Surface critical properties of the three-dimensional clock model

Abstract

Using Monte Carlo simulations and finite-size scaling analysis, we show that the $q$-state clock model with $q=6$ on the simple cubic lattice with open surfaces has a rich phase diagram; in particular, it has an extraordinary-log phase, besides 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 and the special transition separating the ordinary and the extraordinary-log transition are obtained.

12 pages, 8 figures

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• Boundary operator expansion and extraordinary phase transition in the tricritical O(N) model
• Extraordinary surface critical behavior induced by symmetry-protected topological state of a two-dimensional quantum magnet
• Comprehensive studies on the universality of BKT transitions -- Machine-learning study, Monte Carlo simulation, and Level-spectroscopy method
• Boundary criticality for the Gross-Neveu-Yukawa models
• Accurate boundary bootstrap for the three-dimensional O($N$) normal universality class
• Long-Range Order in a Strictly Short-Range Quasi-2D XY Model: When Critical Fluctuations Matter