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 $\underset{0}{\overset{L}{\int }}dr{e}^{iqr}=L{\delta }_{q,0}$. using de Parseval theorem for the Fourier series

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

In the last step we used that ${\hat{h}}_q^{*}(t)={\hat{h}}_{-q}(t)$.

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

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

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

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

• The mean width square grows at short times and saturates at long times:

$⟨w_2(t)⟩=\frac{T}{L\nu }\sum _{q\ne 0}\frac{1-e^{-2\nu q^{2}t}}{q^2}=\{\begin{array}{ll}T\sqrt{\frac{2t}{\pi \nu }},& t\ll L^2,\\ \frac{T}{\nu }\frac{L}{12},& t\gg L^2.\end{array}$

Directed polymers in random media

Let us consider polymers $x(\tau )$ of length $t$, starting in $0$ and ending in $x$ and at thermal equlibrium at temperature $T$. The partition function of the model writes as

$Z(x,t)=\underset{x(0)=0}{\overset{x(t)=x}{\int }}{𝒟}x(\tau )\exp [-\frac{1}{T}\underset{0}{\overset{t}{\int }}d\tau \frac{1}{2}({\partial }_{\tau }x{)}^2+V(x(\tau ),\tau )]$

For simplicity, we assume a white noise, $\overline{V(x,\tau )V(x^{\prime },{\tau }^{\prime })}=D\delta (x-x^{\prime })\delta (\tau -{\tau }^{\prime })$. Here, the partition function is written as a sum over all possible paths, corresponding to all possible polymer configurations that start at $0$ and end at $x$, weighted by the appropriate Boltzmann factor.

Polymer partition function and propagator of a quantum particle

Let's perform the following change of variables: $\tau =i{t}^{\prime }$. We also identifies $T$ with $\hslash$ and $\tilde{t}=-it$ as the time.

$Z(x,\tilde{t})=\underset{x(0)=0}{\overset{x(\tilde{t})=x}{\int }}{𝒟}x(t^{\prime })\exp [\frac{i}{\hslash }\underset{0}{\overset{\tilde{t}}{\int }}d{t}^{\prime }\frac{1}{2}({\partial }_{t^{\prime }}x{)}^2-V(x(t^{\prime }),t^{\prime })]$

Note that $S[x]=\underset{0}{\overset{\tilde{t}}{\int }}d{t}^{\prime }\frac{1}{2}({\partial }_{t^{\prime }}x{)}^2-V(x(t^{\prime }),t^{\prime })$ is the classical action of a particle with kinetic energy $\frac{1}{2}({\partial }_{\tau }x{)}^2$ and time dependent potential $V(x(\tau ),\tau )$, evolving from time zero to time $\tilde{t}$. From the Feymann path integral formulation, $Z[x,\tilde{t}]$ is the propagator of the quantum particle.

In absence of disorder, one can find the propagator of the free particle, that, in the original variables, writes:

$Z_{\text{free}}(x,t)=\frac{e^{-x^2/(2Tt)}}{\sqrt{2\pi Tt}}$

Feynman-Kac formula

Let's derive the Feyman Kac formula for $Z(x,t)$ in the general case:

• First, focus on free paths and introduce the following probability

$P[A,x,t]=\underset{x(0)=0}{\overset{x(t)=x}{\int }}{𝒟}x(\tau )\exp [-\frac{1}{T}\underset{0}{\overset{t}{\int }}d\tau \frac{1}{2}({\partial }_{\tau }x{)}^2]\delta (\underset{0}{\overset{t}{\int }}d\tau V(x(\tau ),\tau )-A)$

• Second, the moments generating function

$Z_p(x,t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dA{e}^{-pA}P[A,x,t]=\underset{x(0)=0}{\overset{x(t)=x}{\int }}{𝒟}x(\tau )e^{-\frac{1}{T}\underset{0}{\overset{t}{\int }}d\tau \frac{1}{2}({\partial }_{\tau }x{)}^2-p\underset{0}{\overset{t}{\int }}d\tau V(x(\tau ),\tau )}$

• Third, the backward approach. Consider free paths evolving up to $t+dt$ and reaching $x$ :

$Z_p(x,t+dt)=⟨e^{-p\underset{0}{\overset{t+dt}{\int }}d\tau V(x(\tau ),\tau )}⟩=⟨e^{-p\underset{0}{\overset{t}{\int }}d\tau V(x(\tau ),\tau )}⟩e^{-pV(x,t)dt}=[1-pV(x,t)dt+\dots ]{⟨Z_p(x-\mathrm{\Delta }x,t)⟩}_{\mathrm{\Delta }x}$

Here $⟨\dots ⟩$ is the average over all free paths, while $⟨\dots {⟩}_{\mathrm{\Delta }x}$ is the average over the last jump, namely $⟨\mathrm{\Delta }x⟩=0$ and $⟨\mathrm{\Delta }{x}^2⟩=Tdt$.

• At the lowest order we have

$Z_p(x,t+dt)=Z_p(x,t)+dt[\frac{T}{2}\underset{x}{\overset{2}{\partial }}{Z}_p-pV(x,t)Z_p]+O(d{t}^2)$

Replacing $p=1/T$ we obtain the partition function is the solution of the Schrodinger-like equation:

${\partial }_{t}Z(x,t)=-\hat{H}Z=-[-\frac{T}{2}\frac{d^2}{d{x}^2}+\frac{V(x,\tau )}{T}]Z(x,t)$

$Z[x,t=0]=\delta (x)$

Remarks

Remark 1:

This equation is a diffusive equation with multiplicative noise. Edwards Wilkinson is instead a diffusive equation with additive noise. The Cole Hopf transformation allows to map the diffusive equation with multiplicative noise in a non-linear equation with additive noise. We will apply this tranformation and have a surprise.

Remark 2: This hamiltonian is time dependent because of the multiplicative noise $V(x,\tau )/T$. For a time independent hamiltonian, we can use the spectrum of the operator. In general we will have to parts:

• A discrete set of eigenvalues $E_n$ with the eigenstates ${\psi }_n(x)$
• A continuum part where the states ${\psi }_E(x)$ have energy $E$. We define the density of states $\rho (E)$, such that the number of states with energy in ($E,E+dE$) is $\rho (E)dE$.

In this case $Z[x,t]$ can be written has the sum of two contributions:

$Z[x,t]={\left(e^{-\hat{H}t}\right)}_{0\to x}=\sum _n{\psi }_n(0){\psi }_n^{*}(x)e^{-E_{n}t}+\underset{0}{\overset{\mathrm{\infty }}{\int }}dE\rho (E){\psi }_E(0){\psi }_E^{*}(x)e^{-Et}.$