Transport and Temperature 1: Exact spectrum and resistivity for the one-dimensional infinite-$U$ Hubbard model

Understanding charge transport in strongly correlated systems remains a central challenge in condensed matter physics, particularly in light of the ubiquitous linear-in-$T$ resistivity observed in strange metals across many platforms from bulk cuprates to twisted bilayer graphene. Here, we investigate charge transport in the one-dimensional Hubbard model in the infinite-interaction limit. Focusing on the dilute limit with a fixed number of doped holes, we first construct the exact \emph{and explicit - i.e. beyond Bethe ansatz} energy spectrum and then derive a closed-form analytical expression for the charge Drude weight at arbitrary temperatures. We further analyze the low-temperature scaling and identify a linear-in-$T$ correction to the Drude weight. Upon regularizing the singular Drude contribution to the DC conductivity, we find that this behavior corresponds to an effective linear-in-$T$ resistivity, which may provide analytical insight into the emergence of strange-metal transport in two-dimensional strongly correlated systems.