Geometric Upper Critical Dimensions of the Ising Model

TL;DR

This paper reveals that the Ising model exhibits two upper critical dimensions, 4 and 6, with critical clusters showing percolation universality above dimension 6, based on extensive simulations across multiple dimensions.

Contribution

It introduces the concept of dual upper critical dimensions for the Ising model and demonstrates their geometric implications through extensive simulation data.

Findings

Identification of two upper critical dimensions at 4 and 6.

Critical clusters follow percolation universality for dimensions ≥6.

Evidence from simulations in dimensions 4 to 7 and complete graphs.

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 $d \geq d_p$, 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.