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A very brief introduction to directed polymers

A classical problem in the field of disordered systems is an elastic string (or elastic polymer) stretched between two points in a random environment. One is interested in knowing how the polymer geometry (roughness), ground state energy, and excitations change due to the disordered environment. A fundamental question is if the polymer becomes rough for arbitrarily weak disorder, or if there is a phase transition between a weakly disordered (smooth) and a strongly disordered (rough) phase.

In the following I will try explain some basic results on the simple case of a two-dimensional directed polymer (i.e. a “stretched” polymer which does not contain loops or overhangs). I will explain why for large polymer lengths $L$ , moments of the partition sum $Z$ grow as $\ln \overline{Z^n} \propto n^3 L$ . I will show why this indicates a rough polymer with roughness exponent $\zeta=\frac{2}{3}$ . This analysis holds for arbitrarily weak disorder, meaning that in two dimensions disorder is always relevant, and the polymer is always in the rough phase.

Although none of these observations are new, the existing results are often buried in technical literature (there is not even a Wikipedia page describing the directed polymer problem). I hope this self-contained summary will be useful for people interested in a quick introduction to the subject. Some original references where more details can be found are cited at the end of the post.

The directed polymer partition sum, and its moments

We can consider a polymer configuration as a trajectory for a particle moving between two points. We call the polymer directed if the particle always moves from the initial towards the final point, i.e. if the trajectory does not contain loops or overhangs.

Formally, this means that the polymer configuration in $1+d$ dimensions can be parametrized by giving $d$ transversal coordinates $(x_1...x_d)=\vec{x}$ as a function of the longitudinal coordinate $t$ (distance along the path). The partition sum is then given by a path integral

$Z = \int \mathcal{D}[\vec{x}]e^{-\int_0^L \mathrm{d}t \, \left[(\dot{\vec{x}})^2 + \eta(\vec{x},t)\right]}$

$(\dot{\vec{x}})^2$ is the elastic energy term, which is minimized for straight paths. $\eta(\vec{x},t)$ is the energy it costs the polymer to pass through the site $(\vec{x},t)$ (and which is minimized for rough paths picking out the minimal energy sites). We model a disordered environment by taking $\eta$ to be uncorrelated Gaussian white noise:

$\displaystyle \overline{\eta(\vec{x}_1,t_1)\eta(\vec{x}_2,t_2)} = 2c\delta^{(d)}(\vec{x}_1-\vec{x}_2)\delta(t_1-t_2)$

Applying the Feynman-Kac formula, the partition sum $Z(\vec{x},t)$ satisfies the following stochastic differential equation:

$\displaystyle \partial_t Z(\vec{x},t) = \nabla^2 Z(\vec{x},t) +\eta(\vec{x},t) Z(\vec{x},t)$

$Z$ is a random quantity, because $\eta$ is random. Since computing the distribution of $Z$ is difficult, we will consider its moments, defined as

$\displaystyle \Psi_n(\vec{x}_1...\vec{x}_n,t) := \overline{Z(\vec{x}_1,t)\cdots Z(\vec{x}_n,t)}$

The time evolution of $\Psi_n$ is determined by applying the Itô formula to the equation for $Z$ :

$\displaystyle \partial_t\Psi_n(\vec{x}_1...\vec{x}_n,t) = \left[\sum_{j=1}^n \nabla_{\vec{x}_j}^2 + 2c\sum_{j<k}\delta(\vec{x}_j-\vec{x}_k)\right] \Psi_n(\vec{x}_1...\vec{x}_n,t) =: H_n \Psi_n(\vec{x}_1...\vec{x}_n,t)$

This is, up to an overall sign, the Schrödinger equation for $n$ bosons interacting through pairwise pointlike attraction of strength $c$ . In one dimension, this is called the Lieb-Liniger model. It is easy to determine the ground state of this system exactly, which will allow us to obtain the leading behaviour of $Z^n$ for large $t$ .

Solution in one spatial dimension: Bethe ansatz ground state

In $d=1$ , the ground state of the Hamiltonian $H_n$ defined above has energy $E_n = \frac{c^2}{12}n(n^2-1)$ and is given by

$\displaystyle \Psi_n(\vec{x}_1...\vec{x}_n) \propto \exp\left(-\frac{c}{2}\sum_{j<k} |x_j-x_k|\right)$

This is a special case of the general Bethe ansatz ubiquitious in modern theoretical physics. Its application to the directed polymer problem, as discussed here, was recognized by Kardar (see references below).

