L-3

Goal: This lecture is dedicated to a classical model in disordered systems: the directed polymer in random media. It has been introduced to model vortices in superconductors or domain walls in magnetic films. We will focus on algorithms that identify the ground state or compute the free energy at temperature $T$, as well as on the Cole–Hopf transformation that maps this model to the KPZ equation.

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.

Directed Polymers on a lattice

Sketch of the discrete Directed Polymer model. At each time the polymer grows either one step left or one step right. A random energy $V(\tau ,x)$ is associated to each node and the total energy is simply $E[x(\tau )]=\sum _{\tau =0}^{t}V(\tau ,x)$.

We introduce a lattice model for the directed polymer (see figure). In a companion notebook we provide the implementation of the powerful Dijkstra algorithm. Dijkstra allows one to identify the minimal energy among the exponential number of configurations $x(\tau )$: $$E_{\min }=\underset{x(\tau )}{\min }E[x(\tau )].$$

We are also interested in the ground state configuration $x_{\min }(\tau )$. For both quantities we expect scale invariance with two exponents $\theta$, $\zeta$ for the energy and for the roughness: $$E_{\min }=c_{\mathrm{\infty }}t+{\kappa }_1{t}^{\theta }\chi ,x_{\min }(t/2)\sim {\kappa }_2{t}^{\zeta }\tilde{\chi }.$$

Universal exponents: Both $\theta$ and $\zeta$ are independent of the lattice, the disorder distribution, the elastic constants, or the boundary conditions.

Non-universal constants: $c_{\mathrm{\infty }}$, ${\kappa }_1$, ${\kappa }_2$ are of order 1 and depend on the lattice, the disorder distribution, the elastic constants, etc. However $c_{\mathrm{\infty }}$ is independent of the boundary conditions.

Universal distributions: $\chi$, $\tilde{\chi }$ are universal, but depend on the boundary conditions. Starting from 2000, a remarkable connection has been revealed between this model and the smallest eigenvalues of random matrices. In particular, we discuss two different boundary conditions:

• Droplet: $x(\tau =0)=x(\tau =t)=0$. In this case, up to rescaling, $\chi$ is distributed as the smallest eigenvalue of a GUE random matrix (Tracy–Widom distribution $F_2(\chi )$).

• Flat: $x(\tau =0)=0$ while the other end $x(\tau =t)$ is free. In this case, up to rescaling, $\chi$ is distributed as the smallest eigenvalue of a GOE random matrix (Tracy–Widom distribution $F_1(\chi )$).

Entropy and scaling relation

It is useful to compute the entropy $$\text{Entropy}=\ln (\genfrac{}{}{0ex}{}{t}{\frac{t-x}{2}})\approx t\ln 2-\frac{x^2}{2t}+O(x^4).$$ From which one could guess from dimensional analysis $$\theta =2\zeta -1.$$ This relation is actually exact also for the continuum model.

Directed polymers in the continuum

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 at $0$, ending at $x$ and at thermal equilibrium at temperature $T$. The partition function of the model reads $$Z(x,t)=\underset{x(0)=0}{\overset{x(t)=x}{\int }}{𝒟}x(\tau )\exp [-\frac{1}{T}\underset{0}{\overset{t}{\int }}d\tau (\frac{1}{2}({\partial }_{\tau }x{)}^2+V(x(\tau ),\tau ))].$$ Here, the partition function is written as a sum over all possible paths, corresponding to all 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 identify $T$ with $\hslash$ and $\tilde{t}=-it$ as the time. $$Z(x,\tilde{t})=\underset{x(0)=0}{\overset{x(\tilde{t})=x}{\int }}{𝒟}x(t^{\prime })\exp \left[\frac{i}{\hslash }\underset{0}{\overset{\tilde{t}}{\int }}d{t}^{\prime }(\frac{1}{2}({\partial }_{t^{\prime }}x{)}^2-V(x(t^{\prime }),t^{\prime }))\right].$$

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 $\frac{1}{2}({\partial }_{t^{\prime }}x{)}^2$ and time-dependent potential $V(x(t^{\prime }),t^{\prime })$, evolving from time zero to time $\tilde{t}$. From the Feynman path integral formulation, $Z(x,\tilde{t})$ is the propagator of the quantum particle.

