Ising model rescaling

Question

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)

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