L-4

Directed Polymer in finite dimension

State of the Art

The directed polymer in random media belongs to the KPZ universality class.

The behavior of this system is well understood in one transverse dimension and in the mean-field case, more precisely for the directed polymer on the Cayley tree. In particular:

• In $N=1$, one has $\theta =1/3$ and a glassy regime present at all temperatures.

The model is integrable through a non-standard Bethe Ansatz, and the distribution of $E_{\min }$ for a given boundary condition is of the Tracy–Widom type.

• In $N=\mathrm{\infty }$, corresponding to the Cayley tree, an exact solution exists, predicting a freezing transition to a one-step replica symmetry breaking phase ($\theta =0$).

In finite transverse dimensions greater than one, no exact solutions are available. Numerical simulations indicate $\theta >0$ in $N=2$, with a glassy regime present at all temperatures. The case $N>2$ remains particularly intriguing.

Let's do replica!

To make progress in disordered systems, we analyze the moments of the partition function. The first moment provides the annealed average, while the second moment contains information about fluctuations. In particular, the partition function is self-averaging if $$\frac{\overline{Z(x,t{)}^2}}{(\overline{Z(x,t)}{)}^2}=1.$$

In this case, the annealed and quenched averages coincide in the thermodynamic limit. This condition is sufficient but not necessary. What is necessary is to show that for large $t$ $$\frac{\overline{Z(x,t{)}^2}}{(\overline{Z(x,t)}{)}^2}<\text{const}.$$

In the following, we compute these moments via a replica calculation, considering polymers starting at $0$ and ending at $x$.

To proceed, we only need two ingredients:

• The random potential $V(x,\tau )$ is a Gaussian field characterized by

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

• Since the disorder is Gaussian, averages of exponentials can be computed using Wick’s theorem:

$$\overline{\exp (W)}=\exp [\bar{W}+\frac{1}{2}(\overline{W^2}-(\bar{W}{)}^2)],$$ for any Gaussian random variable $W$.

These two properties are all we need to carry out the replica calculation below.

First Moment

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

Due to the short-distance divergence of ${\delta }^N(0)$, $$T^2\overline{W^2}=\int d{\tau }_{1}d{\tau }_2\overline{V(x,{\tau }_1)V(x,{\tau }_2)}=Dt{\delta }_0.$$

Hence, $$\overline{Z(x,t)}=\frac{1}{(2\pi tT{)}^{N/2}}\exp [-\frac{x^2}{2tT}]\exp \left[\frac{Dt{\delta }_0}{2T^2}\right]=Z_{\text{free}}(x,t,T)\exp \left[\frac{Dt{\delta }_0}{2T^2}\right].$$

Second Moment

For the second moment we need two replicas.

• Step 1:

$$\overline{Z(x,t{)}^2}=\int {𝒟}{x}_1\int {𝒟}{x}_2\exp [-\frac{1}{2T}\underset{0}{\overset{t}{\int }}d\tau (({\partial }_{\tau }{x}_1{)}^2+({\partial }_{\tau }{x}_2{)}^2\left)\right]\overline{\exp [-\frac{1}{T}\underset{0}{\overset{t}{\int }}d\tau (V(x_1(\tau ),\tau )+V(x_2(\tau ),\tau )\left)\right]}.$$

• Step 2: Wick’s theorem

$$\overline{Z(x,t{)}^2}=\exp \left[\frac{Dt{\delta }_0}{T^2}\right]\int {𝒟}{x}_1\int {𝒟}{x}_2\exp [-\frac{1}{2T}\underset{0}{\overset{t}{\int }}d\tau (({\partial }_{\tau }{x}_1{)}^2+({\partial }_{\tau }{x}_2{)}^2-\frac{D}{T^2}{\delta }^N[x_1(\tau )-x_2(\tau )]\left)\right].$$

• Step 3: Change of coordinates

Let $X=(x_1+x_2)/2$ and $u=x_1-x_2$. Then $$\overline{Z(x,t{)}^2}=(\overline{Z(x,t)}{)}^2\frac{\underset{u(0)=0}{\overset{u(t)=0}{\int }}{𝒟}u\exp [-\underset{0}{\overset{t}{\int }}d\tau (\frac{1}{4T}({\partial }_{\tau }u{)}^2+\frac{D}{T^2}{\delta }^N[u(\tau )])]}{Z_{\text{free}}(u=0,t,2T)}.$$

