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