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Difference between corelation function and corelation lenght

  1. Aug 14, 2012 #1
    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
    the biggest number of align spins?
  2. jcsd
  3. Aug 14, 2012 #2
    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?
  4. Aug 15, 2012 #3


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    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").
  5. Aug 16, 2012 #4
    Yes, I think it's the same. Just in condensed matter I have maybe


    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?
  6. Aug 16, 2012 #5


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    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 [Broken]
    http://ocw.mit.edu/courses/physics/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2008/lecture-notes/lec3.pdf [Broken]
    http://ocw.mit.edu/courses/physics/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2008/lecture-notes/lec4.pdf [Broken]
    Last edited by a moderator: May 6, 2017
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