Why does Critical Points have fluctuations on all scales (Infinite correlation length)? 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 infinite?" "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.
 A: Take the Ising model in dimension $d\ge 2$ for concreteness. Let $\langle\cdot\rangle_{\beta,+}$ denote the plus state at inverse temperature $\beta$ and zero magnetic field. This is obtained by putting the interaction in a box with boundary condition given by $+1$ spins, and then taking the thermodynamic limit where the box fills the entire space. The OP's definition of critical point is the smallest value $\beta_c$ such that for $\beta>\beta_c$, one has spontaneous magnetization, i.e., $\langle\sigma_0\rangle_{\beta,+}>0$ where $\sigma_0$ is the spin at the origin.
Formally, the probability measure is invariant by flipping all the spins, so a naive expectation would be that $\langle\sigma_0\rangle_{\beta,+}=0$. But this does not happen because somehow the influence of the spins on the boundary far away at infinity is felt deep in the bulk, at the origin. This should therefore signal the existence of very long range correlations, i.e., $\langle\sigma_x\sigma_y\rangle_{\beta,+}-\langle\sigma_x\rangle_{\beta,+}\langle\sigma_y\rangle_{\beta,+}$ decaying slowly, i.e., the correlation length being very big.
Also, another definition of the critical $\beta$ is the divergence of the susceptibility which in the high temperature region is $\sum_{x}\langle\sigma_0\sigma_x\rangle_{\beta,+}$. That too signals slow decay of the correlation.
A: The free energy becomes extremely flat around the minimum at the second-order transition (say when plotted against magnetization). The system can now assume many values of magnetization (at least temporarily) apart from the one at which the free energy is minimum because the free energy cost for doing so is nearly zero (See H.B. Callen chapter 10 for a detailed explanation). It is clear that such change in a quantity which is an average over the entire bulk can only happen if the fluctuation that causes that change happens in a large part of the system simultaneously (that is has a large correlation length). This logic can also explain why correlation length does not diverge for first-order transitions, which always have a free energy barrier between the two competing phases which disallow large-scale fluctuations.
A: I will try to answer this restricting myself to quantum critical points. I'll be trying to connect a gap closing of the system with infinite correlation length. In general a system with a finite gap between the ground state and the excited states will have a finite length correlation lengths. This link, at least in my mind, can offer a way out of the circular reasoning you mentioned. This discussion is a watered-down version of section 3, and the appendix here: https://arxiv.org/pdf/1004.3835.pdf
Suppose we study a system that can be described by a Hamiltonian $H(g)$, where $g$ is some parameter that the Hamiltonian depends on. Consider some physical observable $\hat{O}$, and its ground state expectation value which in general does depend on $g$, $\langle \Psi_0(g)|\hat{O}|\Psi_0(g)\rangle = \langle\hat{O}\rangle_0(g)$. We say a phase transition is when $\langle\hat{O}\rangle_0(g)$ develops a singularity as we change the parameter $g$. This defines an equivalence class, $\Psi_0(0)$ and $\Psi_0(1)$ are in same class if one can connect $H(0)$ and $H(1)$ (for which $\Psi_0(0)$ and $\Psi_0(1)$ are the ground states respectively) without going though a phase transition as defined. Now the point here is that $\langle\hat{O}\rangle_0(g)$ will not develop a singularity as long as $H(g)$ is gaped. If the system is gaped then a small change in $g$ will cause the ground state to change continuously, and hence $\langle\hat{O}\rangle_0(g)$ to change continuously. If we want $\langle\hat{O}\rangle_0(g)$ to develop a singularity then we need the ground state to develop a singularity and this entails a gap closing. One often can relate the gap of the system to a correlation length such that $m \propto \frac{1}{\xi}$, where $m$ is the gap of the system and $\xi$ is a correlation length.
