# Correlation function and spontaneous symmetry breaking

Let's look at ferromagnetic system as an example.

The correlation function is defined: $$G(r,r_{0})=\langle(m_{r_{0}}-\langle m_{r_{0}}\rangle)(m_{r}-\langle m_{r}\rangle)\rangle=\frac{1}{r^{p}}e^{-\frac{|r-r_{0}|}{\xi}}$$. In my understanding, When $$G(r,r')\neq 0$$, it means the system is somewhat ordered and $$G(r,r')=0$$ for disordored system.

When study the impact of dimension on whether spontaneous symmetry happens, we may consider the tranverse correlation function $$G_{\perp}(r)=\int \text{d}^{d}k G_{\perp}(k)e^{ikr}\propto \int \text{d}k k^{d-1}\frac{e^{ikr}}{k^{2}}.$$

When $$d\leq2$$, in the limit $$k\rightarrow0$$, $$G_{\perp}(r)$$ is divergent and hence the ordered state will collapse thus no spontaneous symmetry happens.

I understand that since the susceptibility function, which measure the influence of change of external field in position $$r_{0}$$ on the order parameter in position $$r$$: $$\chi_{r,r0}=\frac{\partial m_{r}}{\partial h_{r_{0}}}=\frac{1}{k_{B}T}G_{r,r_{0}}$$ is also divergent. So a infinisimal fluctutaion of $$h_{r_{0}}$$ will cause amountable change in $$m_{_{r}}$$

Here is the problem, divergece of correlation function seems to keep the system in a disordered system. Yet in the contents in bold font, I think non-zero correlation function means order. These two clearly contradict. Could anyone help to give a clearer physical picture?

Another question maybe relevant to this is: spontaneous symmetry always happens from state with higher sysmmetries to state with lower symmetries. How come a disordered system can have more symmetries than an ordered system?

First, 'ordered' does not mean there's more symmetry. Ordered phase means order parameter select a specific nonzero value. Think about Ising $$Z_2$$ case. If total magnetization is zero, system is $$Z_2$$ symmetric(i.e. if you flip all the spins your magnetization is still zero). However, if nonzero magnetization appears, then $$Z_2$$ is broken. Therefore, ordered phase has 'lower' symmetry, and disordered phase has higher symmetry.
Also, about your bold fonted text, disordered does not exactly mean $$G(r, r')$$ is zero, but exponential decay is sufficient. If it's algebraic decay, we say it's quasi-long range order. If it's constant then it is a long-range order.