I've been reading about the Heisenberg Model:
\begin{equation} H=-J\sum_{\langle i, j\rangle } \hat{\vec{S}}_i .\hat{\vec{S}}_j = -J\sum_{\langle i, j\rangle}\left[\hat{S}^z_i \hat{S}^z_j + \frac{1}{2}\left(\hat{S}^+_i \hat{S}^-_j+\hat{S}^-_i \hat{S}^+_j\right)\right] \end{equation}
Where $\hat{\vec{S}}_i$ is the Spin Vector Operator in the site i. In a bipartite lattice (where you can divide the lattice in two sublattices such that every site is surrounded by sites of the other sublattice) the classical antiferromagnetic ($J<0$) ground state is every spin of sublattice A pointing at one direction while every spin in sublattice B pointing in the opposite direction. In the Heisenberg Model this is not an eigenstate because acting with$\hat{S^+}_i$ and $\hat{S^-}_i$ modify neighbouring spins.
My question is: Is this problem solved? Do we know the exact ground state of this model? I'm familiar with the Holstein-Primakoff transformation and the ondulatory solutions known as magnons but I'm wondering if there is an exact known ground state.
