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 vF↑−vF↓=Δ≠0 is considered. The presence of the Zeeman splitting gives rise to a non-diagonal bosonic model for the densities and canonical momenta conjugated to the densities. As a result the bosonic Hamiltonian cannot be diagonalized by a simple unitary transformation. To deal with the problem, we transform our model into the Lagrangian formalism. We find that for an extended Hubbard model controlled by the nearest neighbor interaction with a strength V gives rise to a term ρ↑Rρ↑L+ρ↓Rρ↓L. As a result, two Luttinger liquids for spin-up and spin-down electrons are obtained. We find the conductances for spin-up G↑ and spin-down G↓ are not equal. The anisotropy in the conductances G↑≠G↓ gives rise to a spin-polarized wire.