Correlation length and inverse-participation-ratio exponents and multifractal structure for Anderson localization

We perform numerical finite-size-scaling calculations on a standard diagonally disordered tight-binding Hamiltonian, with a Gaussian site-energy distribution. We find that the localization-length exponent is ν=0.97±0.05. We also find that ${\mathrm{\pi }}_2$/ν=1.43±0.10, where ${\mathrm{\pi }}_2$ is the inverse-participation-ratio exponent. ${\mathrm{\pi }}_2$/ν can also be interpreted as the fractal dimension of the critical eigenstates. Finally, by looking at higher moments of the critical wave functions, we show that they display a multifractal structure.