Using RGT to Find Critical Temp/Point of 2D Ising Model

[wdlang]

is it possible to use renormalization group theory to get the critical temperature of the 2d ising model?

or even, is it possible to show that there is a critical point?

 

[Mute]

Yes. The theory of phase transitions and critical phenomena is entirely centered around the renormalization group.

For example, you start with the hamiltonian of your system on a lattice and you specify a rule for how to coarse-grain that system (i.e., "zooming out" or integrating out some degrees of freedom), and you observe how the parameters of your hamiltonian change as you repeatedly coarse grain your system.

The RG helps you derive (approximate) equations that tell you how your hamiltonian parameters change at each coarse graining step. The stable fixed points of these equations correspond to the various phases of the system, and the unstable fixed point correspond to the phase transition. (The fixed points are the set of parameter values which, when plugged into the RG equations give back the same values).

For example, in the 2d Ising model your parameters of interest are the coupling $J$ and the temperature $T$ (let's say there's no field). The RG procedure will produce equations

$$J' = f_1(J,T),$$
$$T' = f_2(J,T)$$
where J' and T' are the values after a coarse-graining step. The fixed points of the equations, $J^\ast$ and $T^\ast$, satisfy

$$J^\ast = f_1(J^\ast,T^\ast),$$
$$T^\ast = f_2(J^\ast,T^\ast).$$

There is more than one solution to these equations. One finds that the stable fixed points are $(J,T) = (0,\infty)$, which corresponds to the paramagnetic phase (all spins are oriented randomly) and $(J,T) = (\infty,0)$, corresponding to the ferromagnetic phase (all spins have the same preferred orientation). There is an unstable fixed at a temperature $T_c$, which you can calculate with the RG procedure.

So, in doing so you have determined the phases and the parameter values at which the phase transition occurs.

 

[wdlang]

i know this.

but the problem is, whether this approach can give the correct critical point

 

[wdlang]

i am not satisfied with RG, because in most cases, you run into a non-controlled approximation.

RG is more an idea than a practical computational tool

 

[atyy]

I don't think the RG can give the absolute Tc, and it's usually done relative to the critical temperature T-Tc.

 

[Mute]

I don't think the RG can give the absolute Tc, and it's usually done relative to the critical temperature T-Tc.

The RG procedure can give Tc itself. See, for example, section 9.6 in https://www.amazon.com/dp/0201554097/?tag=pfamazon01-20. He does an RG calculation of the 2d Ising model and finds $J/k_B T_c = (1/4) \ln (1+2\sqrt{2}) \approx 0.34$, compared to Onsager's exact result $J/k_B T_c = (1/4) \ln 3 \approx 0.27$.

i know this.

but the problem is, whether this approach can give the correct critical point

It will give you an approximation to the critical point and the critical exponents. How good an approximation that it depends on how good the approximations you made in your RG procedure are. See, for example, the section of Goldenfeld quoted above.

i am not satisfied with RG, because in most cases, you run into a non-controlled approximation.

RG is more an idea than a practical computational tool

What exactly are you wanting to compute? If you want to find critical exponents and transition temperatures, RG will do that for you, and as long as you pick a half-decent RG procedure, the results should be half-decent. Obviously if you want to compare the results to experiments you are going to have to do some hard work to improve the numerical accuracy, but one can always come up with more refined RG schemes to that end.

If you want to develop a picture of the phase diagram, the RG will do that. These will be harder to compare to experiments of course because transition temperatures and transition lines are non-universal quantities whose details will depend very much on the model. (The critical exponents, by contrast, are more robust and only depend on things like the symmetry of the model, range of interactions and the dimensionality).

Remember, Quantum Electrodynamics is regarded as one of the most accurately tested theories we have, and those calculations are built on the renormalization group (albeit done in high energy physicist style rather than block-spin style, but they're equivalent). You can get accurate results if you put the work into it.

Density matrix renormalization group is also purported to have "high accuracy" (I haven't read much up on that myself, though).

It's true that crude analytic calculations are likely to employ an uncontrolled approximation at some point, but that's a price you sometimes have to pay for doing things analytically, and even then you can get relatively decent results. The 2d Ising calculation in Goldenfeld is somewhat uncontrolled, but the results aren't dramatically different from the exact results. Improved RG schemes can improve on the results of the crude first approximation. (though sometimes, as in the 2d Ising example, the convergence to the exact result is not uniform).

Momentum-shell RG is useful for analytic calculations, especially for finding the mean field theory exponents and the upper critical dimension, and for expansions about the upper critical dimension to estimate the values of critical exponents in lower dimensions.

 

[jesse73]

The answer to both questions is yes. It also depends on the approximations you make during the renormalization process. I think https://www.amazon.com/gp/search?index=books&linkCode=qs&keywords=0199577226&tag=pfamazon01-20 does a good job with renormalization and mean field theory. The chapter on conformal field theories is a bit dry though but not related to the OP question.