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We study a nearest neighbors ferromagnetic spin system on the square latticein which the spin field is constrained to take values in a discretization ofthe unit circle consisting of $N$ equi-spaced vectors, also known as $N$-clockmodel. We find a fast rate of divergence of $N$ with respect to the latticespacing for which the $N$-clock model has the same discrete-to-continuumvariational limit of the $XY$ model, in particular concentrating energy ontopological defects of dimension 0. We prove the existence of a slow rate ofdivergence of $N$ at which the coarse-grain limit does not detect topologicaldefects, but it is instead a $BV$-total variation. Finally, the two differenttypes of limit behaviors are coupled in a critical regime for $N$, whoseanalysis requires the aid of Cartesian currents.

The $N$-clock model: Variational analysis for fast and slow divergence rates of $N$