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arXiv:2205.10708 · Nuclear Physics B

An Ising-type formulation of the six-vertex model

Vladimir V. Bazhanov, С. М. Сергеев

Published Dec 21, 2022• 9 citations

Abstract

We show that the celebrated six-vertex model of statistical mechanics (along with its multistate generalizations) can be reformulated as an Ising-type model with only a two-spin interaction. Such a reformulation unravels remarkable factorization properties for row to row transfer matrices, allowing one to uniformly derive all functional relations for their eigenvalues and present the coordinate Bethe ansatz for the eigenvectors for all higher spin generalizations of the six-vertex model. The possibility of the Ising-type formulation of these models raises questions about the precedence of the traditional quantum group description of the vertex models. Indeed, the role of a primary integrability condition is now played by the star-triangle relation, which is not entirely natural in the standard quantum group setting, but implies the vertex-type Yang-Baxter equation and commutativity of transfer matrices as simple corollaries. As a mathematical identity the emerging star-triangle relation is equivalent to the Pfaff-Saalschütz-Jackson summation formula, originally discovered by J. F. Pfaff in 1797. Plausibly, all vertex models associated with quantized affine Lie algebras and superalgebras can be reformulated as Ising-type models with only two-spin interactions.

DOI: 10.1016/j.nuclphysb.2022.116055

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Using the most elementary methods and considerations, the solution of the star-triangle condition (a2+b2-c2)/2ab = ((a’)^2+(b’)^2-(c’))^2/2a’b’ is shown to be a necessary condition for the extension of the operator coalgebra of the six-vertex model to a bialgebra. A portion of the bialgebra acts as a spectrum-generating algebra for the algebraic Bethe ansatz, with which higher-dimensional representations of the bialgebra can be constructed. The star-triangle relation is proved to be necessary for the commutativity of the transfer matrices T(a, b, c) and T(a’, b’, c’).

Abstract This chapter defines transfer matrices, and the existence of a one-parameter family of commuting transfer matrices is defined as the condition of integrability. The local star-triangle (Yang–Baxter) equation is introduced for vertex, spin, and face models and used to demonstrate the commutation of the transfer matrices. The star-triangle equation is solved for the six-vertex, eight-vertex, SOS, RSOS, hard hexagon, and chiral Potts models. The commutation of the transfer matrix with the related quantum spin chain is derived.

Using the most elementary methods and considerations, the solution of the star-triangle condition a 2 +b 2 c 2 2ab = (a 0 ) 2 +(b 0 ) 2 (c 0 ) 2 2a 0 b 0 is shown to be a necessary condition for the extension of the operator coalgebra of the six-vertex model to a bialgebra. A portion of the bialgebra acts as a spectrum-generating algebra for the algebraic Bethe ansatz, with which higher-dimensional representations of the bialgebra can be constructed. The star-triangle relation is proved to be necessary for the commutativity of the transfer matrices T (a;b;c) andT (a 0 ;b 0 ;c 0 ).

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We introduce and solve a special family of integrable interacting vertex models that generalizes the well known six-vertex model. In addition to the usual nearest neighbour interactions among the vertices, there exist extra hard-core interactions among pairs of vertices at larger distances. The associated row-to-row transfer matrices are diagonalized by using the recently introduced matrix product ansatz. Like for the relation of the six-vertex model with the XXZ quantum chain, the row-to-row transfer matrices of these new models are also the generating functions of an infinite set of commuting conserved charges. Among these charges we identify the integrable generalization of the XXZ chain that contains hard-core exclusion interactions among the spins. These quantum chains have already appeared in the literature. The present paper explains their integrability.