The disorder induced metal-insulator transition is investigated in a three-dimensional simple cubic lattice and compared for the presence and absence of time-reversal and spin-rotational symmetry, i.e. in the three conventional symmetry classes. Large scale numerical simulations have been performed on systems with linear sizes up to L = 100 in order to obtain eigenstates at the band center, E = 0. The multifractal dimensions, exponents Dq and αq, have been determined in the range of −1 ≤ q ≤ 2. The finite-size scaling of the generalized multifractal exponents provide the critical exponents for the different symmetry classes in accordance with values known from the literature based on high precision transfer matrix techniques. The multifractal exponents of the different symmetry classes provide further characterization of the Anderson transition, which was missing from the literature so far.
-We present a new method for the numerical treatment of second order phase transitions using the level spacing distribution function P (s). We show that the quantities introduced originally for the shape analysis of eigenvectors can be properly applied for the description of the eigenvalues as well. The position of the metal-insulator transition (MIT) of the three dimensional Anderson model and the critical exponent are evaluated. The shape analysis of P (s) obtained numerically shows that near the MIT P (s) is clearly different from both the Brody distribution and from Izrailev's formula, and the best description is of the form P (s) = c 1 s exp(−c 2 s 1+β ), with β ≈ 0.2. This is in good agreement with recent analytical results.
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