Spectral Rigidity and Eigenfunction Correlations at the Anderson Transition

TL;DR

This paper investigates the spectral properties of disordered conductors near the Anderson transition, revealing a direct link between multifractal wave functions and level statistics, with implications for understanding critical phenomena.

Contribution

It establishes an exact relation between spectral compressibility and multifractal exponents at the Anderson transition.

Findings

Level variance scales linearly with mean level number

Spectral compressibility is given by /2d times the multifractal exponent 

Critical wave functions exhibit multifractality affecting level correlations

Abstract

The statistics of energy levels for a disordered conductor are considered in the critical energy window near the mobility edge. It is shown that, if critical wave functions are multifractal, the one-dimensional gas of levels on the energy axis is ``compressible'', in the sense that the variance of the level number in an interval is $< (\delta N)^{2} > = \chi <N>$ for $<N> >> 1$. The compressibility, $\chi=\eta/2d$, is given ``exactly'' in terms of the multifractal exponent $\eta=d-D_2$ at the mobility edge in a $d$-dimensional system.