What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution

An 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 <i>Z</i>, 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 <i>Z</i> to be minimally plausible. Specifically, we present six necessary—but not sufficient—conditions that <i>Z</i> must satisfy. These criteria will allow very quick plausibility checks of future claims. As illustrative examples, we discuss previous mistaken “solutions”, unveiling their shortcomings.