Large-$n$ limit of the Hubbard-Heisenberg model

To gain insight into the behavior of the Hubbard model, we define a $\mathrm{SU}\mathrm{}\left(n\right)\mathrm{}$ invariant generalization of the Hubbard-Heisenberg model and, in the large-$n$ limit, solve it in one dimension and in two dimensions on a square lattice. In one dimension the ground state is completely dimerized near half filling. We show that this behavior agrees with a renormalization-group solution of the one-dimensional $\mathrm{SU}\mathrm{}\left(n\right)\mathrm{}$ Hubbard model. In two spatial dimensions we find several different ground states depending on the size of the hopping term $t$, the doping $\delta$, and the biquadratic spin interaction $\tilde{J}$. In particular, the undimerized "flux" or "$s+id$" phase is the ground state at half filling for sufficiently large $t$ or $\tilde{J}$. We study the electronic and spin excitations of the various phases and comment on the relevance of the large-$n$ problem to the high-$T_c$ superconductors.