Parametric motion of energy levels in quantum disordered systems

The parametric motion of one-electron energy levels of disordered systems near the Anderson localization transition is studied. We perform numerical calculations of the distributions of the first (level velocity) and the second (level curvature) derivatives with respect to a change of boundary conditions. The statistics of their fluctuations are scale-invariant at the mobility edge and exhibit critical behavior. We show that the distributions of level velocities and level curvatures undergo a crossover between those of the random-matrix theory and the log-normal statistics. The correlation length exponent is extracted from the finite-size scaling of the mean level curvature.