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arXiv:2412.04515 · International Journal of Modern Physics B

Intertwining vectors and Boltzmann weight matrices of a solid-on-solid model from the 20-vertex model

Pete Rigas

Published Aug 14, 2026• 0 citations
arXivHTMLPDF

Abstract

We initiate a new study on the correspondence between the 20-vertex model and a SOS (Solid-on-Solid) model. In comparison to two previous works of the author which have characterized properties of the transfer and quantum monodromy, matrices of the 20-vertex model from the perspective of the quantum inverse scattering method [T. Ozawa, Commun. Math. Phys. 139, 479 (1991), https://doi.org/10.1007/BF02101876; L. Pauling, J. Am. Chem. Soc. 57, 2680 (1935), https://doi.org/10.1021/JA01315A102], in addition to the structure of nonlocal correlations [H. Boos et al., J. Phys. A: Math. Theor. 43, 415208 (2010), https://doi.org/10.1088/1751-8113/43/41/415208], the forthcoming approach is an adaptation of a study on the Rational 7-vertex model [P. V. Antonenko and P. A. Valinevich, J. Math. Sci. 284, 582 (2024), https://doi.org/10.1007/s10958-024-07374-9]. For both the 7-vertex and 20-vertex models nonlinearity through the Schrödinger’s equation plays a fundamental role not only in quantitatively describing well-posed initial value problems, but also through the intertwining operations underlying Yang–Baxter equations for the Rational and Universal, R-Matrices. Given a decomposition of the Universal R-Matrix obtained in [H. Boos et al., Rev. Math. Phys. 26, 143005 (2014), https://doi.org/10.1142/S0129055X14300052; H. Boos et al., J. Phys. A: Math. Theor. 43, 415208 (2010), https://doi.org/10.1088/1751-8113/43/41/415208] we formalize a novel application of the quantum inverse scattering method which is related to entanglement, and other behaviors of nonlinear Schrödinger’s equations [N. N. Akhmediev, V. M. Eleonskii and N. E. Kulagin, Theor. Math. Phys. 72, 809 (1987), https://doi.org/10.1007/BF01017105; L. D. Carr, C. W. Clark and W. P. Reinhardt, Phys. Rev. A 62, 063611 (2000), https://doi.org/10.1103/PhysRevA.62.063610; D. Keating, N. Reshetikhin and A. Sridhar, Commun. Math. Phys. 391, 1181 (2022), https://doi.org/10.1007/s00220-022-04334-9; K. Nakanishi and T. Ozawa, NoDEA Nonlinear Differ. Equ. Appl. 9, 45 (2002), https://doi.org/10.1007/s00030-002-8118-9; H. Niggemann, A. Klümper and J. Zittartz, Z. Phys. B 104, 103 (1997), https://doi.org/10.1007/s002570050425; L. Pauling, J. Am. Chem. Soc. 57, 2680 (1935), https://doi.org/10.1021/JA01315A102; N. Reshetikhin and A. Sridhar, Commun. Math. Phys. 356, 535 (2017), https://doi.org/10.1007/s00220-017-2983-x; G. Wang et al., Appl. Math. Lett. 172 (2026), https://doi.org/10.1016/j.aml.2025.109720; J. Yang, Phys. Rev. E 98, 042202 (2018), https://doi.org/10.1103/PhysRevE.98.042202; V. E. Zakharov and S. V. Manakov, Theor. Math. Phys. 19, 551 (1974), https://doi.org/10.1007/BF01035568]. Within the setting of the Rational 7-vertex model, Antonenko and Valinevich demonstrated that the intertwining vectors can be used to transform the R-Matrix of a vertex model into the Boltzmann weight matrix of an SOS model. To further develop perspectives on classes of higher-dimensional SOS models, by leveraging previous computations with L-operators due to the author [L. Pauling, J. Am. Chem. Soc. 57, 2680 (1935), https://doi.org/10.1021/JA01315A102], and by also manipulating q-exponentials, and other related factors from a factorization of the Universal R-Matrix due to Boos et al. [J. Phys. A: Math. Theor. 43, 415208 (2010), https://doi.org/10.1088/1751-8113/43/41/415208], analogs of intertwining vectors for the 20-vertex model can be obtained. In comparison to the system of equations that have been obtained for the Rational 7-vertex model, the system that one obtains for the 20-vertex model consists of 9 equations for each entry of the R-Matrix. Furthermore, from the fact that the 20-vertex model contains more possible configurations in the sample space than the Rational 7-vertex model does, the Boltzmann weight matrix of the corresponding SOS model asymptotically depends upon several contributions, a few of which include tensor products of basis elements and spectral parameters over the triangular lattice.

