What is the physical meaning of "correlation length"? I am studying phase transitions right now and trying to understand the physical meaning of the concept correlation length. I saw the equations but I still couldn't quite wrap my head around the physical meaning of it. Like is it the length of the correlation between neighbouring atoms or what? And what does it mean for correlation length to be large or small? System is considered more organized when correlation length is large, right? And what is the meaning of correlation length being infinite, like when it is infinite, can we say that all atoms in the system are correlated with each other?
 A: Consider a scalar field of some kind.  When we want statistical information about this field, we need to perform an ensemble average over every possible configuration the field could have. The relative likelihood of a particular field configuration $\phi$ is given by $e^{-\beta F[\phi]}$, where $F$ is the free energy functional; the ensemble average of some quantity $Q$ is then given by
$$\langle Q\rangle := \frac{\int \mathcal D\phi \ Q e^{-\beta F[\phi]} }{\int \mathcal D\phi \ e^{-\beta F[\phi]}}$$
This is the so-called path integral approach to statistical field theory.  How one actually computes this is not important for the present discussion; it suffices to say that $\langle Q\rangle$ is a (weighted) average of $Q$ over all possible field configurations. For instance, we conventionally use the notation $\langle \phi(\mathbf x)\rangle$ to refer to the ensemble average value of the field at the point $\mathbf x$.
Now consider the quantity
$$D(\mathbf x,\mathbf y) := \langle \phi(\mathbf x)\phi(\mathbf y)\rangle-\langle\phi(\mathbf x)\rangle\langle\phi(\mathbf y)\rangle  = \left<\bigg(\phi(\mathbf x)  - \langle \phi(\mathbf x)\rangle \bigg)\bigg(\phi(\mathbf y)-\langle\phi(\mathbf y)\rangle\bigg) \right>$$
This measures the correlation between the field values at the points $\mathbf x$ and $\mathbf y$. If the field values at $\mathbf x$ and $\mathbf y$ are totally independent, then $\langle\phi(\mathbf x)\phi(\mathbf y)\rangle = \langle \phi(\mathbf x)\rangle\langle\phi(\mathbf y)\rangle$ and so $D(\mathbf x,\mathbf y)=0$ (indeed this follows immediately from the definition of independence of two random variables).  $D(\mathbf x,\mathbf y)>0$ means that $\phi(\mathbf x)$ and $\phi(\mathbf y)$ tend to be either both above average or both below average, while $D(\mathbf x,\mathbf y)<0$ means that when $\phi(\mathbf x)$ is above average then $\phi(\mathbf y)$ tends to be below average and vice-versa.
If a system exhibits translation and rotation invariance, then the two-point correlation function $D(\mathbf x,\mathbf y)$ only depends on the distance $|\mathbf x-\mathbf y|$ between the points.  In many cases, one finds that $D(\mathbf x,\mathbf y)$ takes the asymptotic form
$$D(\mathbf x,\mathbf y) \sim \frac{e^{-|\mathbf x - \mathbf y|/\xi}}{|\mathbf x-\mathbf y|^\alpha}$$
for some $\alpha$; we call $\xi$ the correlation length.  Physically, it measures the average size of fluctuations in the field; if $|\mathbf x-\mathbf y|\ll \xi$ then $D(\mathbf x,\mathbf y)$ is comparatively large, while if $|x-y|\gg \xi$ then $D(\mathbf x,\mathbf y)$ is exponentially suppressed.
When we consider thermal effects, we find that $\xi=\xi(T)$ is a function of temperature.  In the limit $T\rightarrow \infty$, $\xi(T)$ typically tends to zero, indicating that at high temperatures the field is totally "randomized", even for very nearby points. As the temperature decreases toward $T_c$, the thermal "kicks" aren't enough to completely disorder the field and the fluctuations begin to span a larger range of sizes.  If $\xi \rightarrow \infty$, then the correlation function has no exponential decay and we see fluctuations of arbitrarily large size.  Finally, as $T\rightarrow 0$ we usually - but not always - see $\xi\rightarrow 0$ once again; fluctuations become small not because of the relentless thermal agitation we see at high $T$, but rather because the free energy cost of all but the lowest energy configurations becomes prohibitively high due to the factor $e^{-\beta F[\phi]}$.
As a concrete example of a critical phenomenon, if $\phi(\mathbf x)$ refers to the out-of-plane magnetization of a 2D film, then $\xi$ measures the range of sizes of the patches which are magnetized in the same direction. As $\xi\rightarrow \infty$, fluctuations span the entire system size and the film becomes magnetically ordered.
To paraphrase a common expression, an applet is worth a thousand words.  You can explore the effect of temperature on correlation length (i.e. the average size of the fluctuations) using this nice tool from Daniel Schroeder.
A: Here's an example using a crystal.   The same concept applies to many other physical systems.
Consider an ideal crystal.  If you know the position of atom at a, then you can predict the position of atom b no matter how far away it is.   But real materials are not ideal crystals.  There are inclusions, faults, vibrations ... so the atoms are not at their ideal location.  If I know the position of a, I probably know the position of its nearest neighbor to an excellent degree of certainty.  The next nearest neighbor ... slightly less certainty.   The atom a kilometer away, not at all.   The length over which the positions can be reliably predicted is the correlation length.
A: In a magnet, atoms with positive spin and atoms with negative spin will cluster together and the correlation length $\xi$ measures the typical size of these clusters. When $\xi$ first reaches infinity, you do not have a single infinitely big cluster yet because even though exponentially decaying correlations are gone, you still have correlations which decay as a power law. This indicates that there are clusters of all sizes. To truly make all spins the same, you need the coefficient of this power law to disappear as well which only happens at zero temperature.
I discussed this a bit in a previous answer where the context was renormalization.
