# History of XY model - Plane Rotator model

I would like to find out more about the history of the XY model. While for the Ising and Potts model it is easy to find information, it's quite difficult for me to understand when and where the XY model (or the plane rotator model) was first presented.

The KT transition is quite famous (1972), but even in the original paper by Kosterlitz and Thouless there's nothing about history.

When was the XY model formulated for the first time? Who was the first scientist to introduce it?

Finally found!

This evening I was reading an article by Silver et al:

• Silver, H., N. E. Frankel, and B. W. Ninham. “A Class of Mean Field Models.” Journal of Mathematical Physics 13, no. 4 (April 1, 1972): 468–74. doi:10.1063/1.1666002,

where you can find an explicit formula for the free energy of a class of mean field models (including XY model). The only references concerning the hamiltonian $H_N^{(\nu)}=-j \sum_{i,j} \boldsymbol{S}_i \cdot \boldsymbol{S}_j$, with $\boldsymbol{S}_i=\left (S_i(1),\dots,S_i(\nu) \right)$ $\nu$-dimensional vectors, are a few papers by Stanley:

• Stanley, H. E. “Spherical Model as the Limit of Infinite Spin Dimensionality.” Physical Review 176, no. 2 (December 10, 1968): 718–22. doi:10.1103/PhysRev.176.718,

• Stanley, H. E. “Exact Solution for a Linear Chain of Isotropically Interacting Classical Spins of Arbitrary Dimensionality.” Physical Review 179, no. 2 (March 10, 1969): 570–77. doi:10.1103/PhysRev.179.570.

The article of 1968 is more about the Berlin-Kac model. But thanks to it, I found an interesting historical paper written by Kac himself, where he tells the story of his model. I guess it's the first one with continuous spins (isn't it?)! The discovery is dated 1947 and the article I'm referring to is:

• Kac, M. “The Work of T.H. Berlin in Statistical Mechanics …a Personal Reminiscence.” Physics Today 17, no. 10 (October 1964). doi:10.1063/1.3051173.

The article of 1969 written by Stanley talks about the XY model and it refers to it as invoked in connection with certain magnetic insulators. It gives two references, in particular one by Betts and Lee:

• Betts, D. D., and M. H. Lee. “Critical Properties of the XY Model.” Physical Review Letters 20, no. 26 (June 24, 1968): 1507–10. doi:10.1103/PhysRevLett.20.1507.

Here there was no explicit date for the XY model, but a hint: Matsubara and Matsuda introduced in 1956 a lattice model for liquid helium which reduces to the XY model for a hard-core molecular-inter-22 action potential. The article is:

• Matsubara, Takeo, and Hirotsugu Matsuda. “A Lattice Model of Liquid Helium, I.” Progress of Theoretical Physics 16, no. 6 (December 1, 1956): 569–82. doi:10.1143/PTP.16.569.

But I wasn't done. In Stanley1969 I found another trail to follow: For a comprehensive introduction to exactly soluble models of interacting particles in one-dimension, see E. H. Lieb and D. C. Mattis, Mathematical Physics in One Dimension (Academic Press Inc., New York, 1966). The same Lieb and Mattis @Yvan Velenik reported in his answer. So I looked for the book and I finally found the answer (cap. 6, around pag 463). Actually the model was firstly considered by Yoichiro Nambu in a paper dated 1950:

• Nambu, Yôichirô. “A Note on the Eigenvalue Problem in Crystal Statistics.” Progress of Theoretical Physics 5, no. 1 (February 1, 1950): 1–13. doi:10.1143/ptp/5.1.1,

but it was denoted "XY model" for the first time in the article reported by Yvan Velenik:

• Two soluble models of an antiferromagnetic chain, Elliott Lieb, Theodore Schultz and Daniel Mattis, Annals of Physics, Volume 16, Issue 3, December 1961, Pages 407--466.

It seems that the XY model was firstly introduced for two different reasons (magnetic insulators and quantum mechanics) not before 50's and had its name in 1961. I hope the (hi)story is concluded!

As far as I know, this model (well, the quantum version of it) was introduced in the following paper:

• Two soluble models of an antiferromagnetic chain, Elliott Lieb, Theodore Schultz and Daniel Mattis, Annals of Physics, Volume 16, Issue 3, December 1961, Pages 407--466.

The paper can be found here. The model is introduced in Section II.

Let us also address your comment below, concerning the other $O(N)$-models (the XY model being the $O(2)$-model). The $O(3)$-model, often called the Heisenberg model, was introduced much earlier, in the following paper:

• Zur Theorie des Ferromagnetismus, W. Heisenberg, Zeitschrift für Physik, September 1928, Volume 49, Issue 9, pp 619--636.

Finally, the general $O(N)$-models seem to have been introduced in

• Dependence of Critical Properties on Dimensionality of Spins, H. E, Stanley, Phys. Rev. Lett. 20:589 (1968).

However, I wouldn't be surprised if it was independently introduced by several people in different contexts...

• In this case, the O(N) model was introduced later, I suppose. Have you got any references for its history? – fdesmond Mar 16 '17 at 19:33
• I think I found the answer. The first article you reported turns out to be where it was called "XY model". – fdesmond Mar 18 '17 at 20:41