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Let's take the Ising model as an example and study the two point spin spin correlation function: $$\langle s_0 s_r\rangle = \frac{\sum_{\{s_i\}}e^{K\sum_{\langle i ,j\rangle}s_i s_j} s_0 s_r}{\sum_{\{s_i\}}e^{K\sum_{\langle i ,j\rangle}s_i s_j} }.$$ In high temperature, i.e., when $K$ is small, the two point correlation function would decays exponentially: $$G(r)\equiv\langle s_0 s_r\rangle \sim \exp(-r/\xi).$$ In momentum space, the two point correlation function would becomes: $$G(k)\sim \frac{1}{k^2+ 1/\xi^2}.$$ I think that in real space, the meaning of the correlation function is straightforward to understand, but how to understand the form $$G(k)\sim \frac{1}{k^2+ 1/\xi^2}$$ in momentum space directly? What is the physics picture in momentum space?

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If the spins are at positions $\bf R$, it is possible to define a $\bf k$-dependent collective variable $s_{\bf k}$ (Fourier component of the spin vector configuration) as: $$ s_{\bf k}=\sum_{\bf R} e^{i\bf k \cdot R}s_{\bf R} $$ (maybe with a normalization factor depending on the exact choice of definition).

The k-space two-point correlation function is the Fourier transform of the spin-spin correlation function in r-space G($\bf R$,$\bf R'$)= $\left< s_{\bf R}s_{\bf R'} \right>$, which, for a translationally invariant system is also equal to $\left< s_{\bf 0}s_{\bf R'-R} \right>$, so that $$ G({\bf k})=\left< s^*_{{\bf k}}s^~_{{\bf k}} \right> = \left< s_{{\bf k}}s_{{\bf -k}} \right>. $$ From this formula, and taking into account that for non zero wavevectors $s_{{\bf k}}$ can be interpreted as a fluctuation of spin density, the physical meaning of $G({\bf k})$ is of correlation between density fluctuations of the same wavevector.

It is an especially important quantity because it is possible to show that it is the most important factor depending on the spin values and positions, of the neutron scattering cross section. Therefore, it provides a direct method to measure two-point correlations in real magnetic systems.

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If you think about $G$ as a propagator, the exponential decay of the real space $G(r)$ describes a massive particle with mass $m = 1/\xi$. In $k$ space this corresponds to the idea that we can read off particle masses from the poles of the $G(k)$.

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