Monte Carlo study of duality and the Berezinskii-Kosterlitz-Thouless phase transitions of the two-dimensional q-state clock model in flow representations

The two-dimensional q-state clock model for q 5 undergoes two Berezinskii-Kosterlitz-Thouless (BKT) phase transitions as temperature decreases. Here we report an extensive worm-type simulation of the square -lattice clock model for q = 5-9 in a pair of flow representations, from high-and low-temperature expansions, respectively. By finite-size scaling analysis of susceptibilitylike quantities, we determine the critical points with a precision improving over the existing results. Due to the dual flow representations, each point in the critical region is observed to simultaneously exhibit a pair of anomalous dimensions, which are eta(1) = 1/4 and eta(2) = 4/q2 at the two BKT transitions. Further, the approximate self-dual points beta sd(L), defined by the stringent condition that the susceptibilitylike quantities in both flow representations are identical, are found to be nearly independent of system size L and behave as beta sd ? q/2 pi asymptotically at the large -q limit. The exponent eta at beta sd is consistent with 1/q within statistical error as long as q 5. Based on this, we further conjecture that eta(beta sd)= 1/q holds exactly and is universal for systems in the q-state clock universality class. Our work provides a vivid demonstration of rich phenomena associated with the duality and self-duality of the clock model in two dimensions.