7. Multifractality of wavefunctions at the Anderson transition

Taken by Piotr Sierant

The Inverse Participation Ratio (IPR) defined as int{|psi(r)|^4 d^dr} (integral taken over the full d-dimensional space) can be used to measure localization properties of wavefunctions. If the wavefunction is statistically uniformly distributed in a system of size L (along each of the d dimensions), the IPR should scale like 1/L^d. If it is localized with localization length xi, it should scale like 1/xi^d. It turns out that the IPR of actual eigenstates of disordered systems behave differently. This is related to the fractality, or multifractality, of the wavefunctions. This paper explores this phenomenon using the 3d Anderson model in the vicinity of the localized/delocalized transition.

The concepts linked to multifractality are far from obvious, but fruitful and at the heart of current research. You may not understand eveything in the paper, but you will learn useful things. Don't be afraid!