# How to calculate critical temperature of the Ising model?

Can someone name a paper or book which calculates the critical temperature of the Ising model from scratch? It might be a book and should contain the necessary prerequisites. I have had a basic course in stat physics and thermodynamics.

Edit: The two suggested books have 500 pages of preface, is this necessary or is there a more compact source available?

• Almost quantum statistics books include this part. Feb 27 '14 at 15:09
• The exact solution of the 1d Ising model should be easy for someone at your level to digest. The 2d Ising model does not fall in that category and will need a lot more work. arxiv.org/abs/cond-mat/0104398 by Boris Kastening is a nice place to work through the 2d Ising model. Mar 5 '14 at 23:45

## 3 Answers

A definitive volume, one that I learned from during graduate school, is Kerson Huang's (of MIT, emeritus of the Physics Dept.) Statistical Mechanics. The book covers both classical and quantum computations of the partition function and observables from it, as well as thermodynamics, kinetic theory, transport, superfluids, critical phenomena, and the Ising model. Chapters 14 and 15 are devoted to the Ising model.

I suggest you:

Statistical Mechanics: Theory and Molecular Simulation Mark E. Tuckerman

There are all the necessary prerequisites and the discussion about Ising model and critical points. I don't know if there's online.

In contrast to the other answers, I would like to mention that it is possible to compute rigorously the value of the critical temperature of the two-dimensional Ising (and Potts) model, without computing explicitly the free energy (which is in any case not possible for general Potts models). In the Ising case, this has been known for a long time, and there are various proofs. The first result valid for all Potts models is this one. Note that it also establishes sharpness of the phase transition (that is, the fact that correlation decay exponentially fast as soon as $\beta<\beta_c(q)$).

In the case of the two-dimensional Ising model, one can also compute the critical temperature for general doubly periodic lattices. A proof can be found here.