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 over the whole system.
• Localized eigenstates, where the wavefunction is exponentially confined to a finite region.
• Multifractal eigenstates, occurring at the mobility edge of the Anderson transition, where the wavefunction exhibits a complex, scale–dependent structure.

In three dimensions, increasing the disorder strength leads to a transition between a metallic phase with delocalized eigenstates and an insulating phase with localized eigenstates. The critical energy separating these two regimes is called the mobility edge. Eigenstates at the mobility edge display multifractal statistics.

To characterize these different regimes it is useful to introduce the inverse participation ratio (IPR)

$$\mathrm{I}\mathrm{P}\mathrm{R}(q)=\sum _n|{\psi }_n{|}^{2q}.$$

For a system of linear size $L$, one typically observes a scaling

$$\mathrm{I}\mathrm{P}\mathrm{R}(q)\sim L^{-{\tau }_q}.$$

The exponent ${\tau }_q$ characterizes the spatial structure of the eigenstate.

Delocalized eigenstates

In a delocalized state the probability is spread uniformly over the system:

$$|{\psi }_n{|}^2\approx L^{-d}.$$

Therefore

$$\mathrm{I}\mathrm{P}\mathrm{R}(q)=\sum _n|{\psi }_n{|}^{2q}\sim L^d(L^{-d}{)}^q=L^{d(1-q)}.$$

Thus

$${\tau }_q=d(q-1).$$

Localized eigenstates

In a localized state the wavefunction is concentrated within a region of size ${\xi }_{\text{loc}}$.

Roughly

$$|{\psi }_n{|}^2\sim {\xi }_{\text{loc}}^{-d}$$

on ${\xi }_{\text{loc}}^d$ sites and negligible elsewhere. Hence

$$\mathrm{I}\mathrm{P}\mathrm{R}(q)\sim \text{const},{\tau }_q=0.$$

Thus localized states do not scale with system size.

Multifractal eigenstates

At the mobility edge the eigenstates are neither extended nor localized. Instead, the amplitudes fluctuate strongly across the system.

The scaling of the IPR is characterized by a nonlinear function

$${\tau }_q.$$

The corresponding multifractal dimensions are defined as

$$D_q=\frac{{\tau }_q}{q-1}.$$

It is useful to contrast fractal and multifractal scaling.

• In a fractal object, a single exponent describes the scaling of the measure. If the wavefunction amplitudes scale with a single exponent $D$, then

$$\mathrm{I}\mathrm{P}\mathrm{R}(q)\sim L^{-D(q-1)}.$$

All moments are controlled by the same dimension $D$.

• In a multifractal object, different regions of the system scale with different exponents. In this case there is no single fractal dimension: different moments probe different effective dimensions of the wavefunction.

Multifractal spectrum

To describe this structure it is useful to introduce the multifractal spectrum $f(\alpha )$.

We assume that the wavefunction amplitudes scale as

$$|{\psi }_n{|}^2\sim L^{-\alpha }$$

on approximately

$$L^{f(\alpha )}$$

sites.

The function $f(\alpha )$ therefore describes the fractal dimension of the set of sites where the wavefunction has exponent $\alpha$.

Using this representation,

$$\mathrm{I}\mathrm{P}\mathrm{R}(q)=\sum _n|{\psi }_n{|}^{2q}\sim \int d\alpha L^{-\alpha q}{L}^{f(\alpha )}.$$

For large $L$, the integral is dominated by the saddle point, giving

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

Thus $\tau (q)$ and $f(\alpha )$ are related by a Legendre transform.

If ${\alpha }^{*}(q)$ satisfies

$$f^{\prime }({\alpha }^{*}(q))=q,$$

then

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

Multifractal wavefunctions exhibit a smooth spectrum with maximum

$$f({\alpha }_0)=d,$$

which corresponds to the most typical amplitude of the wavefunction.

It is useful to contrast this situation with the case of a simple fractal.

In a fractal object, the measure has a single scaling exponent. In our notation this means that the wavefunction amplitudes scale with a single value ${\alpha }_0$ on a set of fractal dimension $D$:

$$|{\psi }_n{|}^2\sim L^{-{\alpha }_0}$$

on approximately

$$L^D$$

sites, and are essentially negligible elsewhere.

In terms of the multifractal spectrum this corresponds to a trivial spectrum

$$f({\alpha }_0)=D,f(\alpha \ne {\alpha }_0)=-\mathrm{\infty }.$$

By contrast, in a multifractal state the amplitudes are broadly distributed and many values of $\alpha$ contribute. The function $f(\alpha )$ then becomes a smooth curve describing the distribution of scaling exponents of the wavefunction.

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 valid only up to a scale $r_f$, the correlation length of the disorder along the $h$ direction. This defines the Larkin length.

Indeed, starting from

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

we obtain

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

Above this scale the roughness changes and pinning sets in with a critical force

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

In $d=1$ we therefore obtain

$${\ell }_L={\left(\frac{r_f}{\sigma }\right)}^{2/3}.$$