Mean Field Theory: Get Book Recommendations

[sridhar]

I have just started reading up on mean field theory. Any book you could refer??

 

[Chris Hillman]

What books have you read so far? I can think of one book which I would recommend which would be confusing unless you have read more modern books.

 

[sridhar]

I have done upto the mean field treatment for ising model from
"Introduction to stat mech" Chandler which is by no means exhaustive.
I have also read Kerson Huang but again only the Ising model is discussed briefly.
I need a book which deals with few more models in detail.

 

[Chris Hillman]

A-haaa! So you want the Ising model, not mean-field theory in general? Is it possible that your real question is: what is a good introduction to the theory of the Ising model?

For example, if you don't know about coupling from the past you are killing yourself by limiting yourself to mean-field theory!

 

[sridhar]

nono. Not at all. A book which introduces mean field theory. Not at all specific to the Ising model. In fact, I need to see how mean field theory treats Heisenberg and n-potts state(I like calling it that!) models. How it becomes exact for infinite dimensions, etc

 

[Chris Hillman]

Hmm... not sure I can help you there. Is Kenneth Wilson in the house?

Seriously, if you don't know about coupling from the past you won't regret learning about it.

 

[sridhar]

I would love to learn more. The only coupling I know of is nearest neighbour for Ising model. Basically haven't ventured beyon spin 1/2 systems!
I am sure I can find it in a library. I would really appreciate some help with my mean field theory issue. I can't find a book which describes various lattice models and their mean field treatments.

 

[Chris Hillman]

Coupling from the past allows one to obtain a sample from the exact critical distribution, say for the Ising model. The nice undergraduate textbook by Haggstrom, Finite Markov Chains and Algorithmic Applications, London Mathematical Society student texts Vol. 52, University of Cambridge Press, 2002, is basically an introduction to this powerful and amazing technique, which can be used to give reliable Monte Carlo estimates of quantities of interest. There are many eprints on the arXiv which have been inspired by the original paper by Wilson and Propp. (See Fig. 12 in this book for a simulation of the 2x2 Ising model on a 15 by 15 square. If that sounds unimpressive, remember that this simulation uses the exact stationary distribution, not an approximation to it, which basically avoids all the problems of mean-field or whatnot.)