Abstract
Elliott Lieb's ice-type models opened up the whole field of solvable models in statistical mechanics. Here we discuss the “commuting transfer matrix” T,Qequations for these models, writing them in a more explicit and transparent notation that we believe offers new insights. The approach manifests the relationship between the six-vertex and chiral Potts models, and between the eight-vertex and Kashiwara–Miwa models.
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REFERENCES
E. H. Lieb, Exact solution of the problem of the entropy of two-dimensional ice, Phys. Rev. Lett. 18:692–694 (1967).
E. H. Lieb, Exact solution of the problem of the Fmodel of an antiferroelectric, Phys. Rev. Lett. 18:1046–1048 (1967).
E. H. Lieb, Exact solution of the two-dimensional Slater KDP model of a ferroelectric, Phys. Rev. Lett. 19:108–110 (1967).
E. H. Lieb, Residual entropy of square ice, Phys. Rev. 162:162–172 (1967).
B. Sutherland, Exact solution of a two-dimensional model for hydrogen bonded crystals, Phys. Rev. Lett. 19:103–104 (1967).
L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65:117–149 (1944).
R. J. Baxter, Eight-vertex model in lattice statistics, Phys. Rev. Letters 26:832–833 (1971).
R. J. Baxter, Eight-vertex model in lattice statistics, Ann. Phys. (N.Y.) 70:193–228 (1972).
R. J. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain, Ann. Phys. (N.Y.) 76:1–71 (1973).
R. J. Baxter, Exactly Solved Models in Statistical Mechanics(Academic, London, 1982).
R. J. Baxter, The Yang-Baxter equations and the Zamolodchikov model, Physica D 18:321–347 (1986).
V. V. Bazhanov and R. J. Baxter, Partition function of a three-dimensional solvable model, Physica A 194:390–396 (1993).
V. V. Bazhanov and Y. G. Stroganov, Chiral Potts model as a descendant of the sixvertex model, J. Statist. Phys. 59:799–817 (1990).
R. J. Baxter, V. V. Bazhanov, and J. H. H. Perk, Functional relations for transfer matrices of the chiral Potts model, Internat. J. Modern Phys. B 4:803–870 (1990).
T. Deguchi, K. Fabricius, and B. M. McCoy, The sl2loop algebra symmetry of the sixvertex model at roots of unity, J. Statist. Phys. 102:701 (2001).
K. Fabricius and B. M. McCoy, Bethe's equation is incomplete for the XXZ model at roots of unity, J. Statist. Phys. 103:647–678 (2001).
K. Fabricius and B. M. McCoy, Completing Bethe's equations at roots of unity, J. Statist. Phys. 104:575 (2001).
K. Fabricius and B. M. McCoy, Evaluation parameters and Bethe roots for the six-vertex model at roots of unity, LANL pre-print, condmat/010857.
R. J. Baxter, Completeness of the Bethe ansatz for the six and eight-vertex models, J. Statist. Phys. 108:1–48 (2002).
P. A. M. Dirac, The Principles of Quantum Mechanics(Clarendon Press, Oxford, 1958), pp. 36 and 53.
H. Weyl, The Classical Groups: Their Invariants and Representations(Princeton University Press, 1946), p. 116.
M. Kashiwara and T. Miwa, A class of elliptic solutions to the star-triangle relation, Nuclear Phys. B 275:121–134 (1986).
K. Hasegawa and Y. Yamada, Algebraic derivation of the broken ZN-symmetric model, Phys. Lett. A 146:387–396 (1990).
V. A. Fateev and A. B. Zamolodchikov, Self-dual solutions of the star-triangle relations in ZNmodels, Phys. Lett. A 92:37–39 (1982).
R. J. Baxter, J. H. H. Perk, and H. Au-Yang, New solutions of the star-triangle relations for the chiral Potts model, Phys. Lett. A 128:138–142 (1988).
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Baxter, R.J. The Six and Eight-Vertex Models Revisited. Journal of Statistical Physics 116, 43–66 (2004). https://doi.org/10.1023/B:JOSS.0000037215.07702.93
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DOI: https://doi.org/10.1023/B:JOSS.0000037215.07702.93