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The two-dimensional $q$-state clock model for $q \geq 5$ undergoes twoBerezinskii-Kosterlitz-Thouless (BKT) phase transitions as temperaturedecreases. Here we report an extensive worm-type simulation of thesquare-lattice clock model for $q=$5--9 in a pair of flow representations, fromthe high- and low-temperature expansions, respectively. By finite-size scalinganalysis of susceptibility-like quantities, we determine the critical pointswith a precision improving over the existing results. Due to the dual flowrepresentations, each point in the critical region is observed tosimultaneously exhibit a pair of anomalous dimensions, which are $\eta_1=1/4$and $\eta_2 = 4/q^2$ at the two BKT transitions. Further, the approximateself-dual points $\beta_{\rm sd}(L)$, defined by the stringent condition thatthe susceptibility like quantities in both flow representations are identical,are found to be nearly independent of system size $L$ and behave as $\beta_{\rmsd} \simeq q/2\pi$ asymptotically at the large-$q$ limit. The exponent $\eta$at $\beta_{\rm sd}$ is consistent with $1/q$ within statistical error as longas $q \geq 5$. Based on this, we further conjecture that $\eta(\beta_{\rm sd})= 1/q$ holds exactly and is universal for systems in the $q$-state clockuniversality class. Our work provides a vivid demonstration of rich phenomenaassociated with the duality and self-duality of the clock model in twodimensions.

physical review. e

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