Numerical evidence for power-law localisation in weakly disordered systems

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Abstract
A direct diagonalisation scheme is used to determine the wavefunctions of one-, two- and three-dimensional samples with weak gaussian disorder. The inverse participation number is investigated for different energy ranges. Its distinct dependence on the sample size allows the authors to draw conclusions about the localisation of the states considered: in one dimension all states, and in two and three dimensions only the energetically extreme states, are strongly localised on few sites. At the band centre and in its vicinity in three dimensions the states are extended. For all other states in two and three dimensions the wavefunctions are characterised by a power-law dependence.