# 2D Ising model partition function expansion

By combinatorial reasons, for 2D toric Ising model of fixed size I need low-temperature expansion for complex temperature, magnetic field, etc. Is there are some references? More precisely, energy of spin configuration $x_i=+1,-1$ given by $-H=\sum_{edges}(-N+a)(x_i+x_j)-Nx_ix_j$ for some $a\in i\mathbb{R}$, where sum going over all edges in lattice-graph on torus surface, and all what i need is asymptotics in first term of partition function $\sum_{states} e^{-H}$ when $N\to +\infty$

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I don't think Onsager's solution will help you in the presence of a magnetic field. In the low temperature regime there are rigorous expansions for pretty much all the quantities of interest. These go under the names of low temperature cluster expansions, contour expansions, Pirogov-Sinai theory (the most general theory for this kind of things). You can find presentations of these tools especially in the archetypal example of the Ising model in just about any book on Gibbs measures: e.g. the books by Malyshev and Minlos, Preston, Georgii, Sinai, etc. A good place to start is the lecture notes by Velenik. If you do not read French then look up the references in these lecture notes.

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Before learning all this stuff, where i can find precise statement for ising model expansion? Thx –  Bad English Oct 19 '12 at 7:29
You will need a more physical oriented book like Giorgio Parisi "Statistical Field Theory" amazon.com/Statistical-Field-Theory-Advanced-Classics/dp/…. If you need some more rigorous results but not specifically for Ising model, Ruelle's book is the right one books.google.it/…. –  Jon Oct 19 '12 at 7:53

This problem is known in literature due to the Onsager's solution of the 2d Ising model. You can already find this on an old paper by Fisher and Ferdinand. Some more recent papers, available also from arxiv are the following:

http://arxiv.org/abs/cond-mat/0009054

http://arxiv.org/abs/cond-mat/0110287

A paper just available with subscription is Partition function of a finite Ising model on a torus T Morita 1986 J. Phys. A: Math. Gen. 19 L1191.

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