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 $$G_{\perp}(r)=\int \text{d}^{d}k G_{\perp}(k)e^{ikr}\propto \int \text{d}k k^{d-1}\frac{e^{ikr}}{k^{2}}. $$

The textbook I've read says:

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.

Divergence in your calculation is an artifact of assuming nonzero value of order parameter. When you calculate your transverse correlation function you probably assumed specific axis of magnetization, and nonzero value of magnetization along that axis. Therefore, divergence in your transverse correlation function means that assumption is wrong and there's no nonzero magnetization, which is disordered phase. If you calculate the correlation function without assuming the existence of magnetization, you will get zero(or exponential decay).


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