Interacting electrons with spin in a one-dimensional dirty wire connected to leads

We investigate a one-dimensional wire of interacting electrons connected to one-dimensional noninteracting leads in the absence and in the presence of a backscattering potential. The ballistic wire separates the charge and spin parts of an incident electron even in the noninteracting leads. The Fourier transform of nonlocal correlation functions is computed for $T\gg \omega$. In particular, this allows to study the proximity effect, related to the Andreev reflection. In addition, a new type of proximity effect emerges when the wire has normally a tendency towards Wigner crystal formation. The latter is suppressed by the leads below a space-dependent crossover temperature; it gets dominated everywhere by the ${2k}_F$ charge-density wave at $T<\mathrm{}{L}^{\mathrm{}(3/2)\left(K-1\right)}$ for short-range interactions with parameter $K<\mathrm{}1/3\mathrm{}$. The lowest-order renormalization equations of a weak backscattering potential are derived explicitly at finite temperature. A perturbative expression for the conductance in the presence of a potential with arbitrary spatial extension is given. It depends on the interactions, but is also affected by the noninteracting leads, especially for very repulsive interactions, $K<\mathrm{}1/3\mathrm{}$. This leads to various regimes, depending on temperature and on $K$. For randomly distributed weak impurities, the conductance fluctuations, equal to that of $\mathcal{R}=g-{2e}^2/h$, are computed. They depend on the interaction parameters, and are different for electrons with or without spin. But the ratio $\mathrm{Var}\left(\mathcal{R}\right)/{\mathcal{R}}^2$ stays always of the same order: it is equal to $L_T/L\ll 1$ in the high-temperature limit, then saturates at $1/2$ in the low-temperature limit, indicating that the relative fluctuations of $\mathcal{R}$ are universal.