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IN RETROSPECT

Fifty years of vertices on ice

Nature Reviews Physics volume 3, page 727 (2021)Cite this article

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Most materials have a well-defined ground state, that is, when cooled to absolute zero they take on a single lowest-energy configuration, such as a crystal structure, and their entropy goes to zero. But one of the strange properties of water is that it doesn’t do this, and instead has a ‘residual entropy’ that scales with the number of atoms. A model to explain this behaviour was first introduced in 1935 by Linus Pauling, who gave an accurate estimate of the residual entropy of ice, but it was the 1960s and 1970s that saw a flurry of activity towards not just estimating but exactly solving related 2D models, known as vertex models. A key step towards understanding vertex models was made 50 years ago this year, when Rodney Baxter published an exact solution to the eight-vertex model.

Replacing the tetrahedra of water ice with a square lattice and the hydrogen positions with arrows leads to a vertex model. Enforcing the two-in, two-out rule leads to the six-vertex model; incorporating additional all-in, all-out vertices leads to the eight-vertex model (pictured). Exactly solving the eight-vertex model involves calculating its partition function. Baxter was motivated by the earlier work of Elliott Lieb, who had solved the six-vertex model by making an educated guess (the Bethe ansatz) at the transfer matrix (which gives the statistical weight of one row of the lattice having a given configuration and its neighbouring row having another given configuration). However, as Baxter noted in his memoirs, the eight-vertex model “didn’t have the ‘conservation of down arrows’ property that [the six-vertex model] did, so one couldn’t use the Bethe ansatz and was stuck.” However, while on holiday in England, Baxter had noticed that the transfer matrices commuted with each other, a property that made it possible to find a solution, which he worked on while on the ship back to Australia.

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References

Original article

  • Baxter, R. J. Eight-vertex model in lattice statistics. Phys. Rev. Lett. 26, 832–833 (1971)

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Related articles

  • Wang, R. F. et al. Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature 439, 303–306 (2006)

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  • Cugliandolo, L. F. Artificial spin-ice and vertex models. J. Stat. Phys. 167, 499–514 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  • Baxter, R. An Accidental Academic (2017)

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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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Correspondence to Zoe Budrikis.

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Budrikis, Z. Fifty years of vertices on ice. Nat Rev Phys 3, 727 (2021). https://doi.org/10.1038/s42254-021-00393-4

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