The $q$-state clock model with the cosine potential has a single phase transition for $q\le 4$ and two transitions for $q\ge 5$. It is shown by Monte Carlo simulations that the helicity modulus for the five-state clock model $\left(q=5\right)$ does not vanish at the high-temperature transition. This is in contrast to the clock models with $q\ge 6$ for which the helicity modulus vanishes. This means that the transition for the five-state clock model differs from the Kosterlitz-Thouless (KT) transition. It is also shown that this change in the transition is caused by an interplay between the number of angular directions and the interaction potential: by slightly modifying the interaction potential, the KT transition for $q=6$ turns into the same non-KT transition. Likewise, the KT transition is recovered for $q=5$ when the Villain potential is used. Comparisons with other clock-model results are made and discussed.

• Received 4 May 2010

DOI:https://doi.org/10.1103/PhysRevE.82.031102