Critical exponents for Anderson localization

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Abstract
We perform numerical calculations on a simple cubic lattice for a standard diagonally disordered tight-binding Hamiltonian, whose random site energies are chosen from a Gaussian distribution with variance ∑2. From phenomenological renormalization group studies of the localization length, we determine that the critical disorder is σc≡∑c/J=6.00±0.17, which is in good agreement with previous results (J is the nearest neighbor transfer matrix element). From our calculations we can also determine the mobility edge trajectory, which appears to be analytic at the band center. Defining an order parameter exponent β, which determines how the fraction of extended states vanishes as the critical point is approached from below, this implies that β=1/2, in agreement with a previous study. From a finite-size scaling analysis we find that π2/ν=1.43±0.10, where π2 and ν are the inverse participation ratio and localization length critical exponents, respectively. This ratio of exponents can also be interpreted as the fractal dimension (also called the correlation dimension) D2 of the critical wave functions. Generalizations of the inverse participation ratio lead to a whole set of critical exponents πk, and corresponding generalized fractal dimensions Dk=πk/ν(k−1). From finite-size scaling results we find that D3=1.08±0.10 and D4=0.87±0.09. The inequality of the three dimensions D2, D3, and D4 shows that the critical wave functions have a multifractal structure. Using a generalized phenomenological renormalization technique on the participation ratios, we find that ν=0.99±0.04. This result is in agreement with experiments on compensated or amorphous doped semiconductors.