- #1
svletana
- 20
- 1
I'm having some trouble understanding some of the steps done in the uploaded paper. I'ts the 1975 paper by Sherrington where they explain the SK model for spin glass.
Up to equation 5 I understand all steps. I used J_0 = J_0 / N and J^2 = J^2 / \sqrt[]{N} so they are intensive.
I have arrived at the following expression, similar to the one in the paper:
<br /> F_{av} = - k_B T \lim_{n \rightarrow 0} \frac{1}{n} \left[<br /> Tr_{s_i} \exp \left( \sum_{i \neq j} \sum_{\alpha = 1}^{n} \frac{\beta}{2} J_0 s_i^{\alpha} s_j^{\alpha} + \sum_{i \neq j} \sum_{\alpha, \gamma = 1}^n \frac{\beta^2 J^2 s_i^{\alpha} s_j^{\alpha} s_i^{\gamma} s_j^{\gamma}}{8} \right)<br /> - 1 \right]<br />
Afterwards I asume they use the identities
\sum_{i \neq j} s_i^{\alpha} s_j^{\alpha} = \frac{1}{2} \left[ \left(\sum_{i=1}^N s_i^{\alpha} \right)^2 - N \right]
\sum_{i \neq j} s_i^{\alpha} s_j^{\alpha} s_i^{\gamma} s_j^{\gamma} = \frac{1}{2} \left[ \left(\sum_{i=1}^N s_i^{\alpha} s_i^{\gamma} \right)^2 - N \right]
It's the next step I'm having trouble with (equation 6).
First of all, I don't understand what terms they are dropping. It says something vanishes in the thermodynamic limit but I'm not sure what it is.
If I factor out some terms I get this:
<br /> F_{av} = - k_B T \lim_{n \rightarrow 0} \frac{1}{n} \left\{<br /> Tr_{s_i} \exp\left( -\frac{N n \beta^2 J^2}{4} \right) \exp \left[<br /> \sum_{\alpha} \frac{J_0 \beta}{4} \left( \sum_i s_i^{\alpha} \right)^2 +<br /> \sum_{\alpha, \gamma} \frac{\beta^2 J^2}{8} \left( \sum_i s_i^{\alpha} s_j^{\gamma} \right)^2<br /> \right]<br /> -1\right\}<br />which is a little different from what they got, which is:
<br /> F_{av} = - k_B T \lim_{n \rightarrow 0} \frac{1}{n} \left\{<br /> Tr_{s_i} \exp\left( \frac{N n \beta^2 J^2}{4} \right) \exp \left[<br /> \sum_{\alpha} \frac{J_0 \beta}{2} \left( \sum_i s_i^{\alpha} \right)^2 +<br /> \sum_{\alpha, \gamma} \frac{\beta^2 J^2}{2} \left( \sum_i s_i^{\alpha} s_j^{\gamma} \right)^2<br /> \right]<br /> -1\right\}<br />
What steps am I missing?
Homework Statement
andHomework Equations
[/B]Up to equation 5 I understand all steps. I used J_0 = J_0 / N and J^2 = J^2 / \sqrt[]{N} so they are intensive.
I have arrived at the following expression, similar to the one in the paper:
<br /> F_{av} = - k_B T \lim_{n \rightarrow 0} \frac{1}{n} \left[<br /> Tr_{s_i} \exp \left( \sum_{i \neq j} \sum_{\alpha = 1}^{n} \frac{\beta}{2} J_0 s_i^{\alpha} s_j^{\alpha} + \sum_{i \neq j} \sum_{\alpha, \gamma = 1}^n \frac{\beta^2 J^2 s_i^{\alpha} s_j^{\alpha} s_i^{\gamma} s_j^{\gamma}}{8} \right)<br /> - 1 \right]<br />
Afterwards I asume they use the identities
\sum_{i \neq j} s_i^{\alpha} s_j^{\alpha} = \frac{1}{2} \left[ \left(\sum_{i=1}^N s_i^{\alpha} \right)^2 - N \right]
\sum_{i \neq j} s_i^{\alpha} s_j^{\alpha} s_i^{\gamma} s_j^{\gamma} = \frac{1}{2} \left[ \left(\sum_{i=1}^N s_i^{\alpha} s_i^{\gamma} \right)^2 - N \right]
It's the next step I'm having trouble with (equation 6).
The Attempt at a Solution
First of all, I don't understand what terms they are dropping. It says something vanishes in the thermodynamic limit but I'm not sure what it is.
If I factor out some terms I get this:
<br /> F_{av} = - k_B T \lim_{n \rightarrow 0} \frac{1}{n} \left\{<br /> Tr_{s_i} \exp\left( -\frac{N n \beta^2 J^2}{4} \right) \exp \left[<br /> \sum_{\alpha} \frac{J_0 \beta}{4} \left( \sum_i s_i^{\alpha} \right)^2 +<br /> \sum_{\alpha, \gamma} \frac{\beta^2 J^2}{8} \left( \sum_i s_i^{\alpha} s_j^{\gamma} \right)^2<br /> \right]<br /> -1\right\}<br />which is a little different from what they got, which is:
<br /> F_{av} = - k_B T \lim_{n \rightarrow 0} \frac{1}{n} \left\{<br /> Tr_{s_i} \exp\left( \frac{N n \beta^2 J^2}{4} \right) \exp \left[<br /> \sum_{\alpha} \frac{J_0 \beta}{2} \left( \sum_i s_i^{\alpha} \right)^2 +<br /> \sum_{\alpha, \gamma} \frac{\beta^2 J^2}{2} \left( \sum_i s_i^{\alpha} s_j^{\gamma} \right)^2<br /> \right]<br /> -1\right\}<br />
What steps am I missing?