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The clock model places a planar spin at each lattice site but restricts its angle to $q$ equally spaced orientations, ${\theta }_i=2\pi n_i/q$. A common nearest-neighbour Hamiltonian isFor $J>0$, neighbouring spins lower their energy by choosing the same orientation. Thermal fluctuations compete with this alignment through the increasing number of disordered configurations.

The model interpolates between a finite set of orientations and the continuous angles of the XY model as $q$ increases. The number of permitted directions changes the symmetry and the possible collective behavior, so replacing a circle by a finite list is not always an innocuous numerical approximation. A four-state clock model has a particularly simple relation to two Ising models, illustrating how the angular description can conceal a discrete decomposition.

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.

Write ${\mathbf{\sigma }}_i=\mathbf{m}+\delta {\mathbf{\sigma }}_i$ and neglect products of two fluctuations. ThenEach site has $q$ neighbours and each bond is counted once, so the mean-field approximation givesChoose the $x$ axis along $\mathbf{m}$. For the four-state clock model, the single-site partition function isConsequentlyand therefore