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:

$IPR(q)=L^{D_{q}(1-q)}\quad \tau _{q}=D_{q}(1-q)$

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