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

• Received 9 January 1981

DOI:https://doi.org/10.1103/PhysRevB.24.2671