= Four-state clock model
The <four-state clock model> restricts each spin angle to $0,\pi/2,\pi,3\pi/2$. For the nearest-neighbour cosine interaction, rotate the spin axes by $\pi/4$ and write the four unit <vectors> as $\mathbf s_i=(\sigma_i,\tau_i)/\sqrt2$ with independent signs $\sigma_i,\tau_i\in\{-1,+1\}$. Then
$$
\mathbf s_i\cdot\mathbf s_j=\frac12(\sigma_i\sigma_j+\tau_i\tau_j).
$$
Consequently $H=-J\sum_{\langle ij\rangle}\mathbf s_i\cdot\mathbf s_j$ is the sum of two independent <Ising model> <Hamiltonians>, each with coupling $J/2$. At zero applied field and with factorized boundary prescriptions, its <partition function> factorizes accordingly.
This exact change of variables explains why a four-direction spin need not behave like a small-perturbation version of a continuous <XY model>. It also specifies the model being compared: adding interactions between the two signs or using a different angular interaction changes the factorization. The result is an algebraic property of this particular four-state cosine model.
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