L2 ICFP

Interfaces and manifolds

Many physical systems are governed by elastic manifolds embedded in a higher-dimensional medium. Typical examples include domain walls in ferromagnets, dislocations in crystals, vortex lines in superconductors, and propagating fronts.

We introduce the following notation:

• $d$: internal dimension of the manifold
• $N$: dimension of the displacement (or height) field
• $D$: dimension of the embedding space

These satisfy

$D=d+N$

Two important cases are:

• Interfaces ($N=1$):

The configuration is described by a scalar height field $h(\overrightarrow{r},t)$, where $\overrightarrow{r}\in {\mathbb{ℝ}}^d$ is the internal coordinate.

• Directed polymers ($d=1$):

The configuration is described by a vector function $\overrightarrow{x}(t)$ embedded in $D=1+N$ dimensions.

Remark. With this notation, a one-dimensional interface ($d=1$, $N=1$) can be viewed both as an interface and as a directed polymer.

In this lecture we focus on thermal interfaces.

Thermal interfaces: Langevin dynamics

We consider an interface at thermal equilibrium at temperature $T$. Two assumptions are made:

• Overhangs and pinch-off are neglected, so $h(\overrightarrow{r},t)$ is single-valued.
• The dynamics is overdamped; inertial effects are neglected.

The Langevin equation of motion reads

${\partial }_{t}h(\overrightarrow{r},t)=-\mu \frac{\delta E_{\mathrm{p}\mathrm{o}\mathrm{t}}}{\delta h(\overrightarrow{r},t)}+\eta (\overrightarrow{r},t).$

Here $\mu$ is the mobility and $\eta (\overrightarrow{r},t)$ is a Gaussian thermal noise, with

$⟨\eta (\overrightarrow{r},t)⟩=0,⟨\eta (\overrightarrow{r},t)\eta ({\overrightarrow{r}}^{\prime },t^{\prime })⟩=2dD{\delta }^d(\overrightarrow{r}-{\overrightarrow{r}}^{\prime })\delta (t-t^{\prime }).$

The diffusion constant is fixed by the Einstein relation

$D=\mu k_{B}T.$

In the following we set $\mu =k_B=1$.

Elastic energy and Edwards–Wilkinson equation

The elastic energy associated with surface tension can be written as

$E_{\mathrm{p}\mathrm{o}\mathrm{t}}=\nu \int d^{d}r\sqrt{1+(\mathrm{\nabla }h{)}^2}\simeq \text{const.}+\frac{\nu }{2}\int d^{d}r(\mathrm{\nabla }h{)}^2,$

where $\nu$ is the stiffness.

Keeping only the lowest-order term in gradients, the equation of motion becomes the Edwards–Wilkinson (EW) equation:

${\partial }_{t}h(\overrightarrow{r},t)=\nu {\mathrm{\nabla }}^{2}h(\overrightarrow{r},t)+\eta (\overrightarrow{r},t).$

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Symmetries and scaling invariance

The EW equation is invariant under global height shifts $h(\overrightarrow{r},t)\to h(\overrightarrow{r},t)+c$. This symmetry leads to scale invariance of the form

$h(b\overrightarrow{r},b^{z}t)\stackrel{\text{in law}}{\sim }{b}^{\alpha }h(\overrightarrow{r},t),$

where $z$ is the dynamical exponent and $\alpha$ the roughness exponent.

A simple dimensional analysis gives

$b^{\alpha -z}{\partial }_{t}h=b^{\alpha -2}{\mathrm{\nabla }}^{2}h+b^{-d/2-z/2}\eta .$

From this one finds

$z=2,\alpha =\frac{2-d}{2}.$

Thus the interface is rough for $d<2$ and marginal at $d=2$.

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Width of the interface

We now focus on a one-dimensional interface ($d=1$) of size $L$ with periodic boundary conditions.

The squared width is defined as

$w_2(t)=\underset{0}{\overset{L}{\int }}\frac{dr}{L}{(h(r,t)-\underset{0}{\overset{L}{\int }}\frac{dr}{L}h(r,t))}^2.$

Introduce Fourier modes

${\hat{h}}_q(t)=\frac{1}{L}\underset{0}{\overset{L}{\int }}dr{e}^{iqr}h(r,t),h(r,t)=\sum _q{e}^{-iqr}{\hat{h}}_q(t),$

with $q=2\pi n/L$.

Using Parseval’s theorem one finds

$w_2(t)=\sum _{q\ne 0}|{\hat{h}}_q(t){|}^2.$

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Solution in Fourier space

The EW equation in Fourier space reads

${\partial }_t{\hat{h}}_q(t)=-\nu q^2{\hat{h}}_q(t)+{\eta }_q(t),$

with noise correlations

$⟨{\eta }_{q_1}(t^{\prime }){\eta }_{q_2}(t)⟩=\frac{2T}{L}{\delta }_{q_1,-q_2}\delta (t-t^{\prime }).$

The solution is

${\hat{h}}_q(t)={\hat{h}}_q(0)e^{-\nu q^{2}t}+\underset{0}{\overset{t}{\int }}ds{e}^{-\nu q^2(t-s)}{\eta }_q(s).$

Assuming a flat initial condition ${\hat{h}}_q(0)=0$, one finds

$⟨{\hat{h}}_q(t){\hat{h}}_{-q}(t)⟩=\{\begin{array}{ll}\frac{T(1-e^{-2\nu q^{2}t})}{L\nu q^2},& q\ne 0,\\ \frac{2T}{L}t,& q=0.\end{array}$

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Growth and saturation of the width

The mean squared width evolves as

$⟨w_2(t)⟩=\frac{T}{L\nu }\sum _{q\ne 0}\frac{1-e^{-2\nu q^{2}t}}{q^2}.$

In the continuum limit this gives

$⟨w_2(t)⟩=\{\begin{array}{ll}T\sqrt{\frac{2t}{\pi \nu }},& t\ll L^2,\\ \frac{T}{\nu }\frac{L}{12},& t\gg L^2.\end{array}$

At short times the interface roughens algebraically, while at long times the width saturates due to the finite system size.