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The level curvature distribution function is studied both analytically andnumerically for the case of T-breaking perturbations over the orthogonalensemble. The leading correction to the shape of the curvature distributionbeyond the random matrix theory is calculated using the nonlinearsupersymmetric sigma-model and compared to numerical simulations on theAnderson model. It is predicted analytically and confirmed numerically that thesign of the correction is different for T-breaking perturbations caused by aconstant vector-potential equivalent to a phase twist in the boundaryconditions, and those caused by a random magnetic field. In the former case itis shown using a nonperturbative approach that quasi-localized states in weaklydisordered systems can cause the curvature distribution to be nonanalytic. In$2d$ systems the distribution function $P(K)$ has a branching point at K=0 thatis related to the multifractality of the wave functions and thus should be ageneric feature of all critical eigenstates. A relationship between thebranching power and the multifractality exponent $d_{2}$ is suggested. Evidenceof the branch-cut singularity is found in numerical simulations in $2d$ systemsand at the Anderson transition point in $3d$ systems.Comment: 34 pages (RevTeX), 8 figures (postscript

Level curvature distribution and the structure of eigenfunctions in disordered systems