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

The two-dimensional (2D) $q$-state clock model for $q \geq 5$ undergoes two Berenzinskii-Kosterlitz-Thouless (BKT) phase transitions as temperature decreases. Here we report an extensive worm-type simulation of the square-lattice clock model for $q=5$ 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 ($\eta_1=1/4$, $\eta_2 = 4/q^2$) at the two BKT transitions. Further, the approximate self-dual points $\beta_{\rm sd}(L)$, 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 $\beta_{\rm sd} \simeq q/2\pi$ asymptotically at the large-$q$ limit. The exponent $\eta$ at $\beta_{\rm sd}$ is consistent with $1/q$ within statistical errors as long as $q \geq 5$. Based on this, we further conjecture that $\eta(\beta_{\rm sd}) = 1/q$ 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.