Magnetic-Field Dependence of the Localization Length in Anderson Insulators

Abstract

Using the conventional scaling approach as well as the renormalization group analysis in $d=2+ε$ dimensions, we calculate the localization length $ξ(B)$ in the presence of a magnetic field $B$. For the quasi 1D case the results are consistent with a universal increase of $ξ(B)$ by a numerical factor when the magnetic field is in the range $\ell\ll{\ell_{\!{_H}}}\altξ(0)$, $\ell$ is the mean free path, ${\ell_{\!{_H}}}$ is the magnetic length $\sqrt{\hbar c/eB}$. However, for $d\ge 2$ where the magnetic field does cause delocalization there is no universal relation between $ξ(B)$ and $ξ(0)$. The effect of spin-orbit interaction is briefly considered as well.