Transfer matrices

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 6 December 2011

Lattice models and Transfer matrices

Let A be a vector space with basis {fμ} and let V be a vector space with basis { eα}.

As operators on AAV, R12(u,v) L13(v) L23(u) = L23(u) L13(v) R12(u,v) and this implies that, as operators on AA VN,

R12(u,v) T13(v) T23(u) = T23(u) T13(v) R12(u,v) (1.1)
Multiplying on the left by R12(u,v) -1 and taking TrAA gives 𝒯(u) 𝒯(v) = 𝒯(v) 𝒯(u).

Remark: In many applications R(u,v) = R(u-v) and the quantum Yang-Baxter equation (QYBE) gives R12(u-v) R13(v) R23(u) = R23(u) R13(v) R12(u-v) , with L(u) =R(u) gives (1.1).

Notes and References

This page is based on section 5.2 of [dG]. The formula ??? appears as ??? in [TF] who quote [Bax].


[Bax] R. Baxter, One-dimensional anisotropic Heisenberg chain, Ann. Phys. 70 (1972), 323-337.

[dG] J. de Gier, Random Tilings and Solvable Lattice Models, Ph.D Thesis, University of Amsterdam, 1998.

[TF] L.A. Takhtajan and L.D. Faddeev, The quantum method of the inverse problem and the Heisenberg XYZ model, Russian Math Surveys 34 5 (1979), 11-68.

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