Static Conductance and Scaling Theory of Localization in One Dimension

It is shown that the Ohmic dc conductance of a one-dimensional system equals to either $G_p=\left(\frac{e^2}{\pi \hslash }\right)T$ or $G_c=\frac{\left(\frac{e^2}{\pi \hslash }\right)T}{\mathrm{}(1-T)\mathrm{}}$, 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 $\beta \equiv \frac{d〈\ln g^{\prime }〉}{d\ln L}$, where $L$ is the length of the system, depends only on $〈\ln g^{\prime }〉$, so that $\beta =〈\ln g^{\prime }〉$, with $g^{\prime }=\left(\frac{\pi \hslash }{e^2}\right)G_p$.