Here, $$Z_{\text{free}}^2(x,t,T)=Z_{\text{free}}(X=x,t,T/2)Z_{\text{free}}(u=0,t,2T),Z_{\text{free}}(u=0,t,2T)=(4\pi Tt{)}^{-N/2}.$$

Two-replica propagator

Define the propagator $$W(0,t)=\underset{u(0)=0}{\overset{u(t)=0}{\int }}{𝒟}u\exp [-\underset{0}{\overset{t}{\int }}d\tau (\frac{1}{4T}({\partial }_{\tau }u{)}^2+\frac{D}{T^2}{\delta }^N[u(\tau )])].$$

By the Feynman–Kac formula, $${\partial }_{t}W(x,t)=-\hat{H}W(x,t),\hat{H}=-T{\mathrm{\nabla }}^2-\frac{D}{T^2}{\delta }^N[u].$$

For $N\le 2$, the attractive potential always produces a bound state with energy $E_0<0$. Hence, at long times $$W(x,t)\sim e^{|E_0|t}.$$ This divergence implies that the quenched free energy is smaller than the annealed one at all temperatures.

For $N>2$, the low-energy behavior depends on $D/T^2$:

• At high temperature, the spectrum is positive and continuous. Annealed and quenched averages coincide, and $\theta =0$.
• At low temperature, bound states appear. There is no replica symmetry breaking, but the quenched free energy is smaller than the annealed one. Numerical simulations indicate $\theta >0$.

Overlap Distribution and Replica Symmetry Breaking

The structure of states can be further characterized through the overlap between two configurations $\alpha$ and $\gamma$, defined as $$q_{\alpha ,\gamma }=\frac{1}{L^d}\sum _{i=1}^{L^d}{\sigma }_i^{\alpha }{\sigma }_i^{\gamma }.$$

For spin glasses, the overlap takes values in the interval $(-1,1)$. This definition can be naturally extended to directed polymers, where the overlap is identified with the fraction of monomers shared by two polymer configurations.

In systems exhibiting one-step replica symmetry breaking (1RSB), the distribution $P(q)$ of the overlap between two configurations sampled from the Gibbs measure sharply distinguishes the two phases.

At high temperature ($\beta <{\beta }_c$), the system is replica symmetric and the overlap distribution is concentrated at zero: $$P(q)=\delta (q).$$

At low temperature ($\beta >{\beta }_c$), the system exhibits one-step replica symmetry breaking, and the overlap distribution becomes $$P(q)=\frac{{\beta }_c}{\beta }\delta (q)+(1-\frac{{\beta }_c}{\beta })\delta (1-q).$$

This picture is realized, for instance, in the Random Energy Model and on the Cayley tree.

Finite-dimensional systems

In finite dimensions, the nature of the low-temperature phase is controlled by the fluctuations of the ground-state energy, characterized by an exponent $\theta$: $$\overline{(E_{\min }-\stackrel{‾}{E_{\min }}{)}^2}\sim L^{2\theta },$$ where $L$ is the linear size of the system and $L^d$ the number of degrees of freedom.

When $\theta <0$, the critical temperature vanishes in the thermodynamic limit, implying the absence of a glass transition. This is the case, for instance, of the Edwards–Anderson spin glass in two dimensions.

When $\theta >0$, one must consider the fluctuations of the free energy $F(L,\beta )$ at finite temperature. Several representative cases can then be distinguished.

• Directed polymer in $N=1,2$:

The fluctuations of the ground-state energy are governed by a positive, temperature-independent exponent $\theta$. The system is glassy at all temperatures, but the glassy phase is dominated by a single ground state. As a consequence, $$P(q)=\delta (1-q),$$ since excitations with vanishing overlap with the ground state are energetically prohibitive.

• Directed polymer in $N=3$:

The exponent $\theta$ depends on temperature: it vanishes above the glass transition and becomes strictly positive below it. Accordingly, $$P(q)=\delta (q).$$ at high temperature, while $$P(q)=\delta (1-q).$$ at low temperature. Even in the glassy phase, the system is controlled by a unique ground state, and no one-step replica symmetry breaking occurs.

• Directed polymer on the Cayley tree:

The behavior is analogous to that of the Random Energy Model. The exponent $\theta =0$ in both phases, and the low-temperature phase is characterized by one-step replica symmetry breaking. At high temperature, $$P(q)=\delta (q),$$ while at low temperature $$P(q)=\frac{{\beta }_c}{\beta }\delta (q)+(1-\frac{{\beta }_c}{\beta })\delta (1-q).$$