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

TL;DR

This study investigates the three-point correlation function of the backbone in the 2D Potts model, using advanced Monte Carlo simulations to validate theoretical predictions and explore universal properties across critical and tricritical regimes.

Contribution

The paper introduces a novel numerical approach to compute the three-point amplitude ratios for the backbone, confirming their relation to FK clusters and universal behavior in the Potts model.

Findings

$R_{FK}$ matches conformal field theory predictions

$R_{BB}$ is larger than $R_{FK}$ at criticality

$R_{BB}$ and $R_{FK}$ coincide at tricriticality

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 biconnected 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.