Surface critical properties of the three-dimensional clock model

TL;DR

This study uses Monte Carlo simulations to explore the phase diagram of the 3D q=6 clock model with open surfaces, revealing an extraordinary-log phase and detailed surface critical behaviors.

Contribution

It uncovers the existence of an extraordinary-log phase in the 3D clock model and explains its origin through emergent O(2) symmetry at the surface.

Findings

Identification of an extraordinary-log phase in the 3D clock model

Numerical evidence of emergent O(2) symmetry at the surface

Characterization of surface critical behaviors and phase transitions

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.