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 Kc=0.5186. However, the exact solution by Onsager gives Kc=0.4407.

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

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?

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

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 Kc=0.5186. However, the exact solution by Onsager gives Kc=0.4407.

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

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 Kc=0.5186. However, the exact solution by Onsager gives Kc=0.4407.

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