One-dimensional electron systems interacting with long-range Coulomb forces (quantum wires) show a Wigner crystal structure. We investigate in this paper the transport properties of such a Wigner crystal in the presence of impurities. Contrary to what happens when only short-range interactions are included, the system is dominated by 4
kF scattering on the impurities. There are two important length scales in such a problem: one is the pinning length above which the (quasi-)long-range order of the Wigner crystal is destroyed by disorder. The other length, ${\ensuremath{\xi}}_{\mathrm{cr}}$, is the length below which Coulomb interactions are not important and the system is behaving as a standard Luttinger liquid with short-range interactions. We obtain the frequency and temperature dependence of the conductivity. We show that such a system is very similar to a classical charge density wave pinned by impurities, but with important differences due to quantum fluctuations and long-range Coulomb interactions. Finally, we discuss our results in comparison with experimental systems.
We use Density Functional Theory to study interacting spinless electrons on a one-dimensional quantum ring in the density range where the system undergoes Wigner crystallization. The Wigner transition leads to a drastic ``collective'' electron localization due to the Wigner crystal pinning, provided a weak impurity potential is applied. To reveal this localization we examine a persistent current in a ring penetrated by a magnetic flux. Using the DFT-OEP method we calculated the current as a function of the interaction parameter r_S. We find that in the limit of vanishing impurity potential the persistent current stays constant up to a critical value of r_S^c=2.05 but shows a drastic exponential decay for larger r_S which reflects a formation of a pinned Wigner crystal. Above r_S^c the amplitude of the electron density oscillations exactly follows the (r_S-r_S^c)^{1/2} behaviour, confirming a second-order phase transition as expected in the mean-field-type OEP approximation.