As in the title, I am trying to understand how the critical fixed point (CFP) can be used to derive the thermodynamic singular behavior of the physical critical point (PCP). The context I have in mind is that of the 2d Ising model in real space renormalization group (RG).
Here I just present the basic notions, since maybe my misunderstanding is give by some false idea I have.
I imagine we have a RG transform that
- leads to analytic recursion relations
- leaves invariant the total free energy
- has a CFP and its associated stable manifold (the critical manifold) whose dimension is given by the number of irrelevant variables relative to that CFP.
- the PCP is the intersection among the critical manifold and the curve of NN (nearest neighbour) Ising model hamiltonians.
- the hamiltonian at the PCP point under the RG transform follows a critical trajectory that ends only asymptotically at the CFP
Now here is what I think should be an argument to show that the CFP entails the PCP:
- scaling laws and critical exponents are asymptotic properties of the PCP, i.e the are valid asymptotically in a neighborhood of the PCP, so I should study how the system behave for values $t=T-T_C\ll1$ and $h\ll1$
- the analytic property of the RG transform allows us to expand in a Taylor series the recursion relations (or the RG differential equation) anywhere. Though the linearization procedure is accurate only in a neighborhood of the CFP, since this is a fixed point of the transformation. This justify our curiosity for the CFP
- I take as starting point $(t,h)$ near the PCP, and apply the RG transform. Since the transformation is analytic we expect that the renormalized point $(t',h')$ will not have a much greater distance from the relative renormalized critical point, respect the distance among the starting point and the PCP.
- We hope that the assumption in 3. holds till we arrive in a neighborhood of the CFP. That is we hope that our RG transform will get us a trajectory that is almost "parallel" to the critical trajectory. To justify this I guess is not trivial since we can't take the linearization of the RG to be accurate in describing the evolution of the starting point in parameter space, the full RG transformation is needed.
- if 4. is true then we know that points matched by the RG transformation describe the same thermodynamics, so the thermodynamic behaviour derived in the neighborhood of the CFP entails the one of the PCP neighborhood.
- from this the whole linearization procedure around the CFP can be used to derive the scaling form of the free energy and the whole critical exponents machinery.
Now I am far from being sure that this is correct.
Usual exposures I have seen in books don't really explain this and maybe avoid the problem by not keeping a net distinction between the CFP anf PCP. Usually, for what I have seen, their explanation consists just in the linearization of the RG transformation near the CFP and how from this we can get the free energy scaling laws and all the critical exponents. This is done using simply the invariance of the total free energy $F$ and hence the following equation for the free energy per particle $f$, $f[t,h]=b^{-2n}f[b^{ny_t}t, b^{ny_h}h]$. Though I think that to make sense, the starting point given by $t$ and $h$, in their context, should be a point near the CFP and not the PCP, otherwise the linearization isn't right.
Now the question per se is in the title how critical fixed point entails the singular behaviour of the physical critical point. I explained what I think I know, though I am far from sure this is correct. Besides, since my knowledge is only superficial, I feel like that even if it happens that what I said is correct ( I have some doubts), there is a lot of space o maneuver to explain the concept in a more rigorous, complete way. Hence Avoiding this to become a check my reasoning question.
I guess it is hard to be quantitative since I imagine that one should take a specific instance of RG transform, and try to prove 3. and 4.