Geometric Upper Critical Dimensions of the Ising Model

The upper critical dimension of the Ising model is known to be d_\rm c=4, above which critical behavior is regarded to be trivial. We hereby argue from extensive simulations that, in the random-cluster representation, the Ising model simultaneously exhibits two upper critical dimensions at (d_\rm c=4,~d_\rm p=6), and critical clusters for d \geq d_\rm 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.