|
|
| Line 13: |
Line 13: |
|
| |
|
| ==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 |
| | |
| <center>
| |
| <math>
| |
| \frac{\overline{Z(x,t)^2}}{ (\overline{Z(x,t)})^2}=1
| |
| </math>
| |
| </center>
| |
| | |
| then the partition function is self-averaging, and
| |
| | |
| <center>
| |
| <math>
| |
| \overline{\ln Z(x,t)} =\ln\overline{Z(x,t)}
| |
| </math>.
| |
| </center>
| |
| | |
| The condition above is sufficient but not necessary. It is enough that
| |
| | |
| <center>
| |
| <math>
| |
| \frac{\overline{Z(x,t)^2}}{ (\overline{Z(x,t)})^2} < \text{const}
| |
| </math>,
| |
| </center>
| |
| | |
| when <math>t \to \infty</math>, 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 <math>0</math> and ending at <math>x</math>. We recall that:
| |
| | |
| * <math>V(x,\tau)</math> is a Gaussian field with
| |
| <center> <math>
| |
| \overline{V(x,\tau)}=0, \quad \overline{V(x,\tau) V(x',\tau')} = D \delta^d(x-x') \delta(\tau-\tau')
| |
| </math></center>
| |
| | |
| * From Wick's theorem, for a generic Gaussian field <math> W </math>, we have
| |
| <center><math>
| |
| \overline{\exp(W)} = \exp\left[\overline{W} +\frac{1}{2} \left(\overline{W^2}-\overline{W}^2\right)\right]
| |
| </math></center>
| |
| | |
| ===The first moment===
| |
| The first moment of the partition function is straightforward to compute and corresponds to a single replica:
| |
| | |
| <center>
| |
| <math>
| |
| \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]}
| |
| </math>
| |
| </center>
| |
| | |
| 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:
| |
| | |
| <center>
| |
| <math>
| |
| \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]
| |
| </math>
| |
| </center>
| |
| | |
| === The second moment ===
| |
| | |
| For the second moment, there are two replicas:
| |
| | |
| * Step 1: The second moment is
| |
| | |
| <center>
| |
| <math>
| |
| \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]}
| |
| </math>
| |
| </center>
| |
| | |
| * Step 2: Using Wick's theorem, we obtain
| |
| | |
| <center>
| |
| <math>
| |
| \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]
| |
| </math>
| |
| </center>
| |
| and we can write:
| |
| <center>
| |
| <math>
| |
| \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]
| |
| </math>
| |
| </center>
| |
| * Step 3: Changing coordinates <math>X=(x_1+x_2)/2; \; u=x_1-x_2</math>, we get
| |
| | |
| <center>
| |
| <math>
| |
| \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)}
| |
| </math>
| |
| </center>
| |
| 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>
| |
| | |
| ===The two replica propagator===
| |
| | |
| Let us define the propagator:
| |
| <center> <math>
| |
| 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]
| |
| </math>
| |
| </center>
| |
| Using the Feynman-Kac formula, we can write the following equation:
| |
| <center>
| |
| <math>
| |
| \partial_t W(x,t) = - \hat H W(x,t)
| |
| </math>
| |
| </center>
| |
| | |
| Here, the Hamiltonian is given by:
| |
| <center>
| |
| <math>
| |
| \hat H = - T \nabla^2 - \frac{D}{T^2} \delta^d[u]
| |
| </math>
| |
| </center>
| |
| | |
| === 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 <math>d \leq 2</math>:
| |
| 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:
| |
| <center>
| |
| <math>
| |
| W(x,t) \sim e^{ |E_0| t}
| |
| </math>
| |
| </center>
| |
| This implies that at all temperatures, in the limit <math>t \to \infty</math>:
| |
| <center>
| |
| <math>
| |
| \overline{\ln Z(x,t)} \ll \ln\overline{Z(x,t)}
| |
| </math>
| |
| </center>
| |
| | |
| * For <math>d > 2</math>:
| |
| 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>:
| |
| <center>
| |
| <math>
| |
| \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}
| |
| </math>
| |
| </center>
| |
| 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'''.
| |
![{\displaystyle E_{\min }[x]-E_{\min }[x']}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2c662761092e4a23c7637663414b6036af9ebd)
