L-4

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Goal : final lecture on KPZ and directed polymers at finite dimension. We will show that 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} a "glass transition" takes place.


KPZ : from to the Cayley tree

We know a lot about KPZ, but still we have much to understand:

  • In we found and a glassy regime present at all temperatures. The stationary solution of the KPZ equation describes, at long times, the fluctions of quantities like . However it does not identify the actual distribution of for a given . In particular we have no idea from where Tracy Widom comes from.
  • In , there is an exact solution for the Cayley tree that predicts a freezing transition to an 1RSB phase ().
  • In finite dimension, but larger than 1, there are no exact solutions. Numerical simulations find in . The case is very interesting.

Let's do replica!

To make progress in disordered systems we have to go through the moments of the partition function. For simplicity we consider polymers starting in and ending in . We recall that

  • is a Gaussian field with
  • From the Wick theorem, for a generic Gaussian field we have

The first moment

The first moment of the partition function is

Note that the term has a short distance divergence due to the delta-function. Hence we can write:

The second moment

Exercise: L-4

  • Step 1: The second moment is
  • Step 2: Use Wick and derive:
  • Step 3: Now change coordinate and get:

Discussion

Hence, the quantity can be computed.

  • The denominator is the free propagator and gives a contribution .
  • Let us define the numerator

Remark 1: From T-I, remember that if

the partition function is self-averaging and . The condition above is sufficient but not necessary. It is enough that , when , to have the equivalence between annealed and quenched averages.

Remark II: From L-3, we derive using Feynman-Kac, the following equation

Here the Hamiltonian reads:

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

The single particle potential is time independent and actractive .

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 W(x,t) = \langle x|\exp\left( - \hat H t\right) |0\rangle }

At large times the behaviour is dominatated by the low energy part of the spectrum.

  • In an actractive potential always gives a bound state. In particular the ground state has a negative 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 large times
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 W(x,t) = e^{ |E_0| t} }

grows exponentially. This means that at all temperature, when

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{\ln Z(x,t)} \ll \ln\overline{Z(x,t)} }
  • 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 part of the spectrum is controlled by the strength of the prefactor . At high temperature we have a continuum positive spectrum, at low temperature bound states exist. Hence, when 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 t\to \infty}
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 \begin{cases} \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{cases} }

This transition, in , is between a high temeprature, 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} phase and a low temeprature ,math> \theta>0</math> no RSB phase.