Galois conjugation, droplet growth, and the differential transcendence of the Ising model in nonzero field

Abstract

The square-lattice Ising model is exactly solvable in zero magnetic field, but the scope of exact solutions at nonzero field remains unresolved. We show that, at every fixed dimensionless field $H=h/(k_BT)\ne 0$, its free energy and magnetization obey no algebraic differential equation in the coupling $K=J/(k_BT)$. This excludes Onsager-type formulas and broad classes of special-function expressions. Previous work left open a countable set of fields with algebraic-integer fugacities. We eliminate these exceptions using Galois conjugation and the large-order growth of droplets of reversed spins. At conjugate fugacities outside the unit circle, nearly square droplets produce coefficient growth incompatible with an algebraic differential equation. We prove that cancellations cannot remove this growth in the full pressure series. The free-energy result also holds for the Lieb lattice and square lattices with equal-length chain decorations and uniform coupling and field. In every dimension $d\ge2$, we additionally resolve fields for which $e^{-2|H|}$ has a real algebraic conjugate greater than one.