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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle IPR(q)=\sum_n |\psi_n|^{2 q} \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

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

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q=1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(\alpha_1)=1} and $f(\alpha _{1})=\alpha _{1}$.

Larkin model

In your homewoork you solved a toy model for the interface. Consider a collection of L monomers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_1,h_2,\ldots, h_L } in 1d with periodic boundary condition:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial_t h_i(t) = h_{i+1}(t)+h_{i-1}(t) -2 h_i(t) + F_i }

For simplicity, we assume $F_i$ iid Gaussian numbers with zero mean a variance D: ${\overline {F_{i}}}=0,\quad {\overline {F_{i}^{2}}}=\sigma ^{2}$. You proved that the roughness exponent of this model is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta_L=(4-d)/2=3/2} and the force per unit length acting of the interface is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f= \sigma/\sqrt{L}}

In the real model for depinning the disorder is however a non-linear function of h. The idea of Larkin is that linearization is correct only up, $r_{f}$ the length of correlation of the disorder along the h direction .