Backbone three-point correlation function in the two-dimensional Potts model

TL;DR

This work probes the three-point connectivity of backbone versus FK clusters in the 2D $Q$-state Potts model using the FK representation and an efficient O$(n)$ loop Monte Carlo approach on the hexagonal lattice. By measuring the universal three-point amplitude ratios $R_ ext{FK}$ and $R_ ext{BB}$ and performing careful finite-size scaling, the authors validate the exact CFT prediction for $R_ ext{FK}$ and uncover a regime-dependent relationship with $R_ ext{BB}$: $R_ ext{BB}>R_ ext{FK}$ on the critical branch, while $R_ ext{BB}=R_ ext{FK}$ along the tricritical branch, suggesting a common universality class at tricriticality. The analysis covers both critical ($x_-$) and tricritical ($x_+$) regimes up to $Q=4$, with detailed fitting strategies to extract infinite-volume constants and discussion of potential logarithmic corrections at the four-state point. These results strengthen the geometric understanding of FK and backbone structures and point toward theoretical avenues for deriving $R_ ext{BB}(g)$ via imaginary Liouville or conformal loop ensemble techniques, while also leaving open the precise nature of corrections at tricriticality for $Q=4$.

Abstract

We study the three-point correlation function of the backbone in the two-dimensional $Q$-state Potts model using the Fortuin-Kasteleyn (FK) representation. The backbone is defined as the bi-connected skeleton of an FK cluster after removing all dangling ends and bridges. To circumvent the severe critical slowing down in direct Potts simulations for large $Q$, we employ large-scale Monte Carlo simulations of the O$(n)$ loop model on the hexagonal lattice, which is regarded to correspond to the Potts model with $Q=n^2$. Using a highly efficient cluster algorithm, we compute the universal three-point amplitude ratios for the backbone ($R_\text{BB}$) and FK clusters ($R_\text{FK}$). Our computed $R_\text{FK}$ exhibits excellent agreement with exact conformal field theory predictions, validating the reliability of our numerical approach. In the critical regime, we find that $R_\text{BB}$ is systematically larger than $R_\text{FK}$. Conversely, along the tricritical branch, $R_\text{BB}$ and $R_\text{FK}$ coincide within numerical accuracy, strongly suggesting that $R_\text{BB}=R_\text{FK}$ holds throughout this regime. This finding mirrors the known equality of the backbone and FK cluster fractal dimensions at tricriticality, jointly indicating that both structures share the same geometric universality.

Figures (7)

• Figure 1: (Color online) Schematic phase diagram of the dilute Potts model in the $(D,1/K)$ plane. The solid line denotes the critical line separating the ferromagnetic and paramagnetic phases, while the dashed line indicates the first-order transition. The blue arrows represent the directions of the RG flow. The green dot marks the stable critical fixed point that governs the critical line, and the red dot marks the tricritical fixed point, which is unstable.
• Figure 2: (Color online) Correspondence between the phase diagrams of the square-lattice dilute Potts model and the hexagonal-lattice O$(n)$ loop model. (a) For the dilute $Q$-state Potts model on the square lattice, the quantity $u \equiv e^{D}/(e^{K}-1)$ is calculated from known critical parameters $(K_c, D_c)$ (yellow) and tricritical parameters $(K_t, D_t)$ (green) Qian_05. The yellow branch represents stable fixed points (critical), while the green branch represents unstable fixed points (tricritical). Solid lines are guides to the eye. (b) For the O$(n)$ loop model on the hexagonal lattice, the analytical branches $x_\pm(n)$ from Eq. \ref{['eq-xpm']} are shown. A continuous phase transition occurs at $x_+$ from a dilute loop phase to a dense, critical phase, whose universality is governed by the stable fixed points at $x_-$. The mapping $Q = n^2$ establishes the correspondence: the stable critical branch of the Potts model corresponds to the $x_-$ branch, while the unstable tricritical branch corresponds to the $x_+$ critical line.
• Figure 3: (Color online) Three-point structure constant of the Potts model as a function of the Coulomb-gas coupling $g$. The solid line represents the exact result for $R_\text{FK}$ (see Appendix \ref{['sec-app']} for calculation details) Delfino_11Picco_13. The Monte Carlo data (Table \ref{['table:R']}) are shown as green squares for $R_\text{FK}$ and yellow circles for $R_\text{BB}$.
• Figure 4: (color online) The three-point structure constants $R_\text{FK}$ and $R_\text{BB}$ as functions of $r$ on the $x_-$ branch of the O$(n)$ loop model with $Q=n^2$. The panels (a), (b), and (c) are for $Q=1,2,3$, respectively. Data are shown for linear sizes $L=32,64,\ldots,4096,8092$, with each size represented by a distinct color. Dashed lines indicate the fits reported in Table \ref{['table:R']}.
• Figure 5: The three-point structure constants $R_\text{FK}$ and $R_\text{BB}$ as functions of $r$ on the $x_+$ branch of the O$(n)$ loop model with $Q=n^2$. The panels (a), (b), (c), and (d) are for $Q=1,2,3,4$, respectively. Data are shown for linear sizes $L=32,64,\ldots,4096,8092$, with each size represented by a distinct color. Dashed lines indicate the fits reported in Table \ref{['table:R']}.