Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions

Arguments are presented that the T = 0 conductance G of a disordered electronic system depends on its length scale L in a universal manner. Asymptotic forms are obtained for the scaling function β ( G ) = d ln G / d ln L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429494116/afa305c7-7e11-4618-94aa-661e4acf5b1d/content/eq2868.tif"/> , valid for both G ≪ G c ≃ e 2 / ℏ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429494116/afa305c7-7e11-4618-94aa-661e4acf5b1d/content/eq2869.tif"/> and G ≫ G c https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429494116/afa305c7-7e11-4618-94aa-661e4acf5b1d/content/eq2870.tif"/> . In three dimensions, G_c is an unstable fixed point. In two dimensions, there is no true metallic behavior; the conductance crosses over smoothly from logarithmic or slower to exponential decrease with L.