Consider the 2D classical Ising model. It's understood that there is a critical temperature $T_c$, and that the correlation length $\xi(T)$ defined by: $$\langle \sigma_i \sigma_j \rangle_\mathrm{connected., T} := \langle \sigma_i \sigma_j \rangle_T - \langle \sigma_i \rangle_T \langle \sigma_j \rangle_T \to Z^2 \mathrm{exp}\left(- \frac{|i-j|}{\xi(T)}\right)$$ is finite away from $T_c$ but diverges as $1/(T - T_c)$ near the critical temperature. At the critical temperature, it's clear that we should expect scale invariance as the correlation length diverges, formally I expect that: $$\langle \sigma_i \sigma_j \rangle_\mathrm{connected., T_c} \approx \frac{Z^2}{|i - j|^\alpha}$$ for some $\alpha$ (and the $\approx$ is accounting for lattice artifacts at small separations?) What I'm confused about is considering the correlation function: $$f\left(i,j\right) = \lim_{T \to T_c} \frac{\langle \sigma_{\xi(T) \cdot i } \sigma_{\xi(T) \cdot j} \rangle_{\mathrm{connected},T}}{\langle \sigma_{\xi(T) \cdot i_0 } \sigma_{\xi(T) \cdot j_0} \rangle_{\mathrm{connected},T}}$$ where $i_0,j_0$ are constant lattice sites (required so that $f$ is nonzero(?)) What is the behaviour of $f(i,j)$, for large $|i-j|$? Is it exponentially decreasing, or power-law?

(Apologies if this is a duplicate, I couldn't find any times this was asked previously)

  • $\begingroup$ I don't think you have written what you had in mind: if $i\neq j$, the limit will be equal to zero, since the distance between the spins diverges. Of course the gist of your question is clear, although stating the results is not so obvious. For instance, read §3 in this review paper. $\endgroup$ Commented Jan 28, 2023 at 9:36
  • $\begingroup$ @YvanVelenik I didn't realise the limit would probably be zero; I've 'renormalized' my correlation function now so hopefully it's nonzero. I'll take a look at the review paper and see if I can write down the answer, thanks $\endgroup$ Commented Jan 28, 2023 at 15:44

1 Answer 1


As far as I can understand, this question probably concerns the correlation function of two spins in the so-called scaling limit. The "renormalized" coordinates of the spins $i' = \xi(T)i,\ j' = \xi(T)j\ $ satisfy the scaling condition $$ \lim_{T\to T_c\pm 0}|i' - j'||T-T_c| = t = С_{\pm} |i-j| \neq 0 $$ with $C_{\pm} = \lim_{T\to T_c \pm 0} |T-T_c|\xi(T)$. Therefore, the correlation function has the following asymptotics as $T$ approaches $T_c$: $$ \langle \sigma_{i'}\sigma_{j'} \rangle \sim M_{\pm}^2 G_{\pm}(t). $$ The index $\pm$ here refers to two cases, $T <T_c$ and $T >T_c$. $G_{\pm}(t)$ are scaling functions that have exact expressions in terms of a Painleve function of the third kind. $M_{\pm}$ are some functions of $T$, $M_{-}$ coincides with magnetization.

You can find much more details, including the asymptotic properties of G, in the following sources:

Article on Scholarpedia about the Ising model

The book of McCoy and Wu

Article by McCoy and Wu


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