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I've been told for a long time that water boiling near critical temperature and the 3D Ising model near critical temperature are described by the same laws, and give a CFT. This is usually mentioned after the RG flow is introduced, and I understand that both Lagrangians are meant to be fixed points.

Where can I find an explicit proof that the laws defining water boiling near critical temperature and the 3D Ising model are the same? I assume this means showing they have the same Lagrangian.

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  • $\begingroup$ I think an explicit proof may not be available, but I can hopefully clear up a little how it is that microscopic lagrangians are matched to their CFT fixed points at criticality in lower dimensions. The idea is that by classifying, say, the unitary CFTs with central charge $\leq$ 1 and by applying the c-theorem you can analyze the RG flow properties of your UV theory to match it with one CFT in a class that you know it has to flow to. This can be used to figure out, say, which minimal models a $\phi^{2m}$ theory flows to: see the paper "RG Flow in N=1 Discrete Series." $\endgroup$ – Diffycue Jun 4 '19 at 21:10
  • $\begingroup$ In 3d you don't have a c- or an a-theorem so probably you have to be a bit sketchier in your argument for which CFTs are the IR fixed points of which UV lagrangians. I think the way they typically do this is by analyzing symmetries, pinning down the form of the spectrum (say, in the Ising model, knowing that there are three primary operators $1, \, \sigma, \, \epsilon$), and comparing simulated critical exponents with calculable bounds on those exponents in the CFT description. I'm similarly interested as you to know if they've got any more rigid explanation of this though $\endgroup$ – Diffycue Jun 4 '19 at 21:13
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If you understand the key content of the Renormalization Group theory RGT), there is no need they should have the same Lagrangian, (well, in a context of Statistical Mechanics it would be more appropriate to speak in term of Hamiltonian, but this is a side remark).

The explanation of the existence of universality classes provided by RGT is based on the fact that fixed points in an abstract space, where each point corresponds to a different system at different physical conditions, do control the critical behavior of all the systems in the same basin of attraction. Therefore, lattice systems with the same symmetry and spatial dimensionality but with different interactions do have the same critical exponents if the controlling critical point is the same.

On the one hand, a lattice gas model captures most of the relevant physics of a fluid system, whatever is the atomic Hamiltonian and this is is an argument in favor of the same critical point of the Ising model. On the other hand, more specialized implementation of RGT for fluid systems have been proposed in the literature (for example the implementation based on correlation functions by Reatto, Parola and coworkers: Parola, A., and L. Reatto. "Hierarchical reference theory of fluids and the critical point." Physical Review A 31.5 (1985): 3309.) arriving again to the conclusion that the universality class of liquid-gas transition is the same of an Ising model at the same dimensionality.

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