There are a 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. Most previous studies have assumed that, for finite onsite repulsion $U>0$, the global symmetry of the model is $\mathrm{SO}\left(4\right)=[\mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right)]/{\mathbb{Z}}_2$. However, the actual global symmetry is larger and given by $[\mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right)]/{\mathbb{Z}}_2^2$, which may equivalently be written as $[\mathrm{SO}\left(4\right)\times \mathrm{U}\left(1\right)]/{\mathbb{Z}}_2$ or as $\mathrm{SO}\left(3\right)\times \mathrm{SO}\left(3\right)\times \mathrm{U}\left(1\right)$. The main goal of this paper is to extract physical insight, including further exact results for the Hubbard model on bipartite lattices with spatial dimension $d>1$, from its full global symmetry. The hidden U(1) symmetry beyond SO(4), which has been ignored in most previous studies, becomes an explicit symmetry within an exact quasiparticle representation of the model for $u=U/t>0$. The generator of this symmetry has eigenvalues $S_{\tau }\in \{0,\frac{1}{2},1,\frac{3}{2},2,...,\frac{N_a}{2}\}$. Importantly, this exact quasiparticle representation naturally leads to the emergence of $N_s=(N_a-2S_{\tau })$ physical spins $1/2$ and $N_{\eta }=2S_{\tau }$ physical $\eta$-spins $1/2$, associated with the two SU(2) symmetries in $[\mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right)]/{\mathbb{Z}}_2^2$. The spin and $\eta$-spin multiplet and singlet configurations of the $4^{N_a}$ exact energy and momentum eigenstates are generated from the vacuum by quasiparticle creation and annihilation operators. 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 $d>1$ 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.

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• 2-dimensional systems
• 3-dimensional systems
• Hubbard model