A Monte Carlo study of the duality and BKT phase transitions of the two-dimensional $q$-state clock model in flow representations

Abstract

The two-dimensional (2D) q-state clock model for q ≥ 5 undergoes two Berenzinskii-Kosterlitz-Thouless (BKT) phase transitions as temperature decreases. Here we report an extensive wormtype simulation of the square-lattice clock model for q = 5 to 9 in a pair of flow representations, respectively from the high-and low-temperature expansions. By finite-size scaling analysis of susceptibility-like quantities, we determine the critical points with a precision improving over the existing results. Thanks to the dual flow representations, each point in the critical region is observed to simultaneously exhibit a pair of anomalous dimensions, which are (η1 = 1/4, η2 = 4/q 2 ) at the two BKT transitions. Further, the approximate self-dual points β sd (L), defined by the stringent condition that the susceptibility-like quantities in both flow representations are identical, are found to be nearly independent of system size L and behave as β sd q/2π asymptotically at the large-q limit. The exponent η at β sd is consistent with 1/q within statistical errors as long as q ≥ 5. Our work provides a vivid demonstration of rich phenomena associated with the duality and self-duality of the clock model in two dimensions.