Geometric Upper Critical Dimensions of the Ising Model

Abstract

The upper critical dimension of the Ising model is known to be $d_c=4$, above which critical behavior is regarded as trivial. We hereby argue from extensive simulations that, in the random-cluster representation, the Ising model simultaneously exhibits two upper critical dimensions at $(d_c=4,d_p=6)$, and critical clusters for $d\ge d_p$, except the largest one, are governed by exponents from percolation universality. We predict a rich variety of geometric properties and then provide strong evidence in dimensions from 4 to 7 and on complete graphs. Our findings significantly advance the understanding of the Ising model, which is a fundamental system in many branches of physics.

arXiv comment6 pages, 4 figures

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• Logarithmic Finite-Size Scaling of the Four-Dimensional Ising Model
• Anomalous dimensions from conformal field theory: Generalized ${\varphi }^{2n+1}$ theories
• Percolation in the two-dimensional Ising model
• Universality of the complete-graph Potts model with $0<q\le 2$
• Anomalous criticality coexists with giant cluster in the uniform forest model