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The two dimensional square lattice Ising model reads $$E[\sigma]=-J\displaystyle\sum_{<ij>}\sigma_i\sigma_j-h\displaystyle\sum_i\sigma_i,$$

where $E$ is the energy, $\sigma_i$ is the spin at lattice position $i$, each taking the value $1$ or $-1$ and $J$, $h$ are coupling constants. Now there is a statement (see Conformal Field Theory by Di Francesco etc., pp.63) that the case $J>0$ is equivalent to that of $J<0$ when $h=0$. How to understand that?

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  • $\begingroup$ 1. Always explain your notation! Although one can guess what $E[\sigma]$ and the $\sigma_i$ are, it's not stated anywhere within the post. What does $\sum_{<ij>}$ mean? 2. "there is a statement" - where is that statement? Who made it? $\endgroup$
    – ACuriousMind ♦
    Commented Sep 16, 2016 at 14:55
  • $\begingroup$ @ACuriousMind, I am soryy. I will modify my post. $\endgroup$
    – Wein Eld
    Commented Sep 16, 2016 at 15:07
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    $\begingroup$ This really only works for bipartite lattices, a statement you never make in your question. $\endgroup$
    – Norbert Schuch
    Commented Sep 16, 2016 at 15:13
  • $\begingroup$ @NorbertSchuch I think you mean ``$\sigma_i$ is the spin at lattice position $i$, each taking the value $1$ or $-1$'', right? $\endgroup$
    – Wein Eld
    Commented Sep 16, 2016 at 15:17
  • $\begingroup$ No, I mean that what you say is e.g. not true on a triangular lattice, but only on square, honeycomb, ... $\endgroup$
    – Norbert Schuch
    Commented Sep 16, 2016 at 15:28

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See, e.g., http://physweb.bgu.ac.il/COURSES/StatMechCohen/ExercisesPool/EXERCISES/ex_5721_sol_Y12.pdf . You can divide your lattice in two lattices (the same way as a chess board is divided into white and black "lattices" and, say, replace $s_i$ by $-s_i$ for the "black lattice".

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