Sherrigton-Kirkpatrick model for spin glass question

In summary, the conversation is about understanding a paper on the SK model for spin glass by Sherrington in 1975. The speaker is having trouble with the next step (equation 6) and is unsure about which terms are being dropped and why there is a difference in the exponents between their expression and the one in the paper. They also mention using identities to simplify the expression.
  • #1
svletana
21
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.

Homework Statement

and

Homework Equations

[/B]
Up to equation 5 I understand all steps. I used [itex]J_0 = J_0 / N[/itex] and [itex]J^2 = J^2 / \sqrt[]{N}[/itex] so they are intensive.
I have arrived at the following expression, similar to the one in the paper:

[tex]
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]
[/tex]

Afterwards I asume they use the identities

[tex]\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][/tex]

[tex]\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][/tex]

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:

[tex]
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\}
[/tex]which is a little different from what they got, which is:

[tex]
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\}
[/tex]

What steps am I missing?
 

Attachments

  • sherrington1975.pdf
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  • #2
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.
 

Related to Sherrigton-Kirkpatrick model for spin glass question

1. What is the Sherrington-Kirkpatrick model for spin glass?

The Sherrington-Kirkpatrick model is a mathematical model used to study the behavior of spin glasses, which are materials with randomly arranged magnetic moments. It was proposed by British physicist John Sherrington and American physicist Dan Kirkpatrick in 1975.

2. How does the Sherrington-Kirkpatrick model work?

The Sherrington-Kirkpatrick model uses a mathematical framework called the Ising model to describe the interactions between the magnetic moments in a spin glass. It assumes that each magnetic moment can have only two possible states, "up" or "down", and that these moments interact with each other through a certain energy function.

3. What are the applications of the Sherrington-Kirkpatrick model?

The Sherrington-Kirkpatrick model has been used to study a variety of physical systems, such as neural networks, protein folding, and disordered systems. It has also been applied in fields such as computer science, economics, and sociology.

4. What are the limitations of the Sherrington-Kirkpatrick model?

While the Sherrington-Kirkpatrick model has been successful in explaining certain physical phenomena, it has its limitations. For example, it assumes that the interactions between magnetic moments are the same for all pairs, which may not be the case in real spin glasses. It also does not take into account the effects of temperature and external magnetic fields.

5. How is the Sherrington-Kirkpatrick model relevant to current research?

The Sherrington-Kirkpatrick model is still a topic of active research, particularly in the field of spin glasses. Recent studies have used the model to investigate the properties of complex systems, such as neural networks and financial markets. It continues to be a valuable tool in understanding the behavior of disordered materials and systems.

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