The multifractal character of the electronic states in disordered two-dimensional systems

The nature of electronic states in disordered two-dimensional (2D) systems is investigated. With this aim, we present our calculations of both density of states and d.c. conductivity for square lattices modelling the Anderson Hamiltonian with on-site energies randomly chosen from a box distribution of width W. For weak disorder (W), the eigenfunctions calculated by means of the Lanczos diagonalization algorithm display spatial fluctuations reflecting their (multi)fractal behaviour. For increasing disorder the observed increase of the curdling of the wavefunction reflects its stronger localization. However, as a function of energy, the eigenstates at energy mod E mod /V approximately=1.5 are found to be the least localized over the band. Our d.c. conductivity results suggest a critical fractal dimension d_c*=1.48+or-0.17 to discriminate between me exponentially and the power-law-localized states. Consequences of the localization for transport properties are also discussed.