We begin our countdown of important physics papers with the 1944 article by Lars Onsager, in which he provided the partition function for the two-dimensional Ising model.

Lars Onsager was a Norwegian-born scientist who first came to the attention of the world when, having never attained a PhD, he approached Peter Debye, informing him that his theory of electrolytes was incorrect. Debye took Onsager on as his assistant, and in 1928, at the age of 25, Onsager took a faculty position with Johns Hopkins University. Having no talent for teaching, Johns Hopkins dismissed Onsager after just one semester.

Onsager moved on to Brown, where he developed his namesake reciprocal relations of non-equilibrium thermodynamics. Although their importance was not recognized for some time, it was for this work that he was awarded the Nobel Prize in Chemistry in 1968.

Onsager moved on to Yale, where he was offered a postdoctoral fellow by the Chemistry department. Much to their surprise, Onsager had never received his PhD, and although the Chemistry department offered to accept his existing body of work as his dissertation, he insisted on creating new work for it. The resulting work on Mathieu functions was not standard chemistry fare, the Mathematics department at Yale considered the work so important that they offered Onsager a PhD in mathematics if the chemistry department would not. Ultimately, Lars Onsager earned his PhD in chemistry at the age of thirty.

During his time at Yale, Onsager rose quickly to become the J. Willard Gibbs Professor of Theoretical Chemistry, and in the 1940s he began developing a theory of continuous phase transitions. This is where the story begins.

The Ising model is a simple model of nearest neighbor interactions of spins on lattice sites. The idea behind it is that it provides a relatively simple microscopic description of ferromagnetism, but the exact details of its solutions were unknown for some time. In 1925, Ernst Ising solved the 1D version of the model in his dissertation, and proved that there could be no phase transition at finite temperature. However, the 2D Ising model proved to be much more complicated to solve.

In 1942, at a conference on phase transitions, Onsager stated without proof the partition function for the 2D Ising model, as well as the spontaneous magnetization at zero external field. This perplexed many of those present, as the Ising model in 2D had remained unsolved, and Lars Onsager simply stated the result almost as a challenge for his peers to match.

Two years later, Onsager provided a publication in which he invented what is now the theory of loop groups as a method of solution for the partition function. Working with Bruria Kauffman (who is now married to Willis Lamb), Onsager continued to flesh out his solution of the magnetization, publishing this result in 1949, the same year Kauffman published her solution through fermionization.

Various solutions of the 2D Ising model began to flow out: Chen-Ning Yang published a solution to the magnetization in 1952 [2], and from 1952-1963, papers by Kac, Ward, Potts, Hurst, Green and Kastelyn [3]-[7] established the combinatorial solution now commonly taught in graduate courses. By 1978, with the development of the renormalization group and the publication of an article “Physics in 3.99 dimensions”, Baxter and Enting published an article [8] poking fun at both sides entitled “The 399th Solution to the Ising Model”.

However, Onsager’s original paper is perhaps the most important. In developing his solution, he developed an entire branch of mathematics which now has broad applications in quantum field theory. The solution Onsager provided was the first of many on what is likely the most difficult exactly soluble problem theoretical physics has ever seen (and possibly ever will). For those reasons, the Onsager solution to the 2D Ising model is #10 on the list of the most important publications in physics.

[1] Onsager, Lars. “Crystal Statistics. I. A Two-dimensional Model with an Order-Disorder Phase”. Phys. Rev. 65, 117.

[2] Yang, C.N. “The Spontaneous Magnetization of the Two-dimensional Ising Model”. Phys. Rev. 85, 808.

[3] M. Kac and J.C. Ward. “Combinatorial solution of the 2-dimensional Ising model”. Phys. Rev. 88, 1332.

[4] R.B. Potts and J.C. Ward. “The combinatorial method and the two dimensional Ising model”. Prog. Theo. Phys. 13, 38.

[5] C.A. Hurst and H.S. Green. “New solution of the Ising problem for a rectangular lattice”. J. Chem. Phys. 33, 1059.

[6] Kasteleyn, P.W. “The statistics of dimers on a lattice, the number of dimer arrangements on a quadratic lattice”. Physica 27, 1209.

[7] Kasteleyn, P.W. “Dimer statistics and phase transitions”. J. Math. Phys. 4, 287.

[8] R.J. Baxter and I.G. Enting. “399th solution of the Ising model”. J. Phys. A 11, 2463.