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arXiv:2501.02377 · Nuclear Physics B

Embedding integrable spin models in solvable vertex models on the square lattice

Marcio Jose Martins

Published Feb 20, 2025• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1016/j.nuclphysb.2025.116849

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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.
In this paper we investigate a correspondence among spin and vertex models with the same number of local states on the square lattice with toroidal boundary conditions. We argue that the partition functions of an arbitrary n -state spin model and of a certain specific n -state vertex model coincide for finite lattice sizes. The equivalent vertex model has n 3 non-null Boltzmann weights and their relationship with the edge weights of the spin model is explicitly presented. In particular, the Ising model in a magnetic field is mapped to an eight-vertex model whose weights configurations combine both even and odd number of incoming and outcoming arrows at a vertex. We have studied the Yang-Baxter algebra for such mixed eight-vertex model when the weights are invariant under arrows reversing. We find that while the Lax operator lie on the same elliptic curve of the even eight-vertex model the respective R-matrix can not be presented in terms of the difference of two rapidities. We also argue that the spin-vertex equivalence may be used to imbed an integrable spin model in the realm of the quantum inverse scattering framework. As an example, we show how to determine the R-matrix of the 27-vertex model equivalent to a three-state spin model devised by Fateev and Zamolodchikov.
Abstract In this paper we investigate the integrability properties of a two-state vertex model on the square lattice whose microstates at a vertex always have an odd number of incoming or outcoming arrows. This model was named the odd eight-vertex model by Wu and Kunz (2004 J. Stat. Phys . 116 67) to distinguish it from the well-known eight-vertex model possessing an even number of arrow orientations at each vertex. When the energy weights are invariant under arrow inversion we show that the integrable manifold of the odd eight-vertex model coincides with that of the even eight-vertex model. The form of the -matrix for the odd eight-vertex model is however not the same as that of the respective Lax operator. Altogether we find that these eight-vertex models give rise to a generic sheaf of -matrices satisfying the Yang–Baxter equations resembling intertwiner relations associated to equidimensional representations.
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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.