= Ising ring magnetization fluctuations
{c}
{title2=$\operatorname{Var}(m)=\langle m^2\rangle,\quad m=\frac14\sum_i\sigma_i$}
Define the instantaneous <spin magnetization> $m=\frac14\sum_i\sigma_i$. Global reversal sends $m$ to $-m$ without changing the zero-field <energy>. Pairing every <microstate> with its reversed partner proves $\langle m\rangle=0$ at every positive <temperature>, even when almost all <probability> lies on the two fully aligned patterns. A vanishing mean does not imply weak alignment: the limiting <probability distribution> is half at $m=+1$ and half at $m=-1$.
The <Ising ring magnetization fluctuations> measure this distinction through <variance>. In the $d=0$ class, both patterns have $m^2=1$. Among the $d=2$ patterns, eight have $m^2=1/4$ and four have $m^2=0$; the two $d=4$ patterns also have $m^2=0$. Consequently
$$
\operatorname{Var}(m)=\langle m^2\rangle
=\frac{2e^{4K}+2}{Z}.
$$
It equals $6/17$ at $K=(\ln2)/4$, approaches $1/4$ for four independent fair <Ising spins> at $K\to0$, and approaches $1$ as $K\to\infty$. Unlike the <energy> <variance>, the <spin magnetization> <variance> remains large in the low-<temperature> symmetric mixture.
\Image[/ising-four-spin-ring-correlations.png]
{title=Exact bond correlation, magnetization variance and heat capacity of the four-spin Ising ring}
{description=Original exact finite-ring calculations at $h=0$ and $J>0$, plotted against $K=\beta J$; increasing $K$ means cooling at fixed $J$. The marked point has $K=(\ln2)/4$, <Ising ring bond correlation> $3/17$ and <spin magnetization> <variance> $6/17$. These smooth curves are model predictions.}
The same <thermal fluctuation> controls the field response. Allow $h$ to vary before setting it to zero; differentiating the <Boltzmann distribution> inserts $\beta\sum_i\sigma_i=4\beta m$, giving
$$
\left.\frac{\partial\langle m\rangle}{\partial h}\right|_{h=0}
=4\beta\operatorname{Var}(m).
$$
This is a spin-normalized <magnetic susceptibility> with respect to the <energy>-valued parameter $h$; conversion to physical <magnetization> and <magnetic field> requires the <magnetic moment> and sample <volume> factors.
For finite $N$, $Z$ is a finite positive sum of <exponential functions>. Its <free energy> $-k_BT\log Z$ is a <real analytic function> for $T>0$, so the smooth <heat capacity> peak here is not a critical <phase transition>. <Spontaneous magnetization> concerns the order of limits: first a <thermodynamic limit>, then removal of a symmetry-breaking field. The <four-spin Ising ring> alone establishes neither that limit nor any two-dimensional <Ising model> result.<Statistical Physics by David Tong>[§5.2.3, paragraph beginning with the analytic finite partition sum: phase transitions require the thermodynamic limit]{ref}
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