Exact results for spin and charge transport in the Hubbard model on bipartite lattices with $N_a\gg 1$ sites and arbitrary spatial dimension $d$ are obtained using a recently introduced representation in terms of physical spins and physical $\eta$-spins. This includes results on the role played in spin and charge transport by the $\tau$-translational U(1) symmetry beyond SO(4), within the model's global $\left[\mathrm{SO}\right(4)\times \mathrm{U}(1)]/{\mathbb{Z}}_2$ symmetry. The $N_{s,\pm 1/2}=(N_a/2-S_{\tau }\pm S_s^z)$ physical spins with projection $\pm 1/2$ and the $N_{\eta ,\pm 1/2}=(2S_{\tau }\pm S_{\eta }^z)$ $\eta$-spins with projection $\pm 1/2$ that populate the energy and momentum eigenstates of the Hubbard model on bipartite lattices, where $S_{\tau }$ denotes the eigenvalue of the generator of the $\tau$-translational U(1) symmetry, are found to carry elementary spin $(\alpha =s)$ and charge $(\alpha =\eta )$ currents ${\vec{j}}_{\alpha ,\pm 1/2}=\pm {\vec{C}}_{\alpha }/N_{\alpha }$. Here, $N_{\alpha }=N_{\alpha ,+1/2}+N_{\alpha ,-1/2}$, and an exact theorem for the $\alpha =s,\eta$ vectors ${\vec{C}}_{\alpha }$ is established, simplifying the expressions for their components $C_{\alpha ,\gamma }$ associated with the spatial directions $\gamma =1,\cdots ,d$. The expectation values of the spin and charge current operator corresponding to $S_{\alpha }^z$ for all $4^{N_a}$ energy and momentum eigenstates are simply given by ${\sum }_{\sigma =\pm 1/2}{N}_{\alpha ,\sigma }{\vec{j}}_{\alpha ,\sigma }$. They can also be expressed as $p_{\alpha z}{\vec{C}}_{\alpha }$, where the dependence on $S_{\tau }$ of the $\alpha =s,\eta$ polarization $p_{\alpha }$, which determines the interval of spin and charge polarization, $p_{\alpha z}=2S_{\alpha }^z/N_{\alpha }\in [-p_{\alpha },p_{\alpha }]$, is found to be universal for all bipartite lattices. Our exact results contribute to a deeper understanding of the model's physics and provide valuable insight into the transport properties of condensed-matter materials—such as cuprate superconductors, graphene, and graphene-derived systems—as well as other quantum systems described by the model.

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• Electrical properties
• Magnetotransport
• Exact solutions for many-body systems
• Hubbard model