Exact results for the Hubbard model on bipartite lattices in spatial dimensions $d>1$: Seven theorems from the full [SU(2)$\times$SU(2)$\times$U(1)]/${\mathbb{Z}}_2^2$ symmetry

There are few exact results for the Hubbard model on bipartite lattices of spatial dimension $d>1$. Nevertheless, the Hubbard model with transfer integral $t$ and onsite repulsion $U$ on bipartite lattices with $N_a$ sites, such as the square, honeycomb, cubic, body-centered cubic, face-centered cubic, and diamond lattices, provides the simplest toy model for describing electronic correlations in many condensed-matter systems and is therefore a quantum problem of considerable physical interest. Seven exact theorems that provide new physical insight into the model are established. Overall, the exact framework based on physical spins and physical $\eta$-spins for the Hubbard model on bipartite lattices of spatial dimension $d>1$ introduced in this paper offers a robust foundation for future studies of the model, as well as of the condensed-matter materials, such as cuprate superconductors, graphene and graphene-derived systems, and other quantum systems, that it describes.