Static Conductance and Scaling Theory of Localization in One Dimension

Abstract

It is shown that the Ohmic dc conductance of a one-dimensional system equals to either ${G}_{p}=(\frac{{e}^{2}}{\ensuremath{\pi}\ensuremath{\hbar}})T$ or ${G}_{c}=\frac{(\frac{{e}^{2}}{\ensuremath{\pi}\ensuremath{\hbar}})T}{(1\ensuremath{-}T)}$, depending on whether the system is connected to a perfect one-dimensional conductor or to a classical wire (behaving as a current source), respectively. The parameter $\ensuremath{\beta}\ensuremath{\equiv}\frac{d〈\mathrm{ln}{g}^{\ensuremath{'}}〉}{d\mathrm{ln}L}$, where $L$ is the length of the system, depends only on $〈\mathrm{ln}{g}^{\ensuremath{'}}〉$, so that $\ensuremath{\beta}=〈\mathrm{ln}{g}^{\ensuremath{'}}〉$, with ${g}^{\ensuremath{'}}=(\frac{\ensuremath{\pi}\ensuremath{\hbar}}{{e}^{2}}){G}_{p}$.