Feynman–Kac formula

Let's derive the Feynman–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 moment generating function

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

• 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 $⟨\cdots ⟩$ is the average over all free paths, while $⟨\cdots {⟩}_{\mathrm{\Delta }x}$ 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 that the partition function is the solution of the Schrödinger-like equation: $${\partial }_{t}Z(x,t)=-\hat{H}Z=-[-\frac{T}{2}\frac{d^2}{d{x}^2}+\frac{V(x,t)}{T}]Z(x,t),Z(x,t=0)=\delta (x).$$

Remarks

Remark 1:

This equation is a diffusive equation with multiplicative noise $V(x,t)/T$. Edwards–Wilkinson is instead a diffusive equation with additive noise.

Remark 2:

This Hamiltonian is time dependent because of the multiplicative noise $V(x,t)/T$. For a time independent Hamiltonian, we can use the spectrum of the operator. In general we will have two parts:

• A discrete set of eigenvalues $E_n$ with 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 as 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: $$Z_{\text{free}}(x,t)=\frac{e^{-x^2/(2Tt)}}{\sqrt{2\pi Tt}}.$$

Hints: free particle in 1D

For a free particle in one dimension the Hamiltonian is $\hat{H}=-\frac{T}{2}\underset{x}{\overset{2}{\partial }}$.

Spectrum. The spectrum is purely continuous. The eigenstates are plane waves $${\psi }_k(x)=\frac{1}{\sqrt{2\pi }}{e}^{ikx},E_k=\frac{T{k}^2}{2},$$ with $k\in \mathbb{ℝ}$. The states are delocalized and satisfy Dirac delta normalization $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dx{\psi }_{k^{\prime }}^{*}(x){\psi }_k(x)=\delta (k-k^{\prime }).$$

Energy representation and density of states. For a given energy $E>0$ there are two degenerate states, $${\psi }_E^{\pm }(x)=\frac{1}{\sqrt{2\pi }}{e}^{\pm i\sqrt{2E/T}x}.$$ The density of states is obtained from $$\rho (E)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dk\delta (E-E_k),E_k=\frac{T{k}^2}{2}.$$

Propagator. Using the spectral decomposition one can write $$Z(x,t)=\underset{0}{\overset{\mathrm{\infty }}{\int }}dE\rho (E)\sum _{\sigma =\pm }{\psi }_E^{\sigma }(0){\psi }_E^{\sigma *}(x)e^{-Et}.$$ Evaluating the resulting Gaussian integral yields $$Z_{\text{free}}(x,t)=\frac{e^{-x^2/(2Tt)}}{\sqrt{2\pi Tt}}.$$

Useful identity: $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dx{e}^{-(a{x}^2+bx)}=\sqrt{\frac{\pi }{a}}{e}^{b^2/(4a)},a>0.$$

Cole Hopf Transformation

Replacing

• $T=2\nu$
• $x=r$
• $Z(x,t)=\exp (\frac{\lambda }{2\nu }h(r,t))$
• $-V(x,t)=\lambda \eta (r,t)$

you get $${\partial }_{t}h(r,t)=\nu {\mathrm{\nabla }}^{2}h(r,t)+\frac{\lambda }{2}(\mathrm{\nabla }h{)}^2+\eta (r,t).$$ The KPZ equation!

We can establish a KPZ/Directed polymer dictionary, valid in any dimension. Let us remark that the free energy of the polymer is $$F=-T\ln Z(x,t)=-\frac{1}{\lambda }h(r,t).$$ At low temperature, the free energy approaches the ground state energy $E_{\min }$.

KPZ / Directed Polymer dictionary

KPZ quantity	KPZ scaling	Directed polymer quantity	Directed polymer scaling
$r$	$r\sim t^{1/z}$	$x$	$x\sim t^{\zeta }$
$t$	$h(r,t)\sim t^{\alpha /z}$	$t$	$\overline{(E_{\min }-\stackrel{‾}{E_{\min }}{)}^2}\sim t^{2\theta }$
$h$	$h(r,t)\sim r^{\alpha }$	$F,E_{\min }$	$\overline{(E_{\min }-\stackrel{‾}{E_{\min }}{)}^2}\sim t^{2\theta }$

We conclude that $$\theta =\alpha /z,\zeta =1/z.$$ Moreover, the scaling relation $\theta =2\zeta -1$ is a reincarnation of the Galilean invariance $\alpha +z=2$.