L-9

Multifractality

In the last lecture we discussed that the eigenstates of the Anderson model can be localized, delocalized or multifractal. The idea is to look at the (generalized) IPR

$IPR(q)=\sum _n|{\psi }_n{|}^{2q}\sim L^{-{\tau }_q}$

The exponent ${\tau }_q$ is called multifractal exponent . Normalization imposes ${\tau }_1=0$ and the fact that the wave fuction is defined everywhere that ${\tau }_0=-d$. In general ${\tau }_0$ is the fractal dimension of the object we are considering and it is simply a geometrical property.

• Delocalized eigenstates

In this case, $|{\psi }_n{|}^2\approx L^{-d}$ for all the $L^d$ sites. This gives

${\tau }_q^{\text{deloc}}=d(q-1)$

• Multifractal eigenstates.

This case correspond to more complex wave function for which

we expect

At the transition( the mobility edge) an anomalous scaling is observed:

$IPR(q)=L^{D_q(1-q)}{\tau }_q=D_q(1-q)$

Here $D_q$ is q-dependent multifractal dimension, smaller than $d$ and larger than zero.