Level curvature distribution and the structure of eigenfunctions in disordered systems

The level curvature distribution function is studied both analytically and numerically for the case of T-breaking perturbations over the orthogonal ensemble. The leading correction to the shape of the curvature distribution beyond the random matrix theory is calculated using the nonlinear supersymmetric sigma model and compared to numerical simulations on the Anderson model. It is predicted analytically and confirmed numerically that the sign of the correction is different for T-breaking perturbations caused by a constant vector-potential equivalent to a phase twist in the boundary conditions, and those caused by a random magnetic field. In the former case it is shown using a nonperturbative approach that quasilocalized states in weakly disordered systems can cause the curvature distribution to be nonanalytic. In two-dimensional (2D) systems the distribution function P(K) has a branching point at K = 0 that is related to the multifractality of the wave functions and thus should be a generic feature of all critical eigenstates. A relationship between the branching power and the multifractality exponent d(2) is suggested. Evidence of the branch-cut singularity is found in numerical simulations in 2D systems and at the Anderson transition point in 3D systems.