Sum of all two point correlation functions in the Ising model

Question

The two point correlation function for the Ising model is defined as $left[langlesigma_isigma_jrangle  -langlesigma_iranglelanglesigma_jrangleright]$. Then the sum over $i$ $j$ of that function gives:
begin{equation}
sum_{ij}left[langlesigma_isigma_jrangle  -langlesigma_iranglelanglesigma_jrangleright] = langle M^2rangle - langle Mrangle^2
end{equation}
Where $sigma_i = pm 1$ and $M  =sum_i sigma_i$. The expression can be normalized by dividing it by $M^2 = N^2$ where $N$ is the number of spins of the system. My question is, I am assuming I can justify the long range correlations in the lattice by computing the sum of the correlation functions, is that correct? I should expect it to vanish when most of the two point correlation vanish, and to have  peak when most two point correlation reach its maximum value.

Yvan Velenik · Answer

Let us be a bit more precise, so that the question can be answered accurately.
Let us thus assume that your system is composed of $N^d$ spins attached to the vertices of the box $Lambda_N={1,dots,N}^d$. I am assuming that there is a positive magnetic field $h>0$ acting on the spins (this is technically useful to ensure that we are considering the proper state below, but the field will soon be set equal to $0$). Let me denote by $langlecdotrangle_{N,beta,h}$ the corresponding expectation.
Then, the susceptibility is given by
$$
chi_N(beta,h) = 
frac1{N^d}sum_{i,jinLambda_N} Bigl[ langle sigma_isigma_j rangle_{N,beta,h} - langle sigma_i rangle_{N,beta,h} langle sigma_j rangle_{N,beta,h} Bigr].
$$
We are really interested in the thermodynamic limit of this quantity,
$$
chi(beta,h) = lim_{Ntoinfty} chi_N(beta,h) =
sum_{iinmathbb{Z}^d} Bigl[ langle sigma_0sigma_i rangle_{beta,h} - langle sigma_0 rangle_{beta,h} langle sigma_i rangle_{beta,h} Bigr],
$$
where I used the fact that the resulting infinite-volume state is translation invariant. We can now get rid of the magnetic field and define
$$
chi(beta) = lim_{hdownarrow 0} chi(beta,h)
= sum_{iinmathbb{Z}^d} Bigl[ langle sigma_0sigma_i rangle_{beta}^+ - langle sigma_0 rangle_{beta}^+ langle sigma_i rangle_{beta}^+ Bigr],
$$
where $langle cdot rangle_beta^+$ denotes expectation with respect to the $+$ state.
It is known, in any dimension $d$, that the truncated 2-point function decays exponentially with the distance when $betaneqbeta_{rm c}$. Namely, for all  $betaneqbeta_{rm c}$, there exists $c=c(beta,d)>0$ such that
$$
0leq langle sigma_0sigma_i rangle_{beta}^+ - langle sigma_0 rangle_{beta}^+ langle sigma_i rangle_{beta}^+ leq e^{-c|i|}.
$$
(In other words, the correlation length is finite at all non-critical temperatures.) This immediately implies that the susceptibility is finite away from the critical point:
$$
chi(beta) < inftyqquadforallbetaneqbeta_{rm c}.
$$
Moreover, it is also known that, in all dimensions $dgeq 2$, the 2-point function does not decay exponentially fast when $beta=beta_{rm c}$ (much more precise information is available when $d=2$ and when $d$ is large enough).
Finally, it is known that the susceptibility diverges as $betauparrowbeta_{rm c}$:
$$
lim_{betauparrowbeta_{rm c}} chi(beta) = +infty.
$$

Sum of all two point correlation functions in the Ising model

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