What does it take to solve the 3D Ising model? Minimal necessary conditions for a valid solution

TL;DR

This paper establishes six necessary conditions that any claimed exact solution for the 3D Ising model's partition function must satisfy, providing a quick plausibility check to evaluate future proposed solutions.

Contribution

It introduces a set of six minimal necessary criteria to assess the plausibility of claimed exact solutions for the 3D Ising model's partition function.

Findings

Previous solutions fail to meet these criteria.

The criteria serve as a quick plausibility test.

Clarifies conceptual obstructions in solving the 3D Ising model.

Abstract

Exact solution of the Ising model on the simple cubic lattice is one of the long-standing open problems in rigorous statistical mechanics. Indeed, it is generally believed that settling it would constitute a methodological breakthrough, fomenting great prospects for further application, similarly to what happened when Lars Onsager solved the two dimensional model eighty years ago. Hence, there have been many attempts to find analytic expressions for the exact partition function $Z$, but all such attempts have failed due to unavoidable conceptual or mathematical obstructions. Given the importance of this simple yet paradigmatic model, here we set out clear-cut criteria for any claimed exact expression for $Z$ to be minimally plausible. Specifically, we present six necessary -- but not sufficient -- conditions that $Z$ must satisfy. These criteria will allow very quick plausibility checks of future claims. As illustrative examples, we discuss previous mistaken ``solutions,'' unveiling their shortcomings.