= Finite-ring antiferromagnetic Ising correlation
{title2=$C_n=(t^n+t^{N-n})/(1+t^N),\quad t=-\tanh(\beta J)$}
= Finite antiferromagnetic Ising-ring correlation
{synonym}
The exact zero-field <two-point correlation function> $C_n=\langle S_1S_{n+1}\rangle$ in the finite <canonical ensemble> of an <antiferromagnetic Ising ring>. The <spin-chain transfer matrix> has <eigenvalues> $\lambda_+=2\cosh(\beta J)$ and $\lambda_-=-2\sinh(\beta J)$. In their <eigenbasis>, an <Ising spin> insertion exchanges the two <eigenvectors>. The <trace> with two insertions is therefore $\lambda_-^n\lambda_+^{N-n}+\lambda_+^n\lambda_-^{N-n}$, while the <partition function> is $\lambda_+^N+\lambda_-^N$, proving the formula. Its two terms retain both paths around the ring. It has $C_0=1$ and $C_n=C_{N-n}$. At fixed $n$ and finite positive <inverse temperature> its <thermodynamic limit> is $(-\tanh\beta J)^n$.
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