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).

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

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

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:

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

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:

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

for large $L$. From this, we obtain:

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

This implies that for $\alpha ^{*}(q)$, which satisfies

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

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 $f(\alpha \neq d)=-\infty$. 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 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 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) = \alpha_1} .

Sometimes one writes:

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

Here 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 D_q} is q-dependent multifractal dimension, smaller than 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 d} and larger than zero.

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 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 \overline{F(r)}=0} , 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 \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 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^d}}
• at long times the interface shape 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 \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

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 \overline{(h(r)-h(0))^2}= \int _d^dq (\overline{h(q)h(-q)}(1-\cos(q r) \sim \sigma^2 r^{2 \zeta_L} }

You get

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 \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 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 d=1} we have 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 \ell_L=\left(\frac{r_f}{\sigma} \right)^{2/3}}