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arXiv:hep-th/9305100 · Journal of Mathematical Physics

Vertex operators in solvable lattice models

Omar E. Foda, Michio Jimbo, Tetsuji Miwa, Kei Miki, Atsushi Nakayashiki

Published Jan 1, 1994• 76 citations
arXivHTMLPDF

Abstract

The basic properties of q-vertex operators are formulated in the context of the Andrews–Baxter–Forrester (ABF) series, as an example of face interaction models, the q-difference equations satisfied by their correlation functions are derived, and their connection with representation theory established. The q-difference equations of the Kashiwara–Miwa (KM) series are discussed as an example of edge interaction models. Next, the Ising model, the simplest special case of both ABF and KM series, is studied in more detail using the Jordan–Wigner fermions. In particular, all matrix elements of vertex operators are calculated.

DOI
10.1063/1.530783

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A crash course on the Yang–Baxter equation and its applications in the statistical mechanics of lattice vertex models is presented. Using the simple example of a one-dimensional lattice gas, the basic terminology and standard mathematical procedures of statistical mechanics are illustrated. The coproduct notation is introduced via a discussion of the two-dimensional dimer model, and the transfer matrix formulation is elaborated further. The algebraic Bethe Ansatz (ABA) is introduced in the context of the ice model, leading to the Yang–Baxter equations. The motivation for introducing the ABA is developed, emphasizing the similarity to the quantum oscillator problem, in features such as the generation of “excited states”, and the determination of the eigenvalue spectrum. Finally, these methods from the theory of lattice models are applied to the case of a quantum many-body problem, the Heisenberg chain. From the condition that the monodromy matrices satisfy the Yang–Baxter equation, a complete set of mutually commuting operators is derived, and the Heisenberg chain problem is completely solved.PACS Nos.: 02.20.Uw, 24.10.Cn, 87.10.Hk, 05.20.–y
In this paper we review the theory of the Yang–Baxter equation related to the 6-vertex model and its higher spin generalizations. We employ a 3D approach to the problem. Starting with the 3D R-matrix, we consider a two-layer projection of the corresponding 3D lattice model. As a result, we obtain a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). In the simplest case of the spin s=1/2 this R-matrix naturally reduces to the R-matrix of the 6-vertex model. Taking a special limit in our construction we also obtain new formulas for the Q-operators acting in the representation space of arbitrary (half-)integer spin. Remarkably, this construction can be naturally extended to any complex values of spin s. We also give all functional equations satisfied by the transfer-matrices and Q-operators.
Exploring a mapping among n -state spin and vertex models on the square lattice, we argue that a given integrable spin model with edge weights satisfying the rapidity difference property can be formulated in the framework of an equivalent solvable vertex model. The Lax operator and the R-matrix associated to the vertex model are built in terms of the edge weights of the spin model and these operators are shown to satisfy the Yang-Baxter algebra. The unitarity of the R-matrix follows from an assumption that the vertical edge weights of the spin model satisfy certain local identities known as inversion relation. We apply this embedding to the scalar n -state Potts model and we argue that the corresponding R-matrix can be written in terms of the underlying Temperley-Lieb operators. We also consider our construction for the integrable Ashkin-Teller model and the respective R-matrix is expressed in terms of sixteen distinct weights parametrized by theta functions. We comment on the possible extension of our results to spin models whose edge weights are not expressible in terms of the difference of spectral parameters.
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