Difference between corelation function and corelation lenght

[LagrangeEuler]

For example if we look Ising model correlation function is
$$\langle S_iS_{i+j}\rangle$$
So we can see that if we took into acount just nearest neighbours interraction we also will see that some $$i+j$$-th spin feels i-th spin. So there is some correlation between them.
Am I right? This is correlation function. And what is correlation lenght?

Is it length between
$\uparrow\uparrow\uparrow\uparrow\downarrow\downarrow$
the biggest number of align spins?

 

[LagrangeEuler]

If I watch some spin system, for example Ising model or something, what is for me there correlation lenght? Can you explain this to me?

 

[atyy]

I'm not sure if the usage in condensed matter is the same, but usually the correlation function is more general. When the correlation function is exponential, exp(-x/L), then L is the correlation length. If the correlation function is a power law, x^n, then the correlation length is not defined (or "infinite", eg. http://www-thphys.physics.ox.ac.uk/people/ClaudioCastelnovo/Talks/060418_Harvard.pdf says "power-law phase with infinite correlation length").

 

[LagrangeEuler]

Yes, I think it's the same. Just in condensed matter I have maybe

$$\Gamma=exp(-\frac{an}{L})$$

where $$a$$ is distance between nearest neighbours. I don't understand very well that if I see phase transition in some point then correlation length there is pretty large. So $$\Gamma$$ goes to zero. Right?

And where I can have polynomial dependence? In ordered phase. Can someone explain me that?

 

[atyy]

Yes, I think it's the same. Just in condensed matter I have maybe

$$\Gamma=exp(-\frac{an}{L})$$

where $$a$$ is distance between nearest neighbours. I don't understand very well that if I see phase transition in some point then correlation length there is pretty large. So $$\Gamma$$ goes to zero. Right?

And where I can have polynomial dependence? In ordered phase. Can someone explain me that?

The divergence of the correlation length only occurs at some phase transitions, eg. at the critical point , which is where the boundary between liquid and gas disappears. Kardar has some notes on this. He writes the correlation function as a power law multiplying an exponential (Eq II.46). The correlation function is approximately a power law only for distances less than the correlation length (Eq II.49). As the critical point is approached, the correlation length diverges, and so the correlation function is close to a power law over very large distances (see his comments before Eq II.52).
http://ocw.mit.edu/courses/physics/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2008/lecture-notes/lec2.pdf
http://ocw.mit.edu/courses/physics/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2008/lecture-notes/lec3.pdf
http://ocw.mit.edu/courses/physics/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2008/lecture-notes/lec4.pdf