L-9

Eigenstates

In absence of disorder the eigenstates are delocalized plane waves.

In presence of disorder, three situations can occur and to distinguish them it is useful to introduce the inverse participation ratio, IPR

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

The normalization imposes $\tau _{1}=0$. For $q=0$, $|\psi _{n}|^{2q}=1$, hence, $\tau _{0}=-d$.

• Delocalized eigenstates In this case, $|\psi _{n}|^{2}\approx L^{-d}$. Hence, we expect

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

• 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

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

• Multifractal eigenstates. 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.

Multifractality

The exponent $\tau _{q}$ is called multifractal exponent :

• $\tau _{1}=0$ imposed by normalization.
• $\tau _{0}=-d$ 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.

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

$|\psi _{n}|^{2}\approx L^{-\alpha }\quad {\text{for}}\;L^{f(\alpha )}\;{\text{sites}}$

The exponent $\alpha$ is positive and $f(\alpha )$ is called multifractal spectrum . It is a convex function and its maximum is the fractal dimension of the object, in our case d. We can determine the relation between multifractal spectrum and exponent

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

for large L

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

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

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

A metal has a simple spectrum. Indeed, all sites have $\alpha =d$, hence $f(\alpha =d)=d$ and $f(\alpha \neq d)=-\infty$. Then $\alpha ^{*}(q)=d$ becomes $q$ independent. A multifractal has a smooth spectrum with a maximum at $\alpha _{0}$ with $f(\alpha _{0})=d$. At $q=1$, $f'(\alpha _{1})=1$ and $f(\alpha _{1})=\alpha _{1}$.

Larkin model

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

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

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

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

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

You get

${\overline {(h(\ell _{L})-h(0))^{2}}}=r_{f}^{2}\quad \ell _{L}=\left({\frac {r_{f}}{\sigma }}\right)^{1/\zeta _{L}}$

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 $\ell _{L}=\left({\frac {r_{f}}{\sigma }}\right)^{2/3}$