LBan-II

Introduction: Interfaces and Directed Polymers

The physical properties of many materials are governed by manifolds embedded in them. Examples include: dislocations in crystals, domain walls in ferromagnets or vortex lines in superconductors. We fix the following notation: - $D$: spatial dimension of the embedding medium – $d$: internal dimension of the manifold – $N$: dimension of the displacement (or height) field

These satisfy the relation:

$D=d+N$

We focus on two important cases:

Directed Polymers (d = 1)

The configuration is described by a vector function: $\overrightarrow{x}(t)$, where $t$ is the internal coordinate. The polymer lives in $D=1+N$ dimensions.

Examples: vortex lines, DNA strands, fronts.

Although polymers may form loops, we restrict to directed polymers, i.e., configurations without overhangs or backward turns.

Interfaces (N = 1)

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

Examples: domain walls and propagating fronts

Again we neglect overhangs or pinch-off: $h(\overrightarrow{r},t)$ is single-valued

Note that using our notation the 1D front is both an interface and a directed polymer

Thermal Interfaces

• The dynamics is overdamped, so that we can neglect the inertial term.

The Langevin equation of motion is

${\partial }_{t}h(r,t)=-\mu \frac{\delta E_{pot}}{\delta h(r,t)}+\eta (r,t)$

The first term $-\delta E_{pot}/\delta h(r,t)$ is the elastic force trying to smooth the interface, the mobility $\mu$ is the inverse of the viscosity. The second term is the Langevin noise. It is Guassian and defined by

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

The symbol $⟨\dots ⟩$ indicates the average over the thermal noise and the diffusion constant is fixed by the Einstein relation $D=\mu K_{B}T$. We set $\mu =K_B=1$

The potential energy of surface tension ($\nu$ is the stiffness) can be expanded at the lowest order in the gradient:

$E_{pot}=\nu \int d^{d}r\sqrt{1+(\mathrm{\nabla }h{)}^2}\sim \text{const.}+\frac{\nu }{2}\int d^{d}r(\mathrm{\nabla }h{)}^2$

Hence, we have the Edwards Wilkinson equation:

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

Scaling Invariance

The equation enjoys of a continuous symmetry because $h(r,t)$ and $h(r,t)+c$ cannot be distinguished. This is a condition of scale invariance:

$h(br,b^{z}t)\stackrel{inlaw}{\sim }{b}^{\alpha }h(r,t)$

Here $z,\alpha$ are the dynamic and the roughness exponent respectively. From dimensional analysis

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

From which you get $z=2$ in any dimension and a rough interface below $d=2$ with $\alpha =(2-d)/2$.

Width of the interface

Consider a 1-dimensional line of size L with periodic boundary conditions. We consider the width square of the interface

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

It is useful to introduce the Fourier modes:

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

Here $q=2\pi n/L,n=\dots ,-1,0,1,\dots$ and recall $\underset{0}{\overset{L}{\int }}dr{e}^{iqr}=L{\delta }_{q,0}$.

Solution in the Fourier space

show that the EW equation writes

${\partial }_t{\hat{h}}_q(t)=-\nu q^2{\hat{h}}_q(t)+{\eta }_q(t),\text{with}⟨{\eta }_{q_1}(t^{\prime }){\eta }_{q_2}(t)⟩=\frac{2T}{L}{\delta }_{q_1,-q_2}\delta (t-t^{\prime })$

The solution of this first order linear equation writes

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

Flat initial condition

Assume that the interface is initially flat, namely ${\hat{h}}_q(0)=0$. Show that

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

using de Parseval theorem for the Fourier series

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

• the width square is a random variable and we can compute its mean:

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

and the displacement of a point of the interface that we tagg

 $⟨h(r,t{)}^2⟩=\sum _q⟨{\hat{h}}_q(t){\hat{h}}_{-q}(t)⟩$. Comment about the roughness and the short times growth.