We investigate numerically the power-law random matrix ensembles. Wave functions are fractal up to a characteristic length whose logarithm diverges asymmetrically with different exponents, 1 in the localized phase and 0.5 in the extended phase. The characteristic length is so anomalously large that for macroscopic samples there exists a finite critical region, in which this length is larger than the system size. The Green’s functions decrease with distance as a power law with an exponent related to the correlation dimension.

• Received 27 November 2000

DOI:https://doi.org/10.1103/PhysRevLett.87.056601