L-9

Multifractality

In the last lecture we discussed that the eigenstates of the Anderson model can be localized, delocalized or multifractal. The idea is to look at the (generalized) IPR

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

The exponent ${\tau }_q$ is called multifractal exponent . Normalization imposes ${\tau }_1=0$ and the fact that the wave fuction is defined everywhere that ${\tau }_0=-d$. In general ${\tau }_0$ is the fractal dimension of the object we are considering and 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

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

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

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

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

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

A metal has a simple spectrum. Indeed, all sites have $\alpha =d$, hence $f(\alpha =d)=d$ and $f(\alpha \ne d)=-\mathrm{\infty }$. 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 $q=1$, $f^{\prime }({\alpha }_1)=1$ and $f({\alpha }_1)={\alpha }_1$.

Larkin model

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

Failed to parse (unknown function "\nable"): {\displaystyle \partial_t h(r,t) = \nable^2 r(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 ${\ell }_L={\left(\frac{r_f}{\sigma }\right)}^{2/3}$