True and quasi long-range order in the generalized $q$-state clock model

From consideration of the order-parameter distribution, we propose an observable which makes a clear distinction between true and quasi long-range orders in the two-dimensional generalized $q$-state clock model. Measuring this quantity by Monte Carlo simulations for $q=8$, we construct a phase diagram and identify critical properties across the phase-separation lines among the true long-range order, quasi long-range order, and disorder. Our result supports the theoretical prediction that there appears a discontinuous order-disorder transition as soon as the two phase-separation lines merge.