Anderson localization in a nonlinear-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>σ</mml:mi></mml:math>-model representation

Abstract

It is shown that the behavior of Anderson localization near the mobility edge is described by a noncompact nonlinear $\ensuremath{\sigma}$ model in $2+\ensuremath{\epsilon}$ dimensions. The explicit calculation in perturbation gives one-to-one correspondence with the effective Hamiltonian of the nonlinear $\ensuremath{\sigma}$ model. The renormalization-group $\ensuremath{\beta}$ function and the anisotropic effect are discussed.