# Water boiling and 3D Ising model

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.

• 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." – Diffycue Jun 4 '19 at 21:10
• 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 – Diffycue Jun 4 '19 at 21:13