Quantum diffusion and localization in disordered systems

We describe quantum diffusion of the electrons in a disordered system by requiring that ψ(r, t)² obeys a diffusion equation, where ψ(r, t) is the time-dependent wavefunction. It is found that, regardless of the weakness of the disorder, this requirement leads to electron eigenstates which consist of a power-law component for dimension d> 1 and a logarithmic correction for d = 1, in addition to an extended function. For d≤2, this is correct only below a certain length scale. As a result, even for k_F^l>>1, the conductivity σ is reduced from the Boltzmann conductivity σ_B in agreement with diagrammatic calculations. By expanding the eigenstate in terms of 1/rⁿit is shown that the 1 /rd⁻¹ term is responsible for the reduction of σ in the weak-disorder limit. It is demonstrated that in three dimensions one can extrapolate the formula for the conductivity down to the Anderson transition to obtain σ =g²σ_B[1−(C/g²k_F²l²) (1 − l/L)] where g is the reduction in the density of states due to disorder and C is a dimensionless constant of order unity which depends on some cut-off length.