Localization in disordered systems

A discussion is presented of localization effects on the conductivity σ of a disordered system. It is shown that for any number of dimensions, the value of σ drops well below the Boltzmann conductivity ne 2τ/m when the disorder is increased. Three arguments are reviewed for power-law corrections to the eigenstates ψ. The power-law correction to ψ yields exactly the same reduction in σ as that obtained from diagrammatic perturbative approaches. The critical behaviour of σ near the Anderson transition in three dimensions is discussed and compared with experiment. In particular, a possible explanation is suggested for the critical exponent v = 1/2. In two dimensions, experiments on Si inversion layers seem to contradict the scaling theory. Including higher-order effects of the power-law corrections to ψ accounts for the experimental non-universality of the β function in two dimensions.