Spectral Statistics at the Anderson Transition: Multifractality of Wave Functions and the Violation of the Normalization Sum Rule

The statistics of energy levels of electrons in a random potential is considered in the critical energy window near the mobility edge. It is shown that the multifractality of critical wave functions results in the violation of the normalization sum rule in the thermodynamic limit and leads to the quasi-Poisson linear term in the level number variance. The slope of the linear term is equal to the sum rule deficiency which is expressed in terms of the fractal dimension D(2).