Abstract

Multifractality of wave functions is a remarkable property of Anderson transition critical points in disordered systems. We develop a classification of gradientless composite operators (that includes the leading multifractal operators but is much broader) representing correlation functions of local densities of states (or wave function amplitudes) at Anderson transitions. Our classification is based on the Iwasawa decomposition for the underlying supersymmetric sigma-model field: the operators are represented by “plane waves” in terms of the corresponding “radial” coordinates. We present also an alternative (but equivalent) construction of scaling operators that uses the notion of highest-weight vectors. We further show that the invariance of the sigma-model manifold with respect to a Weyl group leads to numerous exact symmetry relations between the scaling dimensions of the composite operators.