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In retrospect

100 years of the Ising model

Nature Reviews Physics volume 6, page 530 (2024)Cite this article

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In his 1924 PhD thesis, Ernst Ising solved a 1D version of a model for magnetic materials that had been put forward a few years earlier by his supervisor, Wilhelm Lenz. He was hoping to explain some of the properties of ferromagnets, but his work revealed that the model could not support a ferromagnetic ordered state at any temperature. Disappointed, he gave up physics research altogether. Remarkably, 100 years later the model is one of the best-known models in statistical mechanics, with variants of it used to describe not only magnetic systems but also systems as diverse as atoms in alloys, optimization problems, and even land use and public opinion.

Ising had assumed that a similar argument would follow for two and three dimensions, making the model useless for understanding ferromagnetism. In the field of magnetism, attention quickly turned to the Heisenberg model, in which spins are 3D unit vectors. However, by the 1930s, physicists had realized that the Ising model was equivalent to independently derived models for order and disorder in metallic alloys and for adsorption isotherms. It was this wider applicability of the model that caught the attention of physicists, and through the 1930s physicists developed arguments for why the 2D Ising model does have a transition to a magnetized state at low enough temperatures. This work culminated in 1942, with Lars Onsager’s announcement that he had solved — found an exact expression for the partition function of — the 2D Ising model in zero magnetic field. However, to this day, an exact solution for the Ising model in 3D remains unknown.

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  1. Nature Reviews Physics https://www.nature.com/natrevphys/

    Zoe Budrikis

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  1. Zoe Budrikis
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Budrikis, Z. 100 years of the Ising model. Nat Rev Phys 6, 530 (2024). https://doi.org/10.1038/s42254-024-00760-x

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