Monte Carlo Renormalization Flows in the Space of Relevant and Irrelevant Operators: Application to Three-Dimensional Clock Models

We present a way to visualize and quantify renormalization group flows in a space of observables computed using Monte Carlo simulations. We apply the method to classical three-dimensional clock models, i.e., the planar (XY) spin model perturbed by a $Z_q$ symmetric anisotropy field. The method performs significantly better than standard techniques for determining the scaling dimension $y_q$ of the $Z_q$ field at the critical point if it is irrelevant ($q\ge4$). Furthermore, we analyze all stages of the complex renormalization flow, including the cross-over from the U(1) Nambu-Goldstone fixed point to the ultimate $Z_q$ symmetry-breaking fixed point due to the relevance of the $Z_q$ field inside the ordered phase. We expect our method to be particularly useful in the context of quantum-critical points with inherent dangerously irrelevant operators that cannot be tuned away microscopically but whose renormalization flows can be analyzed exactly as we do here for the clock models.