Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions

Abstract

Recently, we argued [Chin. Phys. Lett. $39$, 080502 (2022)] that the Ising model simultaneously exhibits two upper critical dimensions $(d_c=4,d_p=6)$ in the Fortuin-Kasteleyn (FK) random-cluster representation. In this paper, we perform a systematic study of the FK Ising model on hypercubic lattices with spatial dimensions $d$ from 5 to 7, and on the complete graph. We provide a detailed data analysis of the critical behaviors of a variety of quantities at and near the critical points. Our results clearly show that many quantities exhibit distinct critical phenomena for $4<d<6$ and $d\ge 6$, and thus strongly support the argument that $6$ is also an upper critical dimension. Moreover, for each studied dimension, we observe the existence of two configuration sectors, two lengthscales, as well as two scaling windows, and thus, two sets of critical exponents are needed to describe these behaviors. Our finding enriches the understanding of the critical phenomena in the Ising model.

17 pages, 17 figures

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• Kinetic Theory of Random Graphs: from Paths to Cycles
• Finite-size scaling above the upper critical dimension
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• Geometric explanation of anomalous finite-size scaling in high dimensions
• Geometric Upper Critical Dimensions of the Ising Model
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• Finite-Size Scaling of the High-Dimensional Ising Model in the Loop Representation
• Logarithmic Finite-Size Scaling of the Four-Dimensional Ising Model
• Exploring the Nexus between Thermodynamic Phase Transitions and Geometric Fractals through Systematic Lattice Point Classification
• Anomalous dimensions from conformal field theory: Generalized ${\varphi }^{2n+1}$ theories
• Percolation in the two-dimensional Ising model
• Anomalous criticality coexists with giant cluster in the uniform forest model