L-9

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).

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

Delocalized eigenstates

In this case, $|{\psi }_n{|}^2\approx L^{-d}$. Hence, we expect

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

Localized eigenstates

In this case, $|{\psi }_n{|}^2\approx 1/{\xi }_{\text{loc}}^{1/d}$ for ${\xi }_{\text{loc}}^d$ sites and almost zero elsewhere. Hence, we expect

$IPR(q)=\text{const},{\tau }_q=0$

Multifractal eigenstates

The exponent ${\tau }_q$ is called multifractal exponent . It is a non decreasing function with q with some special points:

• ${\tau }_0=-d$ because the wave fuction is defined on all sites, in general ${\tau }_0$ is the fractal dimension of the object we are considering. It is simply a geometrical property.
• ${\tau }_1=0$ imposed by normalization.

To have multifractal behaviour we expect

$|{\psi }_n{|}^2\approx L^{-\alpha }\text{for}{L}^{f(\alpha )}\text{sites}$

The exponent $\alpha$ is positive and $f(\alpha )$ is called multifractal spectrum . Its maximum is the fractal dimension of the object, in our case d. We can determine the relation between multifractal spectrum $f(\alpha )$ and exponent ${\tau }_q$

$IPR(q)=\sum _n|{\psi }_n{|}^{2q}\sim \int d\alpha L^{-\alpha q}{L}^{f(\alpha )}$

for large L

$\tau (q)=\underset{\alpha }{\min }(\alpha q-f(\alpha ))$

This means that for ${\alpha }^{*}(q)$ that verifies $f^{\prime }({\alpha }^{*}(q))=q$ we have

$\tau (q)={\alpha }^{*}(q)q-f({\alpha }^{*}(q))$

Delocalized wave functions have a simple spectrum: For $\alpha =d$, we have $f(\alpha =d)=d$ and $f(\alpha \ne d)=-\mathrm{\infty }$. Then ${\alpha }^{*}(q)=d$ becomes $q$ independent. Multifractal wave functions smooth this edge dependence and display a smooth spectrum with a maximum at ${\alpha }_0$ with $f({\alpha }_0)=d$. At $q=1$, $f^{\prime }({\alpha }_1)=1$ and $f({\alpha }_1)={\alpha }_1$.

Sometimes one writes:

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

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

Larkin model

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

${\partial }_{t}h(r,t)={\mathrm{\nabla }}^{2}h(r,t)+F(r)$

For simplicity, we assume Gaussian disorder $\overline{F(r)}=0$, $\overline{F(r)F(r^{\prime })}={\sigma }^2{\delta }^d(r-r^{\prime })$.

You proved that:

• the roughness exponent of this model is ${\zeta }_L=\frac{4-d}{2}$ below dimension 4
• The force per unit length acting on the center of the interface is $f=\sigma /\sqrt{L^d}$
• at long times the interface shape is

$\overline{h(q)h(-q)}=\frac{{\sigma }^2}{q^{d+2{\zeta }_L}}$

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, $r_f$ the length of correlation of the disorder along the h direction . This defines a Larkin length. Indeed from

$\overline{(h(r)-h(0){)}^2}=\underset{d}{\overset{d}{\int }}q(\overline{h(q)h(-q)}(1-\cos (qr)\sim {\sigma }^2{r}^{2{\zeta }_L}$

You get

$\overline{(h({\ell }_L)-h(0){)}^2}=r_f^2{\ell }_L={\left(\frac{r_f}{\sigma }\right)}^{1/{\zeta }_L}$

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

$f_c=\frac{\sigma }{{\ell }_L^{d/(2{\zeta }_L)}}$

In $d=1$ we have ${\ell }_L={\left(\frac{r_f}{\sigma }\right)}^{2/3}$