Power spectrum for critical statistics: A novel spectral characterization of the Anderson transition

Abstract

We examine the power spectrum of the energy level fluctuations of a family of critical power-law random banded matrices with properties similar to those of a disordered conductor at the Anderson transition. It is shown both analytically and numerically that the Anderson transition is characterized by a power spectrum which presents $1/f^2$ noise for small frequencies but $1/f$ noise for larger frequencies. The analysis of the transition region between these two power-law limits gives an accurate estimation of the Thouless energy of the system. Finally we discuss under what circumstances these findings may be relevant in the context of non-random Hamiltonians.