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 ${\mathbf{s}}_i=({\sigma }_i,{\tau }_i)/\sqrt{2}$ with independent signs ${\sigma }_i,{\tau }_i\in \{-1,+1\}$. ThenConsequently $H=-J{\sum }_{⟨ij⟩}{\mathbf{s}}_i\cdot {\mathbf{s}}_j$ 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.