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

Abstract

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 $T\ensuremath{\gg}\ensuremath{\omega}$. 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 $T<{L}^{(3/2)(K\ensuremath{-}1)}$ for short-range interactions with parameter $K<1/3$. 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, $K<1/3$. This leads to various regimes, depending on temperature and on $K$. For randomly distributed weak impurities, the conductance fluctuations, equal to that of $\mathcal{R}=g\ensuremath{-}{2e}^{2}/h$, are computed. They depend on the interaction parameters, and are different for electrons with or without spin. But the ratio $\mathrm{Var}(\mathcal{R})/{\mathcal{R}}^{2}$ stays always of the same order: it is equal to ${L}_{T}/L\ensuremath{\ll}1$ 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.