Level statistics and localization: A study of the 1D Anderson model

The level statistics and the localization of a particle in a one-dimensional random potential are investigated numerically. First, we study the level spacing distributionP(S) as a function of the system lengthN and of the disorderw of the system. We show that in addition to the localized and delocalized regimes a third regime can be distinguished: For large disorder a resonance appears inP(S), which is caused by a local level repulsion effect. Second, we investigate the distribution of the localization lengths within the Anderson chain as a function ofN andw. Here, we identify the localization length with the rms spread of the wave functions and we show that this measure for the localization of the eigenstates is not a self-averaging quantity.