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arXiv:2511.10393 · Physical review. E

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

Ming Li, Youjin Deng, Jesper Lykke Jacobsen, Jesús Salas

Published Feb 12, 2026• 0 citations
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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_{BB}) and FK clusters (R_{FK}). Our computed R_{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_{BB} is systematically larger than R_{FK}. Conversely, along the tricritical branch, R_{BB} and R_{FK} coincide within numerical accuracy, strongly suggesting that R_{BB}=R_{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.

DOI
10.1103/hsph-48k8

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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_{BB}) and FK clusters (R_{FK}). Our computed R_{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_{BB} is systematically larger than R_{FK}. Conversely, along the tricritical branch, R_{BB} and R_{FK} coincide within numerical accuracy, strongly suggesting that R_{BB}=R_{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.
We present a Monte Carlo study of the backbone and the shortest-path exponents of the two-dimensional QQQ-state Potts model in the Fortuin-Kasteleyn bond representation. We first use cluster algorithms to simulate the critical Potts model on the square lattice and obtain the backbone exponents dB=1.7320(3){d}_{B}=1.732\phantom{\rule{0.16em}{0ex}}0(3)dB​=1.7320(3) and 1.794(2) for Q=2,3Q=2,3Q=2,3, respectively. However, for large QQQ, the study suffers from serious critical slowing down and slowly converging finite-size corrections. To overcome these difficulties, we consider the O(n)\mathrm{O}(n)O(n) loop model on the honeycomb lattice in the densely packed phase, which is regarded to correspond to the critical Potts model with Q=n2Q={n}^{2}Q=n2. With a highly efficient cluster algorithm, we determine from domains enclosed by the loops dB=1.64339(5),1.73227(8),1.7938(3),1.8384(5),1.8753(6){d}_{B}=1.643\phantom{\rule{0.16em}{0ex}}39(5),1.732\phantom{\rule{0.16em}{0ex}}27(8),1.793\phantom{\rule{0.16em}{0ex}}8(3),1.838\phantom{\rule{0.16em}{0ex}}4(5),1.875\phantom{\rule{0.16em}{0ex}}3(6)dB​=1.64339(5),1.73227(8),1.7938(3),1.8384(5),1.8753(6) for Q=1,2,3,2+3,4Q=1,2,3,2+\sqrt{3},4Q=1,2,3,2+3​,4, respectively, and dmin=1.0945(2),1.0675(3),1.0475(3),1.0322(4){d}_{\mathrm{min}}=1.094\phantom{\rule{0.16em}{0ex}}5(2),1.067\phantom{\rule{0.16em}{0ex}}5(3),1.047\phantom{\rule{0.16em}{0ex}}5(3),1.032\phantom{\rule{0.16em}{0ex}}2(4)dmin​=1.0945(2),1.0675(3),1.0475(3),1.0322(4) for Q=2,3,2+3,4Q=2,3,2+\sqrt{3},4Q=2,3,2+3​,4, respectively. Our estimates significantly improve over the existing results for both dB{d}_{B}dB​ and dmin{d}_{\mathrm{min}}dmin​. Finally, by studying finite-size corrections in backbone-related quantities, we conjecture an exact formula as a function of nnn for the leading correction exponent.
For the two-dimensional QQQ-state Potts model at criticality, we consider Fortuin-Kasteleyn and spin clusters and study the average number ${N}_{\ensuremath{\Gamma}}$ of clusters that intersect a given contour $\ensuremath{\Gamma}$. To leading order, ${N}_{\ensuremath{\Gamma}}$ is proportional to the length of the curve. Additionally, however, there occur logarithmic contributions related to the corners of $\ensuremath{\Gamma}$. These are found to be universal and their size can be calculated employing techniques from conformal field theory. For the Fortuin-Kasteleyn clusters relevant to the thermal phase transition, we find agreement with these predictions from large-scale numerical simulations. For the spin clusters, on the other hand, the cluster numbers are not found to be consistent with the values obtained by analytic continuation, as conventionally assumed.
Potts spin systems play a fundamental role in statistical mechanics and quantum field theory and can be studied within the spin, the Fortuin-Kasteleyn (FK) bond or the q-flow (loop) representation. We introduce a Loop-Cluster (LC) joint model of bond-occupation variables interacting with q-flow variables and formulate an LC algorithm that is found to be in the same dynamical universality as the celebrated Swendsen-Wang algorithm. This leads to a theoretical unification for all the representations, and numerically, one can apply the most efficient algorithm in one representation and measure physical quantities in others. Moreover, by using the LC scheme, we construct a hierarchy of geometric objects that contain as special cases the q-flow clusters and the backbone of FK clusters, the exact values of whose fractal dimensions in two dimensions remain as an open question. Our work not only provides a unified framework and an efficient algorithm for the Potts model but also brings new insights into the rich geometric structures of the FK clusters.
The q-state Potts model can be formulated in geometric terms, with Fortuin-Kasteleyn (FK) clusters as fundamental objects. For vanishing external field, the phase transition of the model can be equivalently described as a percolation transition of FK clusters. In this work, we investigate numerically the percolation behaviour along the line of first-order phase transitions of the 3d 3-state Potts model in a non-vanishing external field and find that the percolation strength exhibits a discontinuity along the entire line. The endpoint is also a percolation point for the FK clusters, but the corresponding critical exponents are neither in the Ising nor in the random percolation universality class. (C) 2002 Published by Elsevier Science B.V.
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Abstract We have considered clusters of like spins in the Q -Potts model, the spin Potts clusters (S clusters). Using Monte Carlo simulations, we studied these clusters on a square lattice with periodic boundary conditions for values of Q ∈ [1, 4]. We continue the work initiated by Delfino et al (2013 J. Stat. Mech. P11011) by measuring the universal finite size corrections of the two-point connectivity. The numerical data are perfectly compatible with the conformal field theory (CFT) prediction, thus supporting the existence of a consistent CFT, still unknown, describing the connectivity Potts spin clusters. We provided in particular new insights on the energy field of such theory. For Q = 2, we found a good agreement with the prediction that the Ising spin clusters behave as the Fortuin–Kasteleyn ones at the tri-critical point of the dilute one-Potts model. We show that the structure constants are likely to be given by the imaginary Liouville structure constants, consistently with the results by Delfino et al (2013 J. Stat. Mech. P11011); Ang and Sun (2021 arXiv:2107.01788). For Q ≠ 2 instead, the structure constants we measure do not correspond to any known bootstrap solutions. The validity of our analysis is backed up by the measures of the spin Potts cluster wrapping probability for Q = 3. We evaluate the main critical exponents and the correction to the scaling. A new exact and compact expression for the torus one-point of the Q -Potts energy field is also given.