Multifractality and scaling in disordered mesoscopic systems

We suggest a new method for investigating scaling properties of mesoscopic observables and their distributions in disordered systems showing metal-insulator transition. In such systems quantum interference effects lead to multifractal structure of eigenstates on scales much smaller than the correlation length of the transition which can be described by a set of exponents, thef(α) spectrum. The analysis off(α) spectra can be extended to any scaling variable. Multifractality is an indication for broad distributions of these variables. If the transition is governed by one correlation length only then thef(α) spectra of normalized scaling variables must be universal. The critical exponentv of the correlation length is determined by the value α(0) wheref(α) takes its maximum and the scaling exponent of normalizationx∶v−1=α(0)+x. As an illustrative example we calculate numerically thef(α) spectra of eigenstates in the critical regime of 2d disordered electron systems in high magnetic fields. We find similarf(α) spectra indicating universal log-normal distributions of scaling variables.