We calculate several thermodynamic quantities for repulsively interacting one-dimensional fermions. We solve the Hubbard model at both zero and finite temperatures using the Bethe-ansatz method. For arbitrary values of the chemical potential, we calculate the particle number density, the double occupancy, various compressibilities, and the entropy as a function of temperature and interaction. We find that these thermodynamic quantities show a characteristic behavior so that measurements of these quantities can be used as a detection of temperature, the metal-insulator transition, and metallic and insulating phases in the trap environment. Further, we discuss an experimental scheme to extract these thermodynamic quantities from the column density profiles. The entropy and the compressibility of the entire trapped atomic cloud also reveal characteristic features indicating whether insulating and/or metallic phases coexist in the trap.
We study long-time spin diffusion of harmonically trapped lattice fermions in one dimension. Combining thermodynamic Bethe ansatz approach and local density approximation, we calculate spin current and spin diffusion coefficient driven by the population imbalance. We find spin current is driven by susceptibility effects rather than typical diffusion where magnetization would transport from regions of high magnetization to low. As expected, spin transport is zero through insulating regions and are only present in the metallic regions. In the weak coupling limit, the local spin diffusion coefficient shows maxima at all the insulating regions. Further, we estimate damping rate of diffusion modes in the weak coupling limit within the lower metallic portion of the cloud. The predicted spin current pattern can be probed via currently available experimental techniques.Comment: 7 pages, 4 figure
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