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Suppose we have some lattice classical spin system. The correlation function is defined as

$$\Gamma_{ij}=\langle S_iS_j\rangle -\langle S_i\rangle \langle S_j\rangle $$ It is often said that

$$ \Gamma_{ij}\sim e^{-|i-j|/\xi}$$ where $\xi$ is the correlation length (this is, if I understand correctly, the definition of correlation length). Is there a way to, if not prove, at least argument that this must be the case, while being agnostic to any particular choice of Hamiltonian, beyond assuming that it is local in some way? I understand that intuitively the correlation must decay, but why exponentially?

Also, can I interpret this definition of $\xi$ literally? Suppose the system is translationally invariant and $\gamma=\Gamma_{ii}$ is the same for every $i$, can I say $$\Gamma_{ij}=\gamma e^{-|i-j|/\xi} $$ and hence

$$\xi= \frac{|i-j|}{\log\gamma-\log\Gamma_{ij}}$$ for every $i,j$? I find it hard to believe that this quantity doesn't ever depend on $i,j$. If this is indeed true, there must be some locality requirements on the Hamiltonian. What are they?

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    $\begingroup$ The correlation length is the typical length over which information propagates. From this point of view, it is pretty clear that a finite correlation length will result in exponential decay of the correlations. This can be made precise pretty generally in perturbative regimes (very low temperatures, very high temperature, strong magnetic field, etc.). Nonperturbative results require very specific classes of models (essentially, models with nice geometric representations of correlations, such as the Ising or Potts models). $\endgroup$ Jan 23, 2020 at 18:06
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    $\begingroup$ Concerning your second question, it is not true that $\Gamma_{ij} = \gamma e^{-|i-j|/\xi}$. First, the quantity in the exponent is not isotropic in general. More importantly, the prefactor is much more complicated: to leading order in $|i-j|$, it is given by a (direction dependent) function of the order $|i-j|^{-(d-1)/2}$, where $d$ is the dimension of the lattice. This is the so called Ornstein-Zernike behavior and can be proved rigorously in perturbative regimes for a rather general class of models and non-perturbatively for simple (Ising, Potts, ...) models. $\endgroup$ Jan 23, 2020 at 18:06
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    $\begingroup$ [There are exceptions to the Ornstein-Zernike behavior, most notably the nearest-neighbor Ising model in 2 dimensions, where the prefactor is of order $|i-j|^{-2}$.] $\endgroup$ Jan 23, 2020 at 18:07
  • $\begingroup$ A discussion of the perturbative arguments leading to exponential decay at low temperatures can be found in this answer. It is nonrigorous and restricted to the Ising model, but it can be made rigorous and variants can be used to prove similar results for a large class of models. $\endgroup$ Jan 23, 2020 at 18:10
  • $\begingroup$ @YvanVelenik Would it be fair to say "In many systems, the correlation function is $\Gamma_{ij}=f(|i-j|)e^{-|i-j|/\xi}$ for some function $f$ and number $\xi$ called the correlation length" being clear that this definition of $\xi$ is valid only for those systems, and for other systems where this is not valid it is defined differently, or not at all? $\endgroup$ Jan 24, 2020 at 9:01

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As mentioned in the comments, I don't think that in every system the correlation function behaves like this. In fact, this is not the case when analyzing a point of phase transitions. This is one of the things that characterize phase transitions! There also other classes of systems that do not show this behavior. However, we can say that this is the general case, and in fact a definition for the correlation length.

To see that, let's start from the following assumption - there is a correlation length in the system. That is, a length scale that when examining chunks of the system smaller than that scale it is correlated, and above it is not correlated. Then the correlation function cannot have a power-law form $\Gamma_{i,j} \propto |r_i-r_j|^{-\alpha}$ simply because this is scale-free! This is the case in conformal symmetric systems (which characterize some phase transitions) and in some special 1d systems.

So looking for a decaying function that shows a characteristic scale we of course have $\Gamma_{i,j} \propto \exp[-|r_i-r_j|/\xi]$. We can add other stuff on top of it, but this will be the dominant behavior. This is usually how we define the correlation length, and then we can see how it behaves when we change system parameters, and usually it will blow up to infinity at the point of phase transition, which exactly means that the other terms (the scale-free) are the dominant ones. I think that one of the definitions of a phase-transition is "the point in which there is no correlation length in the system".

Now that $\propto$ sign plays an important role here. In general the correlation function will be quite complicated and calculating it exactly in a correlated system will be an intractable problem. However, we can approximate its behavior using some methods, and we will always be interested in the dominant behavior, which far from the phase transition will be encoded in this exponential form.

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