标题： 二维$q$状态钟模型在流表示中的对偶性和Berezinskii-Kosterlitz-Thouless相变的蒙特卡罗研究 显示英文标题

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

摘要： 二维$q$状态钟模型对于$q \geq 5$在温度降低时会发生两次贝雷津-科斯特里茨-索利斯（BKT）相变。在这里，我们报告了对平方格子钟模型的广泛虫型模拟，对于$q=$5--9 在一对流表示中进行，分别来自高温和低温展开。通过易磁化率类似量的有限尺寸标度分析，我们以比现有结果更高的精度确定了临界点。由于双重流表示，临界区域中的每个点都被观察到同时表现出一对异常维度，在两次 BKT 相变中分别为$\eta_1=1/4$和$\eta_2 = 4/q^2$。 此外，由严格的条件定义的近似自对偶点$\beta_{\rm sd}(L)$，即在两种流表示中的类似磁化率量相等，被发现几乎与系统尺寸$L$无关，并在大-$q$极限下表现出$\beta_{\rm sd} \simeq q/2\pi$的渐近行为。 指数$\eta$在$\beta_{\rm sd}$处与$1/q$在统计误差范围内一致，只要$q \geq 5$。 基于此，我们进一步猜想$\eta(\beta_{\rm sd}) = 1/q$精确成立，并且对于处于$q$-状态时钟普适类的系统是普遍的。 我们的工作提供了对二维时钟模型对偶性和自对偶性相关丰富现象的生动演示。 显示英文摘要

摘要： The two-dimensional $q$-state clock model for $q \geq 5$ 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 the high- and low-temperature expansions, respectively. By finite-size scaling analysis of susceptibility-like 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 $\eta_2 = 4/q^2$ 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 error 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.