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

We now reanalyze the previous problem in the presence of quenched disorder. Instead of discussing the case of interfaces, we will focus on directed polymers. Let us consider polymers $x(\tau )$ of length $t$. The energy associated with a given polymer configuration can be written as

$E[x(\tau )]=\underset{0}{\overset{t}{\int }}d\tau [\frac{1}{2}{\left(\frac{dx}{d\tau }\right)}^2+V(x(\tau ),\tau )]$

The first term describes the elastic energy of the polymer, while the second one is the disordered potential, which we assume to be

$\overline{V(x,\tau )}=0,\overline{V(x,\tau )V(x^{\prime },{\tau }^{\prime })}=D\delta (x-x^{\prime })\delta (\tau -{\tau }^{\prime }).$

where 'D' is the disorder strength.

Polymer partition function and propagator of a quantum particle

Let us consider polymers starting in $0$, ending in $x$ and at thermal equilibrium at temperature $T$. The partition function of the model writes as

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

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.

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.

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, 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 $V(x,\tau )/T$ . Edwards Wilkinson is instead a diffusive equation with additive noise.

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:

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

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

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