Finite-ring antiferromagnetic Ising correlation ($C_n=(t^n+t^{N-n})/(1+t^N),t=-\tanh (\beta J)$)

The exact zero-field two-point correlation function $C_n=⟨S_1{S}_{n+1}⟩$ 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 $(-\tanh \beta J{)}^n$.

At fixed odd $N$, the zero-temperature canonical ensemble equally weights the $2N$ ground states of an antiferromagnetic Ising ring. The single frustrated bond in an antiferromagnetic Ising ring lies on a chosen $n$-bond path with probability $n/N$. The Ising spin product is $(-1{)}^n$ if the path excludes that bond and $-(-1{)}^n$ if it includes it, proving the displayed two-point correlation function. The loss of staggered alignment upon averaging persists at zero temperature and is caused by geometric frustration, rather than a positive density of thermal defects. The expression remains symmetric under $n\mapsto N-n$ for odd $N$.