PreprintCC BY 4.0arXiv
Droplets obstruct Onsager-type solutions of the Ising model in a magnetic field
Abstract
Onsager solved the two-dimensional Ising model in zero field, but no solution is known in a magnetic field. We prove that in any dimension d≥2, at every fixed dimensionless field outside a countable set, the free energy obeys no algebraic differential equation in the coupling, even locally. The obstruction arises from droplet geometry: the maximal droplet volume grows as the d/(d−1) power of its boundary size, while an integer band in the pressure coefficients protects the extremal contributions against cancellation. Our result excludes Onsager-type solutions and closed forms in elliptic, hypergeometric, or Painlevé functions of the coupling at these fields.
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BibTeX
@misc{luo2026droplets,
title = {{Droplets obstruct Onsager-type solutions of the Ising model in a magnetic field}},
author = {Ran Luo and Jiahua Tian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20569},
eprint = {2610.20569},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20569},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Droplets obstruct Onsager-type solutions of the Ising model in a magnetic field
AU - Ran Luo
AU - Jiahua Tian
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20569
UR - https://arxiv.org/abs/2610.20569
AB - Onsager solved the two-dimensional Ising model in zero field, but no solution is known in a magnetic field. We prove that in any dimension d≥2, at every fixed dimensionless field outside a countable set, the free energy obeys no algebraic differential equation in the coupling, even locally. The obstruction arises from droplet geometry: the maximal droplet volume grows as the d/(d−1) power of its boundary size, while an integer band in the pressure coefficients protects the extremal contributions against cancellation. Our result excludes Onsager-type solutions and closed forms in elliptic, hypergeometric, or Painlevé functions of the coupling at these fields.
ER -