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I'm working on a variation of the Ising Model for my undergraduate thesis and I need the exact correlation function of neighbouring sites $<\sigma_{1,1}\sigma_{1,2}>$ to compare with my results. So far I've checked on many papers and books and the only close answer is in this one "Onsager and Kaufman's calculation of the spontaneous magnetization of the Ising model" by R.J. Baxter. However, they give a very complicated way of obtaining the correlation function of any two spins so the first neighbours corelation is not easy to derive.

I also thought that it would be easy to just do $\frac{1}{N}\frac{\partial}{\partial J}F$ since that would drop a $\sigma_i\sigma_j$ and would do the trick perfectly. However, I can't find a closed expression for the free energy F in any book. Every one that I look is expressed in terms of unsolved integrals or sums.

I would thank anybody who can give me a clue or the name of a paper where I can find this more easily.

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  • $\begingroup$ rewrite it in terms of fermions (i.e. use Jordan-Wigner) and then use Wick's theorem $\endgroup$
    – Ruben Verresen
    Commented Dec 29, 2017 at 22:25
  • $\begingroup$ I always thought that the relation between the fermions and the original Ising Variables was not trivial i.e. it is not obvious that the two point function for fermions is equivalent to the correlation between Ising variables... Is that the case? Thanks! $\endgroup$
    – P. C. Spaniel
    Commented Dec 29, 2017 at 22:28
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    $\begingroup$ @P.C.Spaniel : why do you expect a closed form expression (for the limiting free energy density) to exist? There is no reason why we should be able to express it directly in terms of well-known functions... The expression involving the integral is actually not so ugly and should certainly do for a numerical comparison, for example. $\endgroup$
    – Yvan Velenik
    Commented Apr 27, 2018 at 15:26

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There is an exact solution to the partition function of the 2d classical Ising model on a finite-size lattice with periodic boundary conditions in Kaufman's original paper Phys. Rev. 76, 1232 (1949) (see Eq. (39) in this paper). You can obtain your derivative numerically using the exact partition function.

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