Anderson localization in an anisotropic model

The phase diagram of the Anderson-localization problem in an anisotropic model which has both two-dimensional (2D) and 3D scattering is studied using a diagrammatic method. It is found that the mobility edge follows the relation θ=a exp(-b/${\mathit{W}}_{\mathit{c}}^2$) in the anisotropy limit, where θ is the anisotropy parameter which interpolates a system of two-dimensionally coupled pure 1D chains, each having a different site energy, and a system of 3D isotropic randomness. ${\mathit{W}}_{\mathit{c}}$ is the critical randomness, and a and b are two Fermi-energy-dependent constants. This behavior is different from the results of a 1D-to-3D crossover model studied previously. In that model, there exists a nonzero critical anisotropy ${\mathrm{\theta }}_{\mathit{c}}$ below which all states are localized. Physical reasons are given to explain this difference.