Is there an renormalisation for 2d ising yielding the accurate critical coupling, why?

Question

2d-ising model is a classical model to which renormalisation may be applied to obtain information about criticality.

The partition function has the form $$Z=\sum_{\sigma} e^{-H(\sigma,K)}$$ where $$H(\sigma,K)=\sum_{<ij>}K\sigma_i\sigma_j$$

For example, using perturbative block-transformation at l=2 (block spin made up of 2^2=4 elementary spins) and expand block interaction up to 1st order gives a fixed point for $K=R(K)$ at $K_c=0.5186$. However, the exact solution by Onsager gives $K_c=1/2*ln(1+\sqrt 2)\approx0.4407$

The question follows that, is there a renormalisation function $K'=R(K)$ will give the same $K_c$ as Onsager's solution? If not, why?

Answer 1

Using CTMRG (corner transfer matrix renormalisation) with some fitting techniques (here is a related thesis), it was determined that $T_c \approx 2.26920$, whereas $T_{c_{exact}}≈ 2.26919$. However in this approach, the object being renormalised is the transfer matrix/tensor and appear less intuitive than bond-energy based renormalisation. It's might still be possible to recover a recursion formula for R(K).