Is the 4D Ising model a second-order phase transition?

Question

I have been reading several papers on the Higgs, the 4D Ising model, and $\varphi^4$ theory, and I noticed that some papers support a weak first-order transition scenario. For example, in arXiv:2111.08962, the results of lattice simulations of the 4D Ising model are used to discuss the possibility of a weak first-order transition.

Of course, I understand that the Higgs is not a single-component scalar and interacts with fermions and other fields, making its nature less straightforward. However, I believe it has been rigorously proven (e.g., Aizenman 1986) that the 4D Ising model undergoes a strict second-order phase transition in the thermodynamic limit.

Why, then, do studies on the Higgs and related topics support the computational results of a weak first-order transition? Or is it that the 4D Ising model has not been strictly proven to undergo a second-order phase transition?

Answer 1

The Ising model has a continuous phase transition in every dimension $d\geq 2$ (that is, the magnetization vanishes continuously at the critical temperature). This was proved by Aizenman and Fernandez for dimensions $d\geq 4$ and by Aizenman, Duminil-Copin and Sidoravicius for the 3-dimensional case. The 2-dimensional case follows from the exact computation of the spontaneous magnetization by Yang, using the techniques pioneered by Onsager (although there exist softer proofs nowadays).

It has also been proved by Gunaratnam, Panagiotis, Panis and Severo that the magnetization vanishes continuously at the critical temperature in the (nearest-neighbor) $\varphi^4$ model on $\mathbb{Z}^d$ with $d\geq 3$.