antisymmetric irreducible representations

Motivated by recent experimental progress in the context of ultra-cold multi-colour fermionic atoms in optical lattices, this thesis investigates the properties of the antiferromagnetic SU(N) Heisenberg models with fully antisymmetric irreducible representations labelled by a Young tableau with $m$ boxes on each site of various lattices. Thanks to an intensive use of variational Monte Carlo method based on Gutzwiller projected fermionic wave functions, we aim to determine the ground state and its related properties of systems which are beyond the capabilities of other numerical methods. This typically means that we will study large lattices with many particles per site for $N>2$. The first part of this thesis will introduce the basic theoretical tools connected to the SU(N) Heisenberg model. A very short review of group theory introducing Young tableaux, will clarify the concept of fully antisymmetric irreducible representation of the SU(N) Lie algebra. Some properties of the Heisenberg model will be discussed and a mean-field solution will be given. The numerical methods used during this thesis and their limitations wil also be introduced. We will investigate the properties of 1-dimensional systems in the second part of this work. In particular, the chains have been studied for arbitrary $N$ and $m$ with non-abelian bosonisation leading to predictions about the nature of the ground state (gapped or critical) in most but not all cases. We have been able to verify these predictions for a representative number of cases with $N\leq10$ and $m\leq N/2$, and we have shown that the opening of a gap is associated with a spontaneous dimerisation or trimerisation depending on the value of $m$ and $N$. We have also investigated the marginal cases, where non-abelian bosonisation did not lead to any prediction. In these cases, variational Monte Carlo predicts that the ground state is critical with exponents that are consistent with conformal field theory. The other 1-dimensional system that has been studied is the ladder for which field-theory gives some predictions. For instance, the ground state of the SU(4) case in the fundamental irreducible representation is believed to consist of a 4-site plaquette state for any non-zero inter-leg coupling. We have confirmed the presence of this state, even if in the weak coupling regime, the optimal variational state is gapless. The study has been extended to various $N$ and $m$ values providing interesting results. The last part will present the results for 2-dimensional lattices. The square lattice will first be investigated. A mean-field analysis predicts a plaquette ground state for $3\leq N/m < 5$ and of a chiral ground state for $N/m\geq 5$. By systematically studying the SU(3m) and SU(6m) Heisenberg model, we determined that the mean-field solution is valid even at small $m$ ($m>1$ for SU(3m) and $m>2$ for SU(6m)). Then, for the SU(6m) Heisenberg model with $m$ particles per site on a honeycomb lattice, we make the connection between the $m=1$ case, 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. We were able to show that this system has a clear tendency toward the chiral order as soon as $m\geq 2$. This demonstrates that the chiral phase can be stabilised for not too large values of $m$, opening the way to its experimental realisations in other lattices. We will finally present results for the SU(3m) on the triangular lattice for which we find a plaquette state, when $m>1$.