= Ising ring bond correlation
{c}
{title2=$c_1=\langle\sigma_i\sigma_{i+1}\rangle$}
The <Ising ring bond correlation> is the neighboring <two-point correlation function> $c_1=\langle\sigma_i\sigma_{i+1}\rangle$ in the <four-spin Ising ring>. All four <edges> are equivalent. At zero field each mean <Ising spin> is zero by <spin inversion symmetry>, so $c_1$ also equals the <connected correlation function>. Agreement gives bond product $+1$ and disagreement $-1$; thus $P(\sigma_i=\sigma_{i+1})=(1+c_1)/2$.
Differentiating the finite <partition function> with respect to $K=\beta J$ inserts the sum of four bond products. Therefore
$$
c_1=\frac14\frac{d\log Z}{dK}
=\frac{\sinh(4K)}{\cosh(4K)+3},\qquad
U=\langle H\rangle=-4Jc_1.
$$
Here $U$ is <internal energy>. At $K=(\ln2)/4$, the class-weight calculation gives $c_1=(4-1)/17=3/17$ and $U=-12J/17$. The agreement <probability> for a selected bond is $10/17$, despite the exactly zero mean <spin magnetization>.
A second <derivative> gives the <energy> <variance>, and hence the <heat capacity> $C_H=dU/dT$ at fixed $J$ and $h=0$:
$$
\frac{\operatorname{Var}(H)}{J^2}
=\frac{d^2\log Z}{dK^2}
=\frac{16[1+3\cosh(4K)]}{[\cosh(4K)+3]^2},\qquad
\frac{C_H}{k_B}=K^2\frac{\operatorname{Var}(H)}{J^2}.
$$
To see the <variance> identity directly, a <derivative> of an <expected value> under weights $e^{KS}$ inserts $S-\langle S\rangle$, where $S=\sum_i\sigma_i\sigma_{i+1}=-H/J$. The factor $K^2$ follows from $dK/dT=-K/T$. At the same exact example, $\operatorname{Var}(H)/J^2=1216/289$. The <heat capacity> vanishes both as $K\to0$ and as $K\to\infty$: at high <temperature> its dimensionless factor $K^2$ vanishes, while at low <temperature> excited <microstates> become exponentially rare.
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