Difference between corelation function and corelation lenght

LagrangeEuler

For example if we look Ising model correlation function is
[tex]\langle S_iS_{i+j}\rangle[/tex]
So we can see that if we took into acount just nearest neighbours interraction we also will see that some [tex]i+j[/tex]-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 lenght between
[itex]\uparrow\uparrow\uparrow\uparrow\downarrow\downarrow[/itex]
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/p...18_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

[tex]\Gamma=exp(-\frac{an}{L})[/tex]

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

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

atyy

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...notes/lec2.pdf
http://ocw.mit.edu/courses/physics/8...notes/lec3.pdf
http://ocw.mit.edu/courses/physics/8...notes/lec4.pdf