Scaling studies of the resistance of the one-dimensional Anderson model with general disorder

The resistance ρ of a one-dimensional Anderson model with both diagonal and off-diagonal disorder is studied by analytic and numerical techniques. A recursive method is developed and used to derive an exact scaling law for the average resistance at E=0 for arbitrary disorder, and for E≠0 in the limit of weak disorder. The average resistance grows exponentially with L, the length of the sample, in all cases. The typical resistance ρ~=exp[<ln(1+ρ)>]-1 is also found to grow exponentially with L in all cases, except for purely off-diagonal disorder at E=0, where <ln(1+ρ)>~L. An explanation is given for the existence of this special case and it is shown that all our results are consistent with a lognormal probability distribution of the resistance for ρ>>1. Quantitative estimates are made of the reliability of numerically performed averages which show that a numerical average will converge only very slowly to the analytic result. This provides a qualitative explanation of the slower than linear growth of ln<ρ> with L found in several numerical calculations; its consequences for experiment are also explored.