Matrix continued-fraction calculation of localization length

A matrix continued-fraction method is used to study the localization length of the states at the band center of a two-dimensional crystal with disorder given by the Anderson model and with incommensurate charge-density waves. For the disordered case it is found that exponentially localized states, which scale according to the work of MacKinnon and Kramer, become weakly localized as the disorder becomes weaker, and there is some critical disorder for which the localization length does not saturate with the width of the strips, which confirms the results found by Pichard and Sarma. Weakly localized states are also found in one dimension for $\frac{W}{V}\lesssim 1$. In the case of a crystal with a modulation that is incommensurate in one direction and commensurate in the other, the localization in the first direction behaves in a similar fashion as that found by Aubry, Sokoloff, and others for the one-dimensional chain, that is, approximately the same critical modulation strength is found: $\frac{W_c}{V}=2\mathrm{}$. If the modulation is incommensurate in two perpendicular directions, there appears a tendency of increasing localization lengths as the width of two-dimensional strips is increased and an intermediate regime develops between the insulating and metallic regions.