We perform the state-of-the-art tensor network simulations directly in the thermodynamic limit to clarify the critical properties of the $q$-state clock model on the square lattice. We determine accurately the two phase transition temperatures through the singularity of the classical analog of the entanglement entropy, and provide extensive numerical evidences to show that both transitions are of the Berezinskii-Kosterlitz-Thouless (BKT) type for $q\ge 5$ and that the low-energy physics of this model is well described by the ${\mathbb{Z}}_q$-deformed sine-Gordon theory. We also determine the characteristic conformal parameters, especially the compactification radius, that govern the critical properties of the intermediate BKT phase.

中文翻译：

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二维q状态时钟模型的关键属性。

我们直接在热力学极限条件下执行最新的张量网络模拟，以弄清热力学极限的关键特性。 $q$方格上的状态时钟模型。我们通过纠缠熵的经典相似点的奇异点准确地确定了两个相变温度，并提供了广泛的数值证据，表明这两个相变都是Berezinskii-Kosterlitz-Thouless（BKT）类型的。$q\ge 5$ 并且该模型的低能物理学很好地描述了 ${\mathbb{ž}}_q$变形的正弦-戈登理论。我们还确定了控制中间BKT相关键特性的特征共形参数，尤其是压实半径。