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Modern Physics Letters A

EQUIVALENCE OF S3-INVARIANT LATTICE MODEL AND MODEL OF VERTEX-FACE TYPE

Herman E. Boos

Published Oct 30, 1993• 1 citation
Publisher

Abstract

We show that the S3-invariant model proposed in Ref. 2 is equivalent to vertex-face model with two kinds of Boltzmann weights, which appeared to satisfy the generalized Yang-Baxter equation. The more general solution of this equation is constructed in the framework of some appropriate ansatz. This solution depends on the continuous parameter q. The initial model is a particular case of q=±ie∓2π/3i. The integrability of this model is discussed.

DOI
10.1142/s0217732393002087

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We show that the S3-invariant model proposed in Ref. 2 is equivalent to vertex-face model with two kinds of Boltzmann weights, which appeared to satisfy the generalized Yang-Baxter equation. The more general solution of this equation is constructed in the framework of some appropriate ansatz. This solution depends on the continuous parameter q. The initial model is a particular case of q=±ie∓2π/3i. The integrability of this model is discussed.
A new type of Yang-Baxter equation (YBE) for RRR-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the ZNZ_NZN​ symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for N>2N>2N>2, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated RRR-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter RRR-matrix---realized as an intertwiner of cyclic representations of Uq(sl2)U_q(\mathfrak{sl}_2)Uq​(sl2​) at a root of unity---and Shastry's two-parameter RRR-operator obtained from the decorated YBE.
Abstract Vertex models more general than the ice model are possible and often have physical applications. The square lattice admits the general sixteen-vertex model of which the special cases, the eight- and the six-vertex model, are the most relevant and physically interesting, in particular through their connection to the one-dimensional integrable quantum mechanical models and the Bethe ansatz. This chapter introduces power- ful tools to examine vertex models, including the R- and L-matrices to encode the Boltzmann vertex weights and the monodromy and transfer matrices, which encode the integrability of the vertex models (i.e. that transfer matrices of different spectral parameters commute). This integrability is ultimately expressed in the Yang–Baxter relations.
In this paper we study, in the manner of Baxter [R. J. Baxter Exactly Solved Models in Statistical Mechanics (Academic, New York, 1982)], the possible integrable manifolds of the asymmetric eight-vertex model. As expected, they occur when the Boltzmann weights are either symmetric or satisfy the free-fermion condition; but our analysis clarifies the reason why both manifolds need to share a universal invariant. We also show that the free-fermion condition implies three distinct classes of integrable models.