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

Victor Gorbenko, Slava Rychkov, Bernardo Zan

TL;DR

The paper investigates walking RG behavior in the 2d Q-state Potts model for $Q>4$ by analytic continuation from the solvable regime $Q\le 4$, uncovering a pair of complex CFTs $\mathcal{C}$ and $\overline{\mathcal{C}}$ that control the near-critical flow. It combines cluster/spin formulations, the Coulomb gas, and conformal perturbation theory to compute drifting scaling dims and test the walking scenario, including one- and two-loop CPT results and consistency checks like Im-flip. The analysis predicts two $S_5$-symmetric complex CFTs with central charges $c\approx 1.138\pm 0.021 i$ at $Q=5$ and describes a real walking trajectory that passes between them, yielding an exponentially large correlation length and observable drifting dimensions along the flow. The work provides a framework for identifying walking in Monte Carlo simulations and suggests broader applicability to other walking systems and higher dimensions, illustrating how complex fixed points can govern real RG flows while remaining well-defined nonperturbatively.

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.

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

TL;DR

The paper investigates walking RG behavior in the 2d Q-state Potts model for by analytic continuation from the solvable regime , uncovering a pair of complex CFTs and that control the near-critical flow. It combines cluster/spin formulations, the Coulomb gas, and conformal perturbation theory to compute drifting scaling dims and test the walking scenario, including one- and two-loop CPT results and consistency checks like Im-flip. The analysis predicts two -symmetric complex CFTs with central charges at and describes a real walking trajectory that passes between them, yielding an exponentially large correlation length and observable drifting dimensions along the flow. The work provides a framework for identifying walking in Monte Carlo simulations and suggests broader applicability to other walking systems and higher dimensions, illustrating how complex fixed points can govern real RG flows while remaining well-defined nonperturbatively.

Abstract

We study complex CFTs describing fixed points of the two-dimensional -state Potts model with . Their existence is closely related to the weak first-order phase transition and walking RG behavior present in the real Potts model at . 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 , 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 -symmetric complex CFTs with central charges describing the fixed points of a 5-state dilute Potts model with complexified temperature and vacancy fugacity.

Paper Structure

This paper contains 20 sections, 86 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. 2d Potts model for $Q\leqslant 4$
  3. Lattice transfer matrix and local operators
  4. Symmetry
  5. Height representation
  6. Long cylinder partition function and the Coulomb gas
  7. Torus partition function
  8. Spectrum of primaries for $Q\leqslant 4$
  9. Analytic continuation to $Q> 4$
  10. Walking RG flow in $Q>4$ Potts models
  11. One-loop beta-function
  12. Im-flip for other operators
  13. Drifting scaling dimensions
  14. Two-loop beta-function
  15. General arguments about the real flow
  16. ...and 5 more sections

Figures (13)

  • Figure 1: This graph corresponds to the connectivity state described by a partition $P=\{\{2\},\{1,1',3\}, \{2',3'\}\}$.
  • Figure 2: Passing from a cluster configuration to an oriented loop configuration (figure from diFrancesco:1987qf).
  • Figure 3: Coupling constant as a function of $Q\leqslant 4$ for the critical (lower branch, orange) and tricritical (upper branch, blue) Potts model.
  • Figure 4: Vertices of the 6-vertex model.
  • Figure 5: Passing from the oriented loop model to the $F$-model.
  • ...and 8 more figures