= Clock model
{wiki}
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> is
$$
H=-J\sum_{\langle ij\rangle}\cos(\theta_i-\theta_j).
$$
For $J>0$, 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.
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