Is there an renormalisation for 2d ising yielding the accurate critical coupling, why?
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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=∑σe−H(σ,K)Z=∑σe−H(σ,K)
where
H(σ,K)=∑<ij>KσiσjH(σ,K)=∑<ij>Kσiσj
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)K=R(K) at Kc=0.5186Kc=0.5186. However, the exact solution by Onsager gives Kc=1/2∗ln(1+2–√)≈0.4407Kc=1/2∗ln(1+2)≈0.4407
The question follows that, is there a renormalisation function K′=R(K)K′=R(K) will give the same KcKc as Onsager's solution? If not, why?
The partition function has the form
Z=∑σe−H(σ,K)Z=∑σe−H(σ,K)
where
H(σ,K)=∑<ij>KσiσjH(σ,K)=∑<ij>Kσiσj
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)K=R(K) at Kc=0.5186Kc=0.5186. However, the exact solution by Onsager gives Kc=1/2∗ln(1+2–√)≈0.4407Kc=1/2∗ln(1+2)≈0.4407
The question follows that, is there a renormalisation function K′=R(K)K′=R(K) will give the same KcKc as Onsager's solution? If not, why?
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no there is an renormalisation for 2d ising yielding the accurate critical coupling.
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