# Why correlation function decay as power law at critical temperature point?

1. Aug 11, 2011

### ndung200790

In QTF theory of Schroeder,chapter 13.1 saying:
Just at t=0(t=$\frac{T-T_{c}}{T_{c}}$),the correlation should decay as power law.
Define the exponent $\eta$ by the formula:
G(x)=$\frac{1}{x^{d-2+\eta}}$
where d is Euclidien space dimension.
I do not understand why at critical point, the correlation function decay as power law.Please give me a to favour to explain this.
Thank you very much in advance.

2. Aug 12, 2011

### ndung200790

I would like to add that I am saying about the second phase transition in statistical physics using method of Quantum Field Theory

3. Aug 12, 2011

### vkroom

as one approaches a critical point the correlation length starts increasing, i.e. units farther and farther away becomes aware of each others dynamics.
in math terms the correlation functions $\sim e^{-r/\chi}$, where $\chi$ is the correlation length, and $r$ is the distance from your origin. this $\chi$ grows as one approaches the critical point.

this feature reaches its maximum at the critical point where $\chi \rightarrow \infty$, i.e. every part of the system becomes aware of every other part. this is the reason the correlation functions become algebraic from exponential. one can approximately see this effect if one writes $r^{-a} = e^{-\frac{r}{r(a\ln{r})^{-1}}}$. compare it with $e^{-r/\chi}$. one sees that in the critical case there is no length scale like $\chi$. hence correlation functions don't decay over any characteristic length scale. therefore they have algebraic decay, which lacks any length scale.