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 $D_q$ and $\alpha_q$, have been determined in the range of $-1\leq q\leq 2$. 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.

Recently we found an Anderson-type localization-delocalization transition in the QCD Dirac spectrum at high temperature.Using spectral statistics we obtained a critical exponent compatible with that of the corresponding Anderson model.Here we study the spatial structure of the eigenmodes both in the localized and the transition region.Based on previous studies in the Anderson model, at the critical point, the eigenmodes are expected to have a scale invariant multifractal structure.We verify the scale invariance of Dirac eigenmodes at the critical point.

We investigate the Anderson transition found in the spectrum of the Dirac operator of quantum chromodynamics at high temperature, studying the properties of the critical quark eigenfunctions. Applying multifractal finite-size scaling we determine the critical point and the critical exponent of the transition, finding agreement with previous results, and with available results for the unitary Anderson model. We estimate several multifractal exponents, finding also in this case agreement with a recent determination for the unitary Anderson model. Our results confirm the presence of a true Anderson localization-delocalization transition in the spectrum of the quark Dirac operator at high temperature, and further support that it belongs to the 3D unitary Anderson model class.

The phase diagram of the metal-insulator transition in a three-dimensional quantum percolation problem is investigated numerically based on the multifractal analysis of the eigenstates. The large-scale numerical simulation has been performed on systems with linear sizes up to L = 140. The multifractal dimensions, exponents D-q and alpha(q), have been determined in the range of 0 <= q <= 1. Our results confirm that this problem belongs to the same universality class as the three-dimensional Anderson model; the critical exponent of the localization length was found to be nu = 1.622 +/- 0.035. However, the multifractal function f(alpha) and the exponents D-q and alpha(q) produced anomalous variations along the phase boundary, p(c)(Q) (E).

The localization of one-electron states in the large (but finite) disorder limit is investigated. The inverse participation number shows a non--monotonic behavior as a function of energy owing to anomalous behavior of few-site localization. The two-site approximation is solved analytically and shown to capture the essential features found in numerical simulations on one-, two- and three-dimensional systems. Further improvement has been obtained by solving a three-site model.

In this work we investigate the inverse of the celebrated Bohigas-Giannoni-Schmit conjecture. Using two inversion methods we compute a one-dimensional potential whose lowest N eigenvalues obey random matrix statistics. Our numerical results indicate that in the asymptotic limit N -> infinity the solution is nowhere differentiable and most probably nowhere continuous. Thus such a counterexample does not exist.