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.

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 dimension and in the mean-field case, more precisely for the directed polymer on the Cayley tree. In particular:

• In $d=1$, we have $\theta =1/3$ 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 $d=\infty$, for the Cayley tree, an exact solution exists, 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 $d=2$ and a glassy regime present at all temperatures. The case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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. The first moment provide the annealed average and the second moment tell us about the fluctuantions. In particular, the partition function is self-averaging if

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\overline{Z(x,t)^2}}{ (\overline{Z(x,t)})^2}=1 \, . }

In this case annealed and the quenched average coincides in the thermodynamic limit. This strict 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} and ending at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} .

To proceed, we only need two ingredients:

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{V(x,\tau)} = 0, \qquad \overline{V(x,\tau) V(x',\tau')} = D \, \delta^d(x-x') \, \delta(\tau - \tau'). }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\exp(W)} = \exp\!\Big[\overline{W} + \frac{1}{2}\big(\overline{W^2} - \overline{W}^2\big)\Big], }

for any Gaussian random variable $W$.

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

First Moment

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{Z(x,t)} = \int_{x(0)=0}^{x(t)=x} \mathcal{D}x(\tau) \exp\Big[-\frac{1}{T}\int_0^t d\tau \frac{1}{2}(\partial_\tau x)^2\Big] \overline{\exp\Big[-\frac{1}{T} \int_0^t d\tau V(x(\tau),\tau)\Big]} }

Due to the short-distance divergence of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta^d(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,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{Z(x,t)} = \frac{1}{(2\pi t T)^{d/2}} \exp\Big[-\frac{x^2}{2 t T}\Big] \exp\Big[\frac{D t \delta_0}{2 T^2}\Big] = Z_{free}(x,t,T) \exp\Big[\frac{D t \delta_0}{2 T^2}\Big]. }

Second Moment

For the second moment we need two replicas:

Step 1

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{Z(x,t)^2} = \int \mathcal{D}x_1 \int \mathcal{D}x_2 \exp\!\Bigg[-\frac{1}{2T}\int_0^t d\tau \Big((\partial_\tau x_1)^2 + (\partial_\tau x_2)^2\Big)\Bigg] \; \overline{\exp\!\Bigg[-\frac{1}{T} \int_0^t d\tau V(x_1(\tau),\tau) - \frac{1}{T} \int_0^t d\tau V(x_2(\tau),\tau)\Bigg]}. }

Step 2: Wick’s Theorem

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

Step 3: Change of Coordinates

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = (x_1+x_2)/2} and $u=x_{1}-x_{2}$. Then:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{Z(x,t)^2} = (\overline{Z(x,t)})^2 \frac{\displaystyle \int_{u(0)=0}^{u(t)=0} \mathcal{D}u \exp\!\Bigg[-\int_0^t d\tau \frac{1}{4T} (\partial_\tau u)^2 - \frac{D}{T^2} \delta^d[u(\tau)]\Bigg]} {Z_{free}(u=0,t,2T)}. }

Here,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{free}^2(x,t,T) = Z_{free}(X=x,t,T/2) \, Z_{free}(u=0,t,2T), \qquad Z_{free}(u=0,t,2T) = (4 \pi T t)^{d/2}. }

Two-Replica Propagator

Define the propagator:

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

By the Feynman-Kac formula:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial_t W(x,t) = -\hat H W(x,t), \quad \hat H = -T \nabla^2 - \frac{D}{T^2} \delta^d[u]. }

The single-particle potential is time-independent and attractive. Long-time behavior is governed by the low-energy eigenstates.

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \le 2} , the attractive potential always produces a bound state with energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_0<0} . Hence, at long times:

$W(x,t)\sim e^{|E_{0}|t}$

This explosion means that the quenched free energy is smaller than the annealed one at all temperatures.

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d > 2} , The low-energy behavior depends on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D/T^2} :

• High temperature: the spectrum is positive and continuous. Annealed and quenched coincide, the exponent $\theta =0$.

• Low temperature: bound states appear. No replica-symmetry breaking (RSB), but the quenched free energy is smaller than the annealed one. Numerical simulations show Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta>0} .