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I have been studying statistical field theory for a while and I still haven't found a physical explanation for this question. Every answer seems to be kind of circular. Basically something like this: "Why does the correlation length become infinte?" "Because there are fluctuations on all scales?" "and why are there fluctuations on all scales?" because that's what happens in critical points" "and why does it happen in critical points?" "because the correlation length becomes infinite".

So the question is: what is the thing about phase transitions that makes the correlation length go to infinity?

I know that when you do the math in every case, that's what happens but I want a more physical explanation. Also, I have seen the following posts:

Why correlation length diverges at critical point?

Scale invariance at phase transitions

But this answers only give the circular answer or explain that critical points are the fixed points in the RG flow which is also not explaining why this is physically happening but just using the math to prove it is.

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An infinite correlation length is not physical. For one thing, nothing is actually infinite in size! For another, as length goes to infinity, the time for the phase transition to occur across the whole material also goes to infinity.

Certain systems, when cooled slowly, will tend to find a highly ordered state, because this is the state that minimizes free energy, which you can see by looking at the Hamiltonian of said systems.

However, if a system is not allowed to cool slowly enough, it may get 'stuck' in a state where the free energy is not minimized. And the larger the crystals, the more slowly it needs to be cooled.

In the Ising model for example, as the number of lattice sites goes to infinity, the longer the 'cooling' process takes to cause a completely ordered state. If you try to cool it faster, it will just form a spin glass.

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