# How to calculate critical temperature of the Ising model?

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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?

This post imported from StackExchange Physics at 2014-07-13 04:42 (UCT), posted by SE-user user41404
Almost quantum statistics books include this part.

This post imported from StackExchange Physics at 2014-07-13 04:42 (UCT), posted by SE-user qfzklm
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.

This post imported from StackExchange Physics at 2014-07-13 04:42 (UCT), posted by SE-user suresh

I don't understand the -1. Did I say something wrong/nasty?

@suresh, I have also noticed quite some mysterious -1, all on the comments imported from SE.

@suresh: It was a knee jerk downvote from me, reacting to "someone at your level", since Kramers Wannier duality is really accessible for a high-school student. I was stupid, and I removed the downvote, and sorry, long sleepless night, mistake of judgement, etc.

@Ron_Maimon Interesting and correct point and I should have mentioned Kramers-Wannier duality.

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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.

answered Oct 6, 2014 by (1,110 points)
edited Oct 6, 2014
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You can perform high and low temperature series expansion of the partition function. Then, using the Kramers-Wannier duality, it is possible to locate the critical temperature.

You can read about that in Julia Yeomans' textbook Statistical Mechanics of Phase Transitions. It has a short section with 10 pages about it, there are no prerequisites except basic statistical mechanics to learn this.

answered Oct 6, 2014 by (50 points)

That's correct. But note that it is not at all trivial to prove that this indeed gives the critical temperature. To achieve that is precisely what is done in the paper I cite in my answer.

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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.

This post imported from StackExchange Physics at 2014-07-13 04:42 (UCT), posted by SE-user MarkWayne
answered Feb 27, 2014 by (270 points)
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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.

This post imported from StackExchange Physics at 2014-07-13 04:42 (UCT), posted by SE-user LC7
answered Feb 27, 2014 by (10 points)

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