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). $$\mathrm{I}\mathrm{P}\mathrm{R}(q)=\sum _n|{\psi }_n{|}^{2q}\sim L^{-{\tau }_q}.$$

Delocalized eigenstates

In this case, $|{\psi }_n{|}^2\approx L^{-d}$. Hence, we expect $$\mathrm{I}\mathrm{P}\mathrm{R}(q)=L^{d(1-q)},{\tau }_q=d(1-q).$$

Localized eigenstates

In this case, $|{\psi }_n{|}^2\approx 1/{\xi }_{\text{loc}}^d$ on ${\xi }_{\text{loc}}^d$ sites and almost zero elsewhere. Hence, we expect $$\mathrm{I}\mathrm{P}\mathrm{R}(q)=\text{const},{\tau }_q=0.$$

Multifractal eigenstates

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

• ${\tau }_0=-d$, since the wavefunction is defined on all sites. In general, ${\tau }_0$ represents the fractal dimension of the object under consideration and is purely a geometric property.
• ${\tau }_1=0$, imposed by normalization.

To observe multifractal behavior, 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 the multifractal spectrum. Its maximum corresponds to the fractal dimension of the object, which in our case is $d$.

The relation between the multifractal spectrum $f(\alpha )$ and the exponent ${\tau }_q$ is given by: $$\mathrm{I}\mathrm{P}\mathrm{R}(q)=\sum _n|{\psi }_n{|}^{2q}\sim \int d\alpha L^{-\alpha q}{L}^{f(\alpha )}$$ for large $L$. From this, we obtain: $$\tau (q)=\underset{\alpha }{\min }(\alpha q-f(\alpha )).$$

This implies that for ${\alpha }^{*}(q)$, which satisfies $$f^{\prime }({\alpha }^{*}(q))=q,$$ we have $$\tau (q)={\alpha }^{*}(q)q-f({\alpha }^{*}(q)).$$

Delocalized wavefunctions have a simple spectrum: for $\alpha =d$, we find $f(\alpha =d)=d$ and $f(\alpha \ne d)=-\mathrm{\infty }$. This means that ${\alpha }^{*}(q)=d$ is independent of $q$.

Multifractal wavefunctions exhibit a smoother dependence, leading to a continuous spectrum with a maximum at ${\alpha }_0$, where $f({\alpha }_0)=d$. At $q=1$, we have $f^{\prime }({\alpha }_1)=1$ and $f({\alpha }_1)={\alpha }_1$.

Larkin model

In your homework 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 to $r_f$, the correlation length 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 $$\overline{(h({\ell }_L)-h(0){)}^2}=r_f^2,{\ell }_L={\left(\frac{r_f}{\sigma }\right)}^{1/{\zeta }_L}.$$

Above this scale, roughness changes and pinning starts with a critical force $$f_c=\frac{\sigma }{{\ell }_L^{d/(2{\zeta }_L)}}.$$

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