Exact Relations between Multifractal Exponents at the Anderson Transition

Two exact relations between mutlifractal exponents are shown to hold at the critical point of the Anderson localization transition. The first relation implies a symmetry of the multifractal spectrum linking the exponents with indices $q<1/2$ to those with $q>1/2$. The second relation connects the wave-function multifractality to that of Wigner delay times in a system with a lead attached.

Figure 1

Multifractal exponents ${\Delta }_q$ for the PRBM model with $b=4$, 1, 0.3, 0.1. The symmetry (2) with respect to the point $q=1/2$ is evident. A small difference between ${\Delta }_q$ (full line) and ${\Delta }_{1-q}$ (dashed line) is due to numerical errors.

Figure 2

The data of Fig. 1 in terms of the singularity spectrum $f(\alpha )$. Dashed lines represent $f(2-\alpha )+\alpha -1$, demonstrating the validity of Eq. (9).

Figure 3

Singularity spectrum for the PRBM model with $b=0.1$. Dashed line: $\alpha <1$ behavior, Eq. (12); full line: $\alpha >1$ result, Eq. (14), following from the relation (9); circles: numerical data. Some mismatch between the slopes of the two curves at $\alpha =1$ is related to the fact that the formula (12) is valid to the leading order in $b\ll 1$.