To check that this ansatz for $\Psi$ gives, indeed, an eigenstate, let us take two derivatives with respect to $x_j$ :

$\displaystyle \begin{array}{lcl} \partial_{x_j}^2 \Psi_n & = & \partial_{x_j} \left[-\frac{c}{2}\sum_{j \neq k} \text{sgn}(x_j-x_k) \Psi_n\right] \\ & = & \left[-c\sum_{j \neq k} \delta(x_j-x_k)\right] \Psi_n + \left[-\frac{c}{2}\sum_{k \neq j} \text{sgn}(x_j-x_k)\right]^2 \Psi_n \end{array}$

Taking the sum over all $j$ we get

$\displaystyle \sum_{j=1}^n \partial_{x_j}^2 \Psi_n = -2c\sum_{j < k} \delta(x_j-x_k) \Psi_n + \frac{c^2}{4}\sum_{j=1}^n \left[\sum_{j \neq k} \text{sgn}(x_j-x_k)\right]^2 \Psi_n$

To compute the second term, note that it is symmetric under permutations of the $x_j$ . Let us assume (without loss of generality) that $x_1 < x_2 < ... < x_n$ . Then

$\displaystyle \begin{array}{lcl} \sum_{j=1}^n \left[\sum_{j \neq k} \text{sgn}(x_j-x_k)\right]^2 & = & \sum_{j=1}^n \left[\sum_{k>j} (-1)+\sum_{k<j} (+1)\right]^2 \\ & = & \sum_{j=1}^n \left[-(n-j) + j-1\right]^2 \\ & = & \sum_{j=1}^n \left(2j-n-1\right)^2 = \frac{n(n^2-1)}{3} \end{array}$

Thus, we see that as claimed,

$\displaystyle H_n \Psi_n = E_n \Psi_n$

with the energy $E_n = \frac{c^2}{12}n(n^2-1)$ . To see that this is, indeed, the ground state, is beyond the scope of this post (essentially it would require one to take a finite system volume $V$ , construct all excited states using the Bethe ansatz, and check that they all have higher energy. Since one knows the number of eigenstates in a finite volume, this would exclude any lower-energy state).

Believing that this is, indeed, the ground state our argument shows that for large polymer length $L$ , the moments of the partition sum $Z$ grow as:

$\displaystyle \overline{Z^n (L)} \propto e^{\frac{c^2}{12}n(n^2-1) L}$

Implications for the string geometry

The asymptotics of $\overline{Z^n}$ derived above is consistent with a free energy that scales with polymer length $L$ as $\ln Z = F =: f L^\frac{1}{3}$ . Furthermore, the distribution $P(f)$ of the rescaled free energy should have a tail decaying as

$\displaystyle P(f) \propto e^{-f^{\frac{3}{2}}} \quad \text{for } f \gg 1$

To see this, let us compute $\overline{Z^n}$ assuming this form for $P(f)$ :

$\displaystyle \overline{Z^n} = \int \mathrm{d}f e^{n f L^{\frac{1}{3}}-f^{\frac{3}{2}}}$

For large $n$ , we expect $f$ to be large; thus the integral can be approximated by the maximum of the exponent. It occurs at

$f \propto \left(n L^{\frac{1}{3}}\right)^2$

and thus we get

$\displaystyle \ln \overline{Z^n} \propto L n^3$ ,

consistent with the large- $n$ behaviour of the explicit formula for $\overline{Z^n}$ derived above.
Thus, the asymptotics of $\overline{Z^n}$ allowed us to determine the tail of the free energy distribution $P(f)$ and the scaling $F = f L^\frac{1}{3}$ . The scaling of the typical displacement $x$ with polymer length $L$ follows:

$\displaystyle L^\frac{1}{3} \sim F \sim \int_0^L \mathrm{d}t (\dot{x})^2 \sim L \left(\frac{x}{L}\right)^2 \Rightarrow x \sim L^\frac{2}{3}$

Thus, the polymer is rough in the sense that the typical fluctuations don’t remain bounded, but increase with polymer length as

$\displaystyle \overline{\left[x(L)-x(0)\right]^2}\sim L^{2\zeta}$

with the so-called roughness exponent $\zeta= \frac{2}{3}$ .

This whole analysis remains valid at an arbitrarily weak coupling strength $c$ . Thus, the polymer is rough for arbitrarily weak disorder (or, equivalently, arbitrarily high temperatures). In order to observe a true phase transition between weak-disorder and strong-disorder behaviour one needs to go to higher dimensions.

Higher dimensions and phase transitions

As we have now seen, in two dimensions (one longitudinal + one transversal) the directed polymer is always in a rough phase. It turns out (but it would take too long to explain here) that the same holds in three (1+2) dimensions. However, starting from one longitudinal + three transversal dimensions, weak disorder is insufficient to make the polymer rough. In other words, there is a high-temperature phase where the moments of $Z$ scale as $\ln \overline{Z^n} \propto n^2$ , indicating Gaussian fluctuations, i.e. purely thermal roughness.

References

The original argument for obtaining the moments of $Z$ from the Bethe ansatz solution of the Lieb-Liniger model is due to Kardar (see e.g. Nucl. Phys. B 290, 1989 and the references therein). The saddle-point argument for the tail of the free energy distribution is due to Zhang (see e.g. PRB 42, 1990). If you feel some other reference should be included please let me know!

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