# How to understand the two-point correlation function in momentum space?

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?

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.
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 $$G(k)$$.