Walking, Weak first-order transitions, and Complex CFTs II. Two-dimensional Potts model at $Q>4$

Abstract: We study complex CFTs describing fixed points of the two-dimensional $Q$-state Potts model with $Q>4$. Their existence is closely related to the weak first-order phase transition and walking RG behavior present in the real Potts model at $Q>4$. The Potts model, apart from its own significance, serves as an ideal playground for testing this very general relation. Cluster formulation provides nonperturbative definition for a continuous range of parameter $Q$, while Coulomb gas description and connection to minimal models provide some conformal data of the complex CFTs. We use one and two-loop conformal perturbation theory around complex CFTs to compute various properties of the real walking RG flow. These properties, such as drifting scaling dimensions, appear to be common features of the QFTs with walking RG flows, and can serve as a smoking gun for detecting walking in Monte Carlo simulations. The complex CFTs discussed in this work are perfectly well defined, and can in principle be seen in Monte Carlo simulations with complexified coupling constants. In particular, we predict a pair of $S_5$-symmetric complex CFTs with central charges $c\approx 1.138 \pm 0.021 i$ describing the fixed points of a 5-state dilute Potts model with complexified temperature and vacancy fugacity.

Analysis of the Complex CFTs in the Two-Dimensional Potts Model for Q > 4

The paper presents a detailed investigation of complex Conformal Field Theories (CFTs) that emerge at the fixed points of the two-dimensional $Q$-state Potts model, specifically for $Q > 4$. The research contextualizes the existence of these CFTs within the framework of weak first-order phase transitions and the phenomenon known as "walking" Renormalization Group (RG) flows observed when $Q > 4$. The study uses the Potts model, a classic model in statistical physics renowned for its applicability and significance, as a means to empirically evaluate these occurrences.

Central to the exploration is the technique of complexifying coupling constants within lattice models to scrutinize Monte Carlo simulations that could reveal these complex CFTs. The authors propose and predict the emergence of a pair of $S_5$-symmetric complex CFTs, characterized by central charges $c \approx 1.138 \pm 0.021 i$, which depict the critical points of a 5-state dilute Potts model with complexified temperature and vacancy fugacity.

A primary theoretical focus of the study is the concept of 'walking' in RG flows, an essential feature indicating the presence of an exponential hierarchy between UV and IR scales, produced when the RG flow becomes extremely slow within a certain coupling regime. The paper highlights the critical observation that these complex CFTs exhibit 'drifting scaling dimensions,' a hallmark of QFTs with walking behavior, which suggests more subtle signatures within numerical simulations.

From the theoretical perspective, this exploration advances understanding toward a rigorous and systematic examination of the conditions for the emergence of complex CFTs within the Potts model and possibly other systems in statistical physics. Traditionally, Potts models, and specifically the one under scrutiny for $Q > 4$, are believed to undergo first-order phase transitions without conformal symmetry; the research challenges this notion by suggesting the occurrence of complex fixed points.

The study further engages with lattice constructions, specifically the use of loop and cluster formulations along with Coulomb gas approaches to expand on the nonperturbative definitions across a continuous range of $Q$ values. By testing these formulations against the spectrum of low-lying operators and conjectured permutation symmetries for noninteger Q, the authors reinforce the seminal nature of complex CFTs within statistical ensembles.

Ultimately, the work enriches the theoretical landscape by linking a strong numerical evidence base to theoretical paradigms predicated on the fundamental physics of phase transitions. Looking forward, such studies are poised to inform simulations that further reveal the power and utility of complex CFTs and the relationships among RG flows, phase transitions, and the multifaceted nature of conformal symmetry in quantum statistical systems. This progression opens interesting questions about the reach and application of these ideas in higher-dimensional theories and other systems exhibiting similar structural complexities.