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

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 forbids any local term depending on the absolute height and ensures the absence of a characteristic length scale, leading 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 $d=2$.

Solution in Fourier space

We now focus on a one-dimensional interface ($d=1$) of size $L$ with periodic boundary conditions. We use the Fourier decomposition $${\hat{h}}_q(t)=\frac{1}{L}\underset{0}{\overset{L}{\int }}dx{e}^{iqx}h(x,t),h(x,t)=\sum _q{e}^{-iqx}{\hat{h}}_q(t),$$ with wavevectors $$q=\frac{2\pi n}{L},n=\dots ,-1,0,1,\dots$$

For these discrete wavevectors, the Fourier modes satisfy the orthogonality relation $$\underset{0}{\overset{L}{\int }}dx{e}^{i(q_1+q_2)x}=L{\delta }_{q_1,-q_2}.$$

Assuming a spatially and temporally white noise, $⟨\eta (x,t)\eta (x^{\prime },t^{\prime })⟩=2T\delta (x-x^{\prime })\delta (t-t^{\prime })$, one finds that the Fourier components of the noise satisfy $$⟨{\eta }_{q_1}(t^{\prime }){\eta }_{q_2}(t)⟩=\frac{2T}{L}{\delta }_{q_1,-q_2}\delta (t-t^{\prime }).$$

With these definitions, the Edwards–Wilkinson equation becomes diagonal in Fourier space: $${\partial }_t{\hat{h}}_q(t)=-\nu q^2{\hat{h}}_q(t)+{\eta }_q(t).$$

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

The mode $q=0$ corresponds to the spatial average of the height, i.e.\ to the center-of-mass position of the interface. Its fluctuations grow diffusively, $$⟨{\hat{h}}_0(t{)}^2⟩=\frac{2T}{L}t,$$ with a diffusion constant proportional to $1/L$, reflecting the fact that the interface is composed of $L$ degrees of freedom.

The modes with $q\ne 0$ describe internal fluctuations of the interface. The relaxation time of a mode of wavevector $q$ scales as $${\tau }_q\sim \frac{1}{\nu q^2}.$$

Since $q$ has the dimension of an inverse length, this relaxation time suggests the existence of a growing dynamical length scale $$\ell (t)\sim t^{1/z},z=2,$$ such that modes with wavelength smaller than $\ell (t)$ (i.e. $q\gg 1/\ell (t)$) have already equilibrated, while at longer wavelengths the interface still retains memory of the initial flat condition. Form dimensional analysis, the equilibrium decay $\sim 1/(L{q}^2)$ is consistent with the roughness exponent $\alpha =1/2$, as expected for the Edwards–Wilkinson universality class in one dimension.

Width of the interface

The squared width of the interface is defined as $$w_2(t)=\underset{0}{\overset{L}{\int }}\frac{dr}{L}{[h(r,t)-\underset{0}{\overset{L}{\int }}\frac{dr}{L}h(r,t)]}^2.$$

Using the Fourier decomposition and Parseval’s theorem, one finds $$w_2(t)=\sum _{q\ne 0}|{\hat{h}}_q(t){|}^2.$$

Taking the average over the thermal noise yields $$⟨w_2(t)⟩=\frac{T}{L\nu }\sum _{q\ne 0}\frac{1-e^{-2\nu q^{2}t}}{q^2}.$$

For periodic boundary conditions, with $q=2\pi n/L$, this can be rewritten as $$⟨w_2(t)⟩=\frac{TL}{2{\pi }^2\nu }\sum _{n=1}^{\mathrm{\infty }}\frac{1-e^{-8{\pi }^2\nu t{n}^2/L^2}}{n^2}.$$

Long-time behavior

At long times, $t\gg L^2$, all modes have relaxed and the exponential term can be neglected. One obtains $$⟨w_2(t)⟩\sim \frac{TL}{2{\pi }^2\nu }\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}=\frac{T}{\nu }\frac{L}{12}.$$

Thus the width saturates at a value proportional to the system size.

Short-time behavior

At short times, $t\ll L^2$, the sum can be approximated by an integral. Replacing $\sum _n\to \frac{L}{2\pi }\int dq$, one finds $$⟨w_2(t)⟩\simeq \frac{T}{\nu }\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{dq}{2\pi }\frac{1-e^{-2\nu q^{2}t}}{q^2}.$$

Evaluating the integral gives $$⟨w_2(t)⟩\sim T\sqrt{\frac{2t}{\pi \nu }},t\ll L^2.$$