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

We study renormalization group flows in a space of observables computed by Monte Carlo simulations. As an example, we consider three-dimensional clock models, i.e., the XY spin model perturbed by a Z_{q} symmetric anisotropy field. For q=4, 5, 6, a scaling function with two relevant arguments describes all stages of the complex renormalization flow at the critical point and in the ordered phase, including the crossover from the U(1) Nambu-Goldstone fixed point to the ultimate Z_{q} symmetry-breaking fixed point. We expect our method to be 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 as we do here for the clock models.