Abstract
The ice problem on a lattice can be expressed in terms of arrow configurations on bonds with equal numbers of arrows pointing to and away from the vertices. In some cases vertex (ice) models can be expressed as colouring problems. We use the colouring association in conjunction with an old heuristic argument of Pauling to (re)derive the exact results for the residual entropy of ice on the square and triangular lattices.
Similar content being viewed by others
References
Pauling, L.: J. Am. Chem. Soc. 57, 2680 (1935)
Nagle, J.F.: J. Math. Phys. 7, 1484 (1966)
Lieb, E.H.: Phys. Rev. 162, 162 (1967)
Lieb, E.H.: Phys. Rev. Lett. 18, 692 (1967)
Domb, C.: ‘Graph theory and embeddings’ and ‘Ising model’. In: Domb, C., Green, M.S. (eds.) Phase Transitions and Critical Phenomena, vol. III. Academic Press, London (1973)
Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London (1982)
Lieb, E.H., Wu, F.Y.: Two-dimensional ferroelectric models. In: Domb, C., Green, M.S. (eds.) Phase Transitions and Critical Phenomena, vol. I. Academic Press, London (1972)
Baxter, R.J.: J. Math. Phys. 10, 1211 (1969)
Baxter, R.J.: J. Math. Phys. 11, 784 (1970)
Baxter, R.J.: J. Phys. A, Math. Gen. 19, 2821 (1986)
Baxter, R.J.: J. Phys. A, Math. Gen. 20, 5241 (1987)
Fisher, M.E.: Rep. Prog. Phys. 30, 615 (1967)
Edwards, S.F.: Proc. R. Soc. 85, 613 (1965)
Thompson, C.J.: J. Phys. A, Math. Gen. 9, 425 (1976)
Author information
Authors and Affiliations
Corresponding author
Additional information
To mark the 90th birthday of Professor Cyril Domb
Rights and permissions
About this article
Cite this article
Thompson, C.J. Colouring Solutions of the Ice Problem. J Stat Phys 145, 647–651 (2011). https://doi.org/10.1007/s10955-011-0246-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-011-0246-3