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Percolation effects in the Fortuin-Kasteleyn Ising model on the complete graph

Sheng Fang, Zongzheng Zhou, and Youjin Deng
Phys. Rev. E 103, 012102 – Published 4 January 2021
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  • INTRODUCTION
  • SIMULATIONS AND OBSERVABLES
  • RESULTS
  • DISCUSSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    The Fortuin-Kasteleyn (FK) random-cluster model, which can be exactly mapped from the q-state Potts spin model, is a correlated bond percolation model. By extensive Monte Carlo simulations, we study the FK bond representation of the critical Ising model (q=2) on a finite complete graph, i.e., the mean-field Ising model. We provide strong numerical evidence that the configuration space for q=2 contains an asymptotically vanishing sector in which quantities exhibit the same finite-size scaling as in the critical uncorrelated bond percolation (q=1) on the complete graph. Moreover, we observe that, in the full configuration space, the power-law behavior of the cluster-size distribution for the FK Ising clusters except the largest one is governed by a Fisher exponent taking the value for q=1 instead of q=2. This demonstrates the percolation effects in the FK Ising model on the complete graph.

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    • Received 17 August 2020
    • Revised 27 October 2020
    • Accepted 16 November 2020

    DOI:https://doi.org/10.1103/PhysRevE.103.012102

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Classical statistical mechanicsCritical phenomena
    1. Physical Systems
    Classical spin modelsErdos-Renyi graph
    1. Techniques
    Finite-size scalingIsing modelMonte Carlo methods
    Statistical Physics & Thermodynamics

    Authors & Affiliations

    Sheng Fang1, Zongzheng Zhou2,*, and Youjin Deng1,3,†

    • 1Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
    • 2ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematics, Monash University, Clayton, Victoria 3800, Australia
    • 3MinJiang Collaborative Center for Theoretical Physics, College of Physics and Electronic Information Engineering, Minjiang University, Fuzhou 350108, China

    • *eric.zhou@monash.edu
    • †yjdeng@ustc.edu.cn

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    Vol. 103, Iss. 1 — January 2021

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    Images

    • Figure 1
      Figure 1

      The probability distribution of C1, the size of the largest cluster. Here fX1(x) (top) and fX1′(x) are respectively the probability density functions of X1=C1/V3/4 and X1′=C1/V2/3. The black dashed line in the top figure shows the rigorous limiting distribution fX1∞(x), shown in Eq. (3).

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    • Figure 2
      Figure 2

      The probability distribution of C2P, the size of the second largest cluster in the percolation sector. Here fX2P(x) is the probability density function of X2P=C2P/V2/3. The inset plots the average C2P versus V.

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    • Figure 3
      Figure 3

      The cluster-size distribution nP(s,V) in the percolation sector. The slope of the black dashed line is −5/2. The inset plots in log-log scale nP(s,V)s5/2 versus s/V2/3. It shows that the scaling function ñ(x) is consistent with 1/2π if x≪1.

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    • Figure 4
      Figure 4

      The probability distribution of C2I, the size of the second-largest cluster in the Ising sector. Here fX2I(x) is the probability density function of X2I:=C2I/(VlogV). The inset plots the average C2I/(VlogV) versus V.

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    • Figure 5
      Figure 5

      Log-log plot of the cluster-size distribution nI(s,V) versus s in the Ising sector. The slope of the black dashed line is −5/2. The inset shows the log-log plot of nI(s,V)s5/2 versus s/(VlogV), which implies that the scaling function is consistent with 1/2π when s/(VlogV)≪1.

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    • Figure 6
      Figure 6

      Plot to show the leading correction term of the size of the largest cluster C1. A straight line with slope 1/12 is to guide the eye.

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    • Figure 7
      Figure 7

      Plot of the distribution of C2, size of the second-largest cluster. Here fX2(x) (top) and fX2′(x) (bottom) are respectively the probability density function of X2=C2/(Vlog10V) and X2′=C2/V2/3. The curve in the top figure plots h(x), shown in Eq. (A1), which is the limiting distribution of X2 presented in Ref. [5].

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    • Figure 8
      Figure 8

      Log-log plot of n(s,V) versus s for various systems. The slopes of the two dashed lines are 5/2 and 7/3, respectively, corresponding the Fisher exponent taking the percolation and Ising values. The inset plot n(s,V)s5/2 versus s/V3/4 on a log-log scale, and the data suggest the scaling function is constant at 1/2π when s≪V3/4.

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    • Figure 9
      Figure 9

      The (KP,K) diagram and its associated RG flow. The line KP=K corresponds to the FK Ising model, and (KP,c,Kc) is the critical point.

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    • Figure 10
      Figure 10

      Plot to show the distribution of C2I, the size of the second largest cluster in the Ising sector. Here X2I:=C2I/(Vlog10V) and fX2I(x) is its probability density function. It can be seen that, for x<x0 with x0≈0.2, data from various systems are observed to collapse onto a straight line with slope −1/36.

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    • Figure 11
      Figure 11

      Finite-size scaling of the reduced susceptibility χ′ at the critical point pc.

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    • Figure 12
      Figure 12

      The finite-size scaling of the reduced susceptibility χ′ in (a) low-T and (b) high-T critical windows, and the thermodynamic-limit behavior of χ′ in (c) low-T and (d) high-T regions. The same range of vertical axis applies to panels (a) and (b) and to panels (c) and (d).

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