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| ==Let's do replica!== | | ==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 | | To make progress in disordered systems, we need to analyze the moments of the partition function. From Valentina's lecture, recall that if |
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| <center>
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| <math>
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| \frac{\overline{Z(x,t)^2}}{ (\overline{Z(x,t)})^2}=1
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| </math>
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| </center>
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| then the partition function is self-averaging, and
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| <center>
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| <math>
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| \overline{\ln Z(x,t)} =\ln\overline{Z(x,t)}
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| </math>.
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| </center>
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| The condition above is sufficient but not necessary. It is enough that
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| <center>
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| <math>
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| \frac{\overline{Z(x,t)^2}}{ (\overline{Z(x,t)})^2} < \text{const}
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| </math>,
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| </center>
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| when <math>t \to \infty</math>, to ensure the equivalence between annealed and quenched averages.
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| In the following, we compute this quantity, which corresponds to a two-replica calculation. For simplicity, we consider polymers starting at <math>0</math> and ending at <math>x</math>. We recall that:
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| * <math>V(x,\tau)</math> is a Gaussian field with
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| <center> <math>
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| \overline{V(x,\tau)}=0, \quad \overline{V(x,\tau) V(x',\tau')} = D \delta^d(x-x') \delta(\tau-\tau')
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| </math></center>
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| * From Wick's theorem, for a generic Gaussian field <math> W </math>, we have
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| <center><math>
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| \overline{\exp(W)} = \exp\left[\overline{W} +\frac{1}{2} \left(\overline{W^2}-\overline{W}^2\right)\right]
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| </math></center>
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| ===The first moment===
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| The first moment of the partition function is straightforward to compute and corresponds to a single replica:
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| <center>
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| <math>
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| \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]}
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| </math>
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| </center>
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| Note that the term <math> T^2 \overline{W^2} = \int d \tau_1 d\tau_2 \overline{V(x,\tau_1)V(x,\tau_2)}= D t \delta_0</math> exhibits a short-distance divergence due to the delta function. Hence, we can write:
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| <center>
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| <math>
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| \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} \right]
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| </math>
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| </center>
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| === The second moment ===
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| For the second moment, there are two replicas:
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| * Step 1: The second moment is
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| <center>
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| <math>
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| \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]}
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| </math>
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| </center>
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| * Step 2: Using Wick's theorem, we obtain
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| <center>
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| <math>
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| \overline{Z(x,t)^2 } = \exp\left[ \frac{D t \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]
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| </math>
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| </center>
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| and we can write:
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| <center>
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| <math>
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| \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]
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| </math>
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| </center>
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| * Step 3: Changing coordinates <math>X=(x_1+x_2)/2; \; u=x_1-x_2</math>, we get
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| <center>
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| <math>
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| \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)}
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| </math>
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| </center>
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| where we used <math> Z_{\text{free}}^2(x,t,T)=Z_{\text{free}}(X=x,t,T/2)Z_{\text{free}}(u=0,t,2T) </math> with <math>Z_{\text{free}}(u=0,t,2T) = (4 \pi T t)^{d/2} </math>
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| ===The two replica propagator===
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| Let us define the propagator:
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| <center> <math>
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| 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]
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| </math>
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| </center>
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| Using the Feynman-Kac formula, we can write the following equation:
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| <center>
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| <math>
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| \partial_t W(x,t) = - \hat H W(x,t)
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| </math>
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| </center>
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| Here, the Hamiltonian is given by:
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| <center>
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| <math>
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| \hat H = - T \nabla^2 - \frac{D}{T^2} \delta^d[u]
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| </math>
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| </center>
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| === The Spectrum of the Two-Replica Hamiltonian ===
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| 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:
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| * A discrete set of eigenvalues corresponding to bound states, followed by a continuous spectrum.
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| * Only a continuous spectrum.
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| As a funcion of the dimension we distiguish two cases:
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| * For <math>d \leq 2</math>:
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| An attractive potential always leads to the formation of a bound state.The ground state has a negative energy <math>E_0 < 0</math>. At long times, the propagator behaves as:
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| <center>
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| <math>
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| W(x,t) \sim e^{ |E_0| t}
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| </math>
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| </center>
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| This implies that at all temperatures, in the limit <math>t \to \infty</math>:
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| <center>
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| <math>
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| \overline{\ln Z(x,t)} \ll \ln\overline{Z(x,t)}
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| </math>
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| </center>
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| * For <math>d > 2</math>:
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| The low-energy part of the spectrum is controlled by the prefactor <math>\frac{D}{T^2}</math>. At high temperatures, the spectrum remains continuous and positive. At low temperatures, bound states appear. Thus, in the limit <math>t \to \infty</math>:
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| <center>
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| <math>
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| \begin{cases}
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| \overline{\ln Z(x,t)} = \ln\overline{Z(x,t)} \quad \text{for} \quad T > T_c \\
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| \\
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| \overline{\ln Z(x,t)} \ll \ln\overline{Z(x,t)} \quad \text{for} \quad T < T_c
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| \end{cases}
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| </math>
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| </center>
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| This transition, in <math>d = 3</math>, separates a high-temperature phase with <math>\theta = 0</math> and a low-temperature phase with <math>\theta > 0</math> and '''no RSB'''.
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Goal : final lecture on KPZ and directed polymers at finite dimension. We will show that for 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 , we have found 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 . However, it does not determine the actual distribution of for a given . In particular, we have no clear understanding of the origin of the Tracy-Widom distribution.
- In , an exact solution exists for the Cayley tree, predicting a freezing transition to a 1RSB phase ().
- In finite dimensions greater than one, no exact solutions are available. Numerical simulations indicate in . The case 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