Localization and delocalization in the one-dimensional Tomonaga model

To understand what happens in a one-dimensional system when both localization due to impurities and electron-electron interaction ($U_{\parallel }$) are important, we studied the effect of impurities in the Tomonaga model by looking at the perturbation series for the conductivity in the impurity potential $U_{\perp }$. By making an analogy to the two-dimensional Coulomb gas we were able to relate the static conductivity to the effective potential of two test charges infinitely far apart and understand the localization phenomena in terms of Debye screening. Furthermore we found that as the electron-electron interaction becomes attractive enough the electronic state becomes delocalized. This corresponds to a phase transition in the Coulomb gas when the positive and the negative charges are paired up and Debye screening disappears.