Sherrigton-Kirkpatrick model for spin glass question

[svletana]

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.

Homework Statement

and

Homework 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:

$$F_{av} = - k_B T \lim_{n \rightarrow 0} \frac{1}{n} \left[ Tr_{s_i} \exp \left( \sum_{i \neq j} \sum_{\alpha = 1}^{n} \frac{\beta}{2} J_0 s_i^{\alpha} s_j^{\alpha} + \sum_{i \neq j} \sum_{\alpha, \gamma = 1}^n \frac{\beta^2 J^2 s_i^{\alpha} s_j^{\alpha} s_i^{\gamma} s_j^{\gamma}}{8} \right) - 1 \right]$$

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:

$$F_{av} = - k_B T \lim_{n \rightarrow 0} \frac{1}{n} \left\{ Tr_{s_i} \exp\left( -\frac{N n \beta^2 J^2}{4} \right) \exp \left[ \sum_{\alpha} \frac{J_0 \beta}{4} \left( \sum_i s_i^{\alpha} \right)^2 + \sum_{\alpha, \gamma} \frac{\beta^2 J^2}{8} \left( \sum_i s_i^{\alpha} s_j^{\gamma} \right)^2 \right] -1\right\}$$which is a little different from what they got, which is:

$$F_{av} = - k_B T \lim_{n \rightarrow 0} \frac{1}{n} \left\{ Tr_{s_i} \exp\left( \frac{N n \beta^2 J^2}{4} \right) \exp \left[ \sum_{\alpha} \frac{J_0 \beta}{2} \left( \sum_i s_i^{\alpha} \right)^2 + \sum_{\alpha, \gamma} \frac{\beta^2 J^2}{2} \left( \sum_i s_i^{\alpha} s_j^{\gamma} \right)^2 \right] -1\right\}$$

What steps am I missing?

 

[Nate810]

How do I get to the expression they got?The main difference is that in my expression there are exponents of -Nn\beta^2J^2/4 and in theirs there is +Nn\beta^2J^2/4. I'm not sure where this comes from.