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

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. For weak diagonal disorder the analysis of the transition region between these two power-law limits provides with an accurate estimation of the Thouless energy of the system. As disorder increases the Thouless energy looses its meaning and the power spectrum presents a 1 /f 2 decay up to frequencies related to the Heisenberg time of the system. Finally we discuss under what circumstances these findings may be relevant in the context of non-random Hamiltonians.