Transport in a spin polarized Luttinger liquid in the presence of a single impurity

The conductance for a single impurity in a one dimensional gas with electron-electron interaction and Zeeman splitting $v_F^\uparrow - v_F^\downarrow = \Delta \neq 0$ is considered. We find that for an extented Hubbard model, $\rho_\uparrow^R \rho_\uparrow^L + \rho_\downarrow^R \rho_\downarrow^L$, a Luttinger liquid for spin up and spin down is obtained. As a result the conductances for spin up $G_\uparrow$ and spin down $G_\downarrow$ are not equal. The anisotropy in the conductances $G_\uparrow \neq G_\downarrow$ gives rise to a spin polarized wire.