Dangerously Irrelevant Field at Clock Models - A study with the Monte Carlo Based RG Flows

The phase transitions between the paramagnetic phase and the $Z_q$ symmetry-breaking phase in the three dimensional XY model with $Z_q$-anisotropic field is now well accepted as belonging to the 3D XY universality class for $q>q_c$, with the $Z_q$-anisotropic field being irrelevant at the critical point. However, when $T<T_c$ it is always relevant as the symmetry is broken discretely. This is ”dangerous” because it leads to a larger length scale $\xi'$ than the normal correlation length $\xi$, which, by definition, governs the crossover behavior from $U(1)$ to $Z(q)$ symmetry breaking. While RG theory has provided generic pictures of the RG flow in coupling space, we here construct a quantitative Monte Carlo based RG flow using corresponding physical observables, where not only the crossover behaviors can be observed explicitly but also the critical exponents $\nu$ and $\nu'$ can be extracted accordingly. For this purpose, we have firstly restudied the anisotropy order parameter and confirm that its scaling dimension at the critical point is the same as that of the $Z_q$ field. We also clarify unsettled issues in several recent studies of this quantity.