Classification and symmetry properties of scaling dimensions at Anderson transitions

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We develop a classification of composite operators without gradients at Anderson-transition critical points in disordered systems. These operators represent correlation functions of the local density of states (or of wave-function amplitudes). Our classification is motivated by the Iwasawa decomposition for the field of the pertinent supersymmetric σ-model: the scaling operators are represented by "plane waves" in terms of the corresponding radial coordinates. We also present an alternative construction of scaling operators by using the notion of highest-weight vector. We further argue that a certain Weyl-group invariance associated with the σ-model manifold leads to numerous exact symmetry relations between the scaling dimensions of the composite operators. These symmetry relations generalize those derived earlier for the multifractal spectrum of the leading operators.