Non-Kosterlitz-Thouless transitions for the q-state clock models

The q-state clock model with the cosine potential has a single phase transition for q <= 4 and two transitions for q >= 5. It is shown by Monte Carlo simulations that the helicity modulus for the five-state clock model (q=5) does not vanish at the high-temperature transition. This is in contrast to the clock models with q >= 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.