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We study $q$-state clock models of regular and Villain types with $q=5,6$using cluster-spin updates and observed double transitions in each model. Wecalculate the correlation ratio and size-dependent correlation length asquantities for characterizing the existence of Berezinskii-Kosterlitz-Thouless(BKT) phase and its transitions by large-scale Monte Carlo simulations. Wediscuss the advantage of correlation ratio in comparison to other commonly usedquantities in probing BKT transition. Using finite size scaling of BKT typetransition, we estimate transition temperatures and corresponding exponents.The comparison between the results from both types revealed that the existingtransitions belong to BKT universality.

Berezinskii-Kosterlitz-Thouless transition on regular and Villain types of $q$-state clock models