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The correlation length $\xi$ is related to critical temperature $T_c$ as $$ \xi\sim|T-T_{c}|{}^{-\nu}, $$

where $\nu$ is the critical exponent.

  1. Is this the formal definition of correlation length? If not, what is the formal definition of correlation length (for phase transition in the Ising model)?
  2. Can you give a physical understanding of correlation length?
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That is not a definition of correlation length. (It is a definition of the critical exponent.)

The correlation length is defined in terms of the 2-point correlation function of spin observables. Pick points $x$ and $y$ on the lattice, and consider the expectation value $\langle s(x) s(y) \rangle$ of the product of the spin observable at $x$ and the spin observable at $y$. This quantity tells you how strongly correlated the spin at $x$ and the spin at $y$ are, as a function of the temperature, coupling constant, and the distance between $x$ and $y$. If $T > T_c$, then the correlation function dies off exponentially fast in $|x-y|$.

$\langle s(x) s(y) \rangle \sim e^{-\frac{|x-y|}{\xi(T)}}$

The correlation length is, by definition, the constant (in $x$ and $y$, but not in $T$) which tells you how fast the correlation function vanishes.

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    $\begingroup$ So how will you give a formal definition of the correlation length (for Ising model)? $\endgroup$ – cosmicraga Apr 1 '13 at 14:05
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    $\begingroup$ What I wrote above is the formal definition: the correlation length is the exponential decay rate of the 2-point correlation function. $\endgroup$ – user1504 Apr 1 '13 at 14:19
  • $\begingroup$ Correlation length is found out by plotting $\langle s(x) s(y) \rangle $ vs. r and it is the length where the curve first changes its sign by crossing the r axis. $\langle s(x) s(y) \rangle \sim e^{-\frac{|x-y|}{\xi(T)}}=0$. \\ $-\frac{|x-y|}{\xi(T)} = 1$ \\ $|x-y| = -\xi(T)$ is it correct ? $\endgroup$ – cosmicraga Apr 1 '13 at 14:44
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    $\begingroup$ No, that's wrong. $\langle s(x) s(y)\rangle$ never changes sign. You can extract $\xi$ by looking at the ratio $\frac{\langle s(0) s(y)\rangle}{\langle s(0) s(2y)\rangle}$. $\endgroup$ – user1504 Apr 1 '13 at 15:02
  • $\begingroup$ Also, you've botched the math: $e^a = 0$ does not imply $a=1$. $\endgroup$ – user1504 Apr 1 '13 at 15:20
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Just a small addition to what user1504 said: the correlation length can be defined for $T<T_c$ as well, so that $\big\langle\big(s(x)-\langle s(x) \rangle \big)\big(s(y)-\langle s(y) \rangle \big)\big\rangle=e^{-\frac{|x-y|}{\xi}}$

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    $\begingroup$ Yes, +1, thanks for adding this. (I was being lazy, so I only discussed the case $T > T_c$, where $\langle s(x) \rangle = 0$. ) $\endgroup$ – user1504 Apr 1 '13 at 19:36
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The two-dimensional square-lattice Ising model, which is a simplified model of reality exhibits phase transition. Onsager showed that there is a specific temperature, called the Curie temperature or critical temperature, $T_c$ below which the system shows ferromagnetic long-range order. Above it, it is paramagnetic and is disordered.

At zero temperature, every spin is aligned in either +1 (or -1) direction. When we increase the temperature, keeping below $T_c$, some spin of starts orienting themselves in the opposite direction. The typical length scale of cluster forming is called correlation length, $\xi$, and it grows as we increase the temperature and diverges at $T_c$. If we go beyond $T_c$, the correlation length starts decreasing, and at the infinite temperature, it becomes zero.

enter image description here

2-dimensional Ising model simulation on 100x100 lattice. Left to right and top to bottom, the temperature is increasing. At equilibrium, when $T<T_c$, typical configurations in the + phase look like a sea'' of + spins withislands'' of $-$ spins. For larger lattice size, the island'' havelakes'' of + spins. In this picture, + spins are in black, $-$ spins are in white. Each connected white object is a cluster.

Formaly The two-point correlation function defined as \begin{equation}\label{Gamma} \Gamma(i-j)=\langle S_iS_j\rangle-\langle S_i\rangle\langle S_j\rangle \end{equation} The correlation length, $\xi(T)$ is characteristic length at which the value of correlation function $\Gamma(i)$ has decayed to $e^{-1}$: $$\Gamma(i)\sim \exp\Bigg(\frac{|i|}{\xi(T)}\Bigg)$$ And \begin{equation}\label{correlationlength} \xi(T)\sim |T-T_c|^{-\nu} \end{equation} And for $d=2$, $\nu=1$.

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