Using Monte Carlo simulations and finite-size scaling analysis, we show that the $q$-state clock model with $q=6$ on a simple cubic lattice with open surfaces has a rich phase diagram; in particular, it has an extraordinary-log phase, in addition to the ordinary and extraordinary transitions at the bulk critical point. We prove numerically that the presence of the intermediate extraordinary-log phase is due to the emergence of an O(2) symmetry in the surface state before the surface enters the $Z_q$ symmetry-breaking region as the surface coupling is increased at the bulk critical point, while O(2) symmetry emerges for the bulk. The critical behaviors of the extraordinary-log transition, as well as the ordinary transition and the special transition separating the ordinary and extraordinary-log transitions, are obtained.

中文翻译：

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三维时钟模型的表面临界特性

使用蒙特卡罗模拟和有限尺寸缩放分析，我们表明$q$状态时钟模型$q=6$在具有开放表面的简单立方晶格上具有丰富的相图；特别是，它除了在体临界点的普通和非常转变之外，还有一个非常对数阶段。我们在数值上证明了中间非对数相的存在是由于在表面进入之前的表面状态中出现了 O(2) 对称性。$Z_q$随着表面耦合在体积临界点处增加，对称破坏区域，而体积出现 O(2) 对称性。获得了异常对数转换的临界行为，以及区分普通和异常对数转换的普通转换和特殊转换。