When there is no external field, we can derive a functional equation that f ( β β , 0 ) = f ( β β ) satisfies using renormalization. Specifically, let Z N ( β β , J ) be the partition function with N sites. Now we have: Z N ( β β , J ) = ∑ ∑ σ σ e K σ σ 2 ( σ σ 1 + σ σ 3 ) e K σ σ 4 ( σ σ 3 + σ σ 5 ) ⋯ ⋯ where K := β β J . We sum over each of σ σ 2 , σ σ 4 , ⋯ ⋯ , to obtain Z N ( β β , J ) = ∑ ∑ σ σ ( 2 cosh ⁡ ⁡ ( K ( σ σ 1 + σ σ 3 ) ) ) ⋅ ⋅ ( 2 cosh ⁡ ⁡ ( K ( σ σ 3 + σ σ 5 ) ) ) ⋯ ⋯ Now, since the cosh function is even, we can solve A e K ′ σ σ 1 σ σ 3 = 2 cosh ⁡ ⁡ ( K ( σ σ 1 + σ σ 3 ) ) as A = 2 cosh ⁡ ⁡ ( 2 K ) , K ′ = 1 2 ln ⁡ ⁡ cosh ⁡ ⁡ ( 2 K ) . Now we have a self-similarity relation: 1 N ln ⁡ ⁡ Z N ( K ) = 1 2 ln ⁡ ⁡ ( 2 cosh ⁡ ⁡ ( 2 K ) ) + 1 2 1 N / 2 ln ⁡ ⁡ Z N / 2 ( K ′ ) Taking the limit, we obtain f ( β β ) = 1 2 ln ⁡ ⁡ ( 2 cosh ⁡ ⁡ ( 2 K ) ) + 1 2 f ( β β ′ ) where β β ′ J = 1 2 ln ⁡ ⁡ cosh ⁡ ⁡ ( 2 β β J ) .