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 $E_{\min }[x]-E_{\min }[x']$. 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 $d=\infty$, 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 $t\to \infty$, 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

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

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

${\overline {\exp(W)}}=\exp \left[{\overline {W}}+{\frac {1}{2}}\left({\overline {W^{2}}}-{\overline {W}}^{2}\right)\right]$

The first moment

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

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

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:

${\overline {Z(x,t)}}={\frac {1}{(2\pi tT)^{d/2}}}\exp \left[-{\frac {1}{2}}{\frac {x^{2}}{tT}}\right]\exp \left[{\frac {Dt\delta _{0}}{2T^{2}}}\right]=Z_{\text{free}}(x,t,T)\exp \left[{\frac {Dt\delta _{0}}{2T^{2}}}\right]$

The second moment

For the second moment, there are two replicas:

• Step 1: The second moment is

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

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

${\overline {Z(x,t)^{2}}}=\exp \left[{\frac {Dt\delta _{0}}{T^{2}}}\right]\int {\cal {D}}x_{1}\int {\cal {D}}x_{2}\exp \left[-\int _{0}^{t}d\tau {\frac {1}{2T}}[(\partial _{\tau }x_{1})^{2}+(\partial _{\tau }x_{2})^{2}-{\frac {D}{T^{2}}}\delta ^{d}[x_{1}(\tau )-x_{2}(\tau )]\right]$

and we can write:

${\overline {Z(x,t)^{2}}}=({\frac {\overline {Z(x,t)}}{Z_{\text{free}}(x,t,T)}})^{2}\int {\cal {D}}x_{1}\int {\cal {D}}x_{2}\exp \left[-\int _{0}^{t}d\tau {\frac {1}{2T}}[(\partial _{\tau }x_{1})^{2}+(\partial _{\tau }x_{2})^{2}-{\frac {D}{T^{2}}}\delta ^{d}[x_{1}(\tau )-x_{2}(\tau )]\right]$

• Step 3: Changing coordinates $X=(x_{1}+x_{2})/2;\;u=x_{1}-x_{2}$, we get

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

where we used $Z_{\text{free}}^{2}(x,t,T)=Z_{\text{free}}(X=x,t,T/2)Z_{\text{free}}(u=0,t,2T)$ with $Z_{\text{free}}(u=0,t,2T)=(4\pi Tt)^{d/2}$

The two replica propagator

Let us define the propagator:

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

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:

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

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 $d\leq 2$:

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 $t\to \infty$:

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

• 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 $t\to \infty$:

${\begin{cases}{\overline {\ln Z(x,t)}}=\ln {\overline {Z(x,t)}}\quad {\text{for}}\quad T>T_{c}\\\\{\overline {\ln Z(x,t)}}\ll \ln {\overline {Z(x,t)}}\quad {\text{for}}\quad T<T_{c}\end{cases}}$

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