L-4

Goal : final lecture on KPZ and directed polymers at finite dimension. We will show that for $d>2$ a "glass transition" takes place.

KPZ : from 1d to the Cayley tree

We know a lot about KPZ, but there is still much to understand:

• In $d=1$, we have found $\theta =1/3$ and a glassy regime present at all temperatures. The stationary solution of the KPZ equation describes, at long times, the fluctuations of quantities such as $E_{\min }[x]-E_{\min }[x^{\prime }]$. However, it does not determine the actual distribution of $E_{\min }$ for a given $x$. In particular, we have no clear understanding of the origin of the Tracy-Widom distribution.

• In $d=\mathrm{\infty }$, an exact solution exists for the Cayley tree, predicting a freezing transition to a 1RSB phase ($\theta =0$).

• In finite dimensions greater than one, no exact solutions are available. Numerical simulations indicate $\theta >0$ in $d=2$. The case $d>2$ remains particularly intriguing.

Let's do replica!

To make progress in disordered systems, we need to analyze the moments of the partition function. From Valentina's lecture, recall that if

$\frac{\overline{Z(x,t{)}^2}}{(\overline{Z(x,t)}{)}^2}=1$

then the partition function is self-averaging, and

$\overline{\ln Z(x,t)}=\ln \overline{Z(x,t)}$.

The condition above is sufficient but not necessary. It is enough that

$\frac{\overline{Z(x,t{)}^2}}{(\overline{Z(x,t)}{)}^2}<\text{const}$,

when $t\to \mathrm{\infty }$, to ensure the equivalence between annealed and quenched averages.

In the following, we compute this quantity, which corresponds to a two-replica calculation. For simplicity, we consider polymers starting at $0$ and ending at $x$. We recall that:

• $V(x,\tau )$ is a Gaussian field with

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

• From Wick's theorem, for a generic Gaussian field $W$, we have

$\overline{\exp (W)}=\exp [\bar{W}+\frac{1}{2}(\overline{W^2}-\stackrel{2}{\stackrel{‾}{W}})]$

The first moment

The first moment of the partition function is straightforward to compute and corresponds to a single replica:

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

Note that the term $T^2\overline{W^2}=\int d{\tau }_{1}d{\tau }_2\overline{V(x,{\tau }_1)V(x,{\tau }_2)}=Dt{\delta }_0$ exhibits a short-distance divergence due to the delta function. Hence, we can write:

Failed to parse (syntax error): {\displaystyle \overline{Z(x,t) } = \frac{1}{(2 \pi t T)^{d/2}}\exp\left[ -\frac{1}{2} \frac{ x^2}{t T} \right] \exp\left[ \frac{D t \delta_0}{2T^2} \right] = Z_{\text{free}}(x,t,T) \exp\left[ \frac{D t \delta_0}{2T^2} }

The second moment

For the second moment, there are two replicas:

• Step 1: The second moment is

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

• Step 2: Using Wick's theorem, we obtain

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

and we can write:

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

• Step 3: Changing coordinates $X=(x_1+x_2)/2;u=x_1-x_2$, we get

$\overline{Z(x,t{)}^2}=(\overline{Z(x,t)}{)}^2\frac{\underset{u(0)=0}{\overset{u(t)=0}{\int }}{𝒟}u\exp [-\underset{0}{\overset{t}{\int }}d\tau \frac{1}{4T}({\partial }_{\tau }u{)}^2-\frac{D}{T^2}{\delta }^d[u(\tau )]]}{Z_{free}(u,t)}$

The two replica propagator

Hence, the quantity $\frac{\overline{Z(x,t{)}^2}}{(\overline{Z(x,t)}{)}^2}$ can be computed.

• The denominator

$\underset{u(0)=0}{\overset{u(t)=0}{\int }}{𝒟}u\exp [-\underset{0}{\overset{t}{\int }}d\tau \frac{1}{4T}({\partial }_{\tau }u{)}^2]$

is the free propagator and contributes as $\sim (4Tt{)}^{d/2}$.

• Let us define the numerator:

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

Using the Feynman-Kac formula, we can write the following equation:

${\partial }_{t}W(x,t)=-\hat{H}W(x,t)$

Here, the Hamiltonian is given by:

$\hat{H}=-2T{\mathrm{\nabla }}^2-\frac{D}{T^2}{\delta }^d[u]$

The Spectrum of the Two-Replica Hamiltonian

The single-particle potential is time-independent and attractive. Since it is time-independent, we can use the spectral decomposition of the propagator. The long-time behavior is controlled by the low-energy part of the spectrum. In the presence of an attractive potential, we may have:

• A discrete set of eigenvalues corresponding to bound states, followed by a continuous spectrum.
• Only a continuous spectrum.

As a funcion of the dimension we distiguish two cases:

• For $d\le 2$:

An attractive potential always leads to the formation of a bound state.The ground state has a negative energy $E_0<0$. At long times, the propagator behaves as:

$W(x,t)\sim e^{|E_0|t}$

This implies that at all temperatures, in the limit $t\to \mathrm{\infty }$:

$\overline{\ln Z(x,t)}\ll \ln \overline{Z(x,t)}$

• For $d>2$:

The low-energy part of the spectrum is controlled by the prefactor $\frac{D}{T^2}$. At high temperatures, the spectrum remains continuous and positive. At low temperatures, bound states appear. Thus, in the limit $t\to \mathrm{\infty }$:

$\{\begin{array}{l}\overline{\ln Z(x,t)}=\ln \overline{Z(x,t)}\text{for}T>T_c\\ \\ \overline{\ln Z(x,t)}\ll \ln \overline{Z(x,t)}\text{for}T<T_c\end{array}$

This transition, in $d=3$, separates a high-temperature phase with $\theta =0$ and a low-temperature phase with $\theta >0$ and no RSB.