Self-consistent theory of magnetoconductance in two-dimensional Anderson localized systems

The conductivity of noninteracting electrons in a two-dimensional disordered system with weak impurity scattering is studied by using the diagrammatic Green's-function method. Starting from the current-current correlation functions, a self-consistent equation can be constructed for the conductivity σ(ω). In the absence of a magnetic field, we show that $Re{σ}({ω}){~}1{-}{(2{π}{E}F{τ})}^{{-}1}ln({1}{{τ}{ω}})$ for large ${ω}$ and $Re{σ}({ω}){~}{{ω}}²$ as ${ω}{→}0$. This result is in agreement with that of Vollhardt and W\"olfle. In the presence of a weak magnetic field, we show that the dc magnetoconductance ${σ}(0){~}1{-}{(2{π}{E}F{τ})}^{{-}1}ln(1/H)$ for ${l}L{}{l}B{}l$ and ${σ}(0)=0$ for ${l}B{}{l}L{}l$ where ${l}L$, ${l}B$, and $l$ are, respectively, the localization length, radius of the lowest Landau orbit, and mean free path.