L-9

From Disordered Systems Wiki
Jump to navigation Jump to search

Eigenstates

Without disorder, the eigenstates are delocalized plane waves.

In the presence of disorder, three scenarios can arise: Delocalized eigenstates, where the wavefunction remains extended; Localized eigenstates, where the wavefunction is exponentially confined to a finite region; Multifractal eigenstates, occurring at the mobility edge, where the wavefunction exhibits a complex, scale-dependent structure.

To distinguish these regimes, it is useful to introduce the inverse participation ratio (IPR).

Delocalized eigenstates

In this case, . Hence, we expect

Localized eigenstates

In this case, for sites and almost zero elsewhere. Hence, we expect


Multifractal eigenstates

The exponent is called the multifractal exponent. It is a non-decreasing function of with some special points:

  • , since the wavefunction is defined on all sites. In general, represents the fractal dimension of the object under consideration and is purely a geometric property.
  • , imposed by normalization.

To observe multifractal behavior, we expect:

The exponent is positive, and is called the multifractal spectrum. Its maximum corresponds to the fractal dimension of the object, which in our case is . The relation between the multifractal spectrum and the exponent is given by:

for large . From this, we obtain:

This implies that for , which satisfies

we have

Delocalized wavefunctions have a simple spectrum: for , we find and . This means that is independent of .

Multifractal wavefunctions exhibit a smoother dependence, leading to a continuous spectrum with a maximum at , where . At , we have and .

Larkin model

In your homewoork you solved a toy model for the interface:

For simplicity, we assume Gaussian disorder , .

You proved that:

  • the roughness exponent of this model is below dimension 4
  • The force per unit length acting on the center of the interface is
  • at long times the interface shape is

In the real depinning model the disorder is however a non-linear function of h. The idea of Larkin is that this linearization is correct up, the length of correlation of the disorder along the h direction . This defines a Larkin length. Indeed from

You get

Above this scale, roguhness change and pinning starts with a crtical force

In we have