Transfer-matrix inverison identities for exactly solvable lattice-spin models

Paul A. Pearce

doi:10.1103/PhysRevLett.58.1502

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Transfer-matrix inverison identities for exactly solvable lattice-spin models

Paul A. Pearce

Mathematics Department, University of Melbourne, Parkville, Victoria 3052, Australia

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Phys. Rev. Lett. 58, 1502 – Published 13 April, 1987

DOI: https://doi.org/10.1103/PhysRevLett.58.1502

Abstract

A new functional equation satisfied by the commuting row-to-row transfer matrices is derived for the eight-vertex model. Functional equations of the same form are satisfied by hard hexagons, magnetic hard squares, the self-dual Potts models, the Andrews-Baxter-Forrester models, and others. These new functional equations are called inversion identities because they generalize the inversion relation for local transfer matrices. It is conjectured that all solvable models satisfying Yang-Baxter equations possess such inversion identities and that, in general, these functional equations can be solved for the transfer-matrix eigenvalues.

References (11)

.1C R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, London, 1982).
.1C See, e.g., E. Date, M. Jimbo, T. Miwa, and M. Okada, Lett. Math. Phys. (to be published); Y. Akutsu, A. Kuniba, and M. Wadati, ``Inhomogeneous Eight-Vertex SOS Model and Solvable IRF Hierarchies'' (to be published).
.1C R. J. Baxter, Ann. Phys. (N.Y.) 70, 193 (1972), and J. Stat. Phys. 8, 25 (1973); J. D. Johnson, S. Krinsky and B. M. McCoy, Phys. Rev. A 8, 2526 (1973).
.1C R. J. Baxter, J. Phys. A 13, L61 (1980); R. J. Baxter and P. A. Pearce, 15, 897 (1982).
.1C R. J. Baxter and P. A. Pearce, J. Phys. A 16, 2239 (1983).
.1C P. A. Pearce, J. Phys. A 18, 3217 (1985); P. A. Pearce and D. Kim, to be published.
.1C P. A. Pearce, to be published.
.1C G. E. Andrews, R. J. Baxter, and P. J. Forrester, J. Stat. Phys. 35, 193 (1984). The inversion relations are given by equations (A11) and (A12); the derivation of the ABF inversion identity is due to R. J. Baxter, unpublished.
.1C Y. G. Stroganov, Phys. Lett. 74A, 116 (1979); R. J. Baxter, J. Stat. Phys. 28, 1 (1982).
.1C I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965).
.1C R. J. Baxter, in Fundamental Problems in Statistical Mechanics V, edited by E. G. D. Cohen (North-Holland, Amsterdam, 1980).

[c1] .1C R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, London, 1982).

[c2] .1C See, e.g., E. Date, M. Jimbo, T. Miwa, and M. Okada, Lett. Math. Phys. (to be published); Y. Akutsu, A. Kuniba, and M. Wadati, ``Inhomogeneous Eight-Vertex SOS Model and Solvable IRF Hierarchies'' (to be published).

[c3] .1C R. J. Baxter, Ann. Phys. (N.Y.) 70, 193 (1972), and J. Stat. Phys. 8, 25 (1973); J. D. Johnson, S. Krinsky and B. M. McCoy, Phys. Rev. A 8, 2526 (1973).

[c4] .1C R. J. Baxter, J. Phys. A 13, L61 (1980); R. J. Baxter and P. A. Pearce, 15, 897 (1982).

[c5] .1C R. J. Baxter and P. A. Pearce, J. Phys. A 16, 2239 (1983).

[c6] .1C P. A. Pearce, J. Phys. A 18, 3217 (1985); P. A. Pearce and D. Kim, to be published.

[c7] .1C P. A. Pearce, to be published.

[c8] .1C G. E. Andrews, R. J. Baxter, and P. J. Forrester, J. Stat. Phys. 35, 193 (1984). The inversion relations are given by equations (A11) and (A12); the derivation of the ABF inversion identity is due to R. J. Baxter, unpublished.

[c9] .1C Y. G. Stroganov, Phys. Lett. 74A, 116 (1979); R. J. Baxter, J. Stat. Phys. 28, 1 (1982).

[c10] .1C I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965).

[c11] .1C R. J. Baxter, in Fundamental Problems in Statistical Mechanics V, edited by E. G. D. Cohen (North-Holland, Amsterdam, 1980).

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Transfer-matrix inverison identities for exactly solvable lattice-spin models

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