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

Abstract

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 Zq symmetric anisotropy field. The method performs significantly better than standard techniques for determining the scaling dimension yq of the Zq field at the critical point if it is irrelevant (q ≥ 4). 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 Zq symmetry-breaking fixed point due to the relevance of the Zq 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. SUPPLEMENTARY INFORMATION

Monte Carlo study of an improved clock model in three dimensions

“…In addition we make use of the estimates 1/β c,N =5 = 2.20502(1) and 1/β c,N =6 = 2.20201(1) reported in [47]. Note that for N = 6 the result of [47] is fully consistent with ours. Similar to the Caley tree approximation, we see a rapid convergence of β c,N with N → ∞.…”

“…One might also study the low temperature phase of the improved (N + 1)-state clock model. The consequences of the fact that a Z N symmetric perturbation of the O(2) symmetric fixed point is dangerously irrelevant in the low temperature phase are debated in the literature, as can be seen in [47] and references therein.…”

“…This value has to be compared with y N =6 = −2.55(6) and −2.509 (7) given in refs. [47] and [5], respectively.…”

Tunable deconfined quantum criticality and interplay of different valence-bond solid phases*

“…The correlation length and the U(1) length are governed by exponents ν and ν, respectively, and ν > ν. The exponents are related to each other and to the scaling dimension of the clock perturbation in a way that has been controversial [49][50][51][52] and for which new insights were presented very recently [53]. The analogous U(1) length scale has also been studied in the standard J-Q model [14], though not yet at the level of precision as was possible in the classical case.…”

“…In contrast, the "soft" q = 4 clock model with small h 4 exhibits clock-like fluctuations, and its phase transition is in the XY universality class on account of the emergent U(1) symmetry. The exact value of h 4 at which the change in fluctuation paths takes place is not known, because difficulties in analyzing the emergent U(1) symmetry when the scaling dimension y 4 of h 4 is only very slightly negative (y 4 ≈ −0.1) [21,53].…”