DOI
10.1142/s0217979226502413

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We initiate a new study on the correspondence between the 20-vertex model and a SOS (Solid-on-Solid) model. In comparison to two previous works of the author which have characterized properties of the transfer and quantum monodromy, matrices of the 20-vertex model from the perspective of the quantum inverse scattering method [T. Ozawa, Commun. Math. Phys. 139, 479 (1991), https://doi.org/10.1007/BF02101876; L. Pauling, J. Am. Chem. Soc. 57, 2680 (1935), https://doi.org/10.1021/JA01315A102], in addition to the structure of nonlocal correlations [H. Boos et al., J. Phys. A: Math. Theor. 43, 415208 (2010), https://doi.org/10.1088/1751-8113/43/41/415208], the forthcoming approach is an adaptation of a study on the Rational 7-vertex model [P. V. Antonenko and P. A. Valinevich, J. Math. Sci. 284, 582 (2024), https://doi.org/10.1007/s10958-024-07374-9]. For both the 7-vertex and 20-vertex models nonlinearity through the Schrödinger’s equation plays a fundamental role not only in quantitatively describing well-posed initial value problems, but also through the intertwining operations underlying Yang–Baxter equations for the Rational and Universal, R-Matrices. Given a decomposition of the Universal R-Matrix obtained in [H. Boos et al., Rev. Math. Phys. 26, 143005 (2014), https://doi.org/10.1142/S0129055X14300052; H. Boos et al., J. Phys. A: Math. Theor. 43, 415208 (2010), https://doi.org/10.1088/1751-8113/43/41/415208] we formalize a novel application of the quantum inverse scattering method which is related to entanglement, and other behaviors of nonlinear Schrödinger’s equations [N. N. Akhmediev, V. M. Eleonskii and N. E. Kulagin, Theor. Math. Phys. 72, 809 (1987), https://doi.org/10.1007/BF01017105; L. D. Carr, C. W. Clark and W. P. Reinhardt, Phys. Rev. A 62, 063611 (2000), https://doi.org/10.1103/PhysRevA.62.063610; D. Keating, N. Reshetikhin and A. Sridhar, Commun. Math. Phys. 391, 1181 (2022), https://doi.org/10.1007/s00220-022-04334-9; K. Nakanishi and T. Ozawa, NoDEA Nonlinear Differ. Equ. Appl. 9, 45 (2002), https://doi.org/10.1007/s00030-002-8118-9; H. Niggemann, A. Klümper and J. Zittartz, Z. Phys. B 104, 103 (1997), https://doi.org/10.1007/s002570050425; L. Pauling, J. Am. Chem. Soc. 57, 2680 (1935), https://doi.org/10.1021/JA01315A102; N. Reshetikhin and A. Sridhar, Commun. Math. Phys. 356, 535 (2017), https://doi.org/10.1007/s00220-017-2983-x; G. Wang et al., Appl. Math. Lett. 172 (2026), https://doi.org/10.1016/j.aml.2025.109720; J. Yang, Phys. Rev. E 98, 042202 (2018), https://doi.org/10.1103/PhysRevE.98.042202; V. E. Zakharov and S. V. Manakov, Theor. Math. Phys. 19, 551 (1974), https://doi.org/10.1007/BF01035568]. Within the setting of the Rational 7-vertex model, Antonenko and Valinevich demonstrated that the intertwining vectors can be used to transform the R-Matrix of a vertex model into the Boltzmann weight matrix of an SOS model. To further develop perspectives on classes of higher-dimensional SOS models, by leveraging previous computations with L-operators due to the author [L. Pauling, J. Am. Chem. Soc. 57, 2680 (1935), https://doi.org/10.1021/JA01315A102], and by also manipulating q-exponentials, and other related factors from a factorization of the Universal R-Matrix due to Boos et al. [J. Phys. A: Math. Theor. 43, 415208 (2010), https://doi.org/10.1088/1751-8113/43/41/415208], analogs of intertwining vectors for the 20-vertex model can be obtained. In comparison to the system of equations that have been obtained for the Rational 7-vertex model, the system that one obtains for the 20-vertex model consists of 9 equations for each entry of the R-Matrix. Furthermore, from the fact that the 20-vertex model contains more possible configurations in the sample space than the Rational 7-vertex model does, the Boltzmann weight matrix of the corresponding SOS model asymptotically depends upon several contributions, a few of which include tensor products of basis elements and spectral parameters over the triangular lattice.
We initiate a novel application of the quantum inverse scattering method for the 20-vertex model, building upon seminal work from Faddeev and Takhtajan on the study of Hamiltonian systems. In comparison to a previous work of the author in late 2023 which characterized integrability of a Hamiltonian flow for the 6-vertex model from integrability of inhomogeneous limit shapes, formalized in a work of Keating, Reshetikhin and Sridhar, notions similar to those of integrability can be realized for the 20-vertex model by studying new classes of higher-dimensional L-operators. Such L-operators provided by Boos and colleagues have algebraic, combinatorial, and geometric, qualities, all of which impact leading order approximations of correlations, products of L-operators, the transfer matrix, and the quantum monodromy matrix.
We implement the quantum inverse scattering method for the 4-vertex model. In comparison to previous works of the author which examined the 6-vertex, and 20-vertex, models, the 4-vertex model exhibits different characteristics, ranging from L-operators expressed in terms of projectors and Pauli matrices to algebraic and combinatorial properties, including Poisson structure and boxed plane partitions. With far fewer computations with an L-operator provided for the 4-vertex model by Bogoliubov in 2007, in comparison to those for L-operators of the 6, and 20, vertex models, from lower order expansions of the transfer matrix we derive a system of relations from the structure of operators that can be leveraged for studying characteristics of the higher-spin XXX chain in the weak finite volume limit. In comparison to quantum inverse scattering methods for the 6, and 20, vertex models which can be used to further study integrability, and exact solvability, an adaptation of such an approach for the 4-vertex model can be used to approximate, asymptotically in the weak finite volume limit, sixteen brackets which generate the Poisson structure. From explicit relations for operators of the 4-vertex transfer matrix, we conclude by discussing corresponding aspects of the Yang-Baxter algebra, which is closely related to the operators obtained from products of L-operators for approximating the transfer, and quantum monodromy, matrices. The structure of computations from L-operators of the 4-vertex model directly transfers to L-operators of the higher-spin XXX chain, revealing a similar structure of another Yang-Baxter algebra of interest.
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.
The construction of auxiliary matrices for the six-vertex model at a root of unity is investigated from a quantum group theoretic point of view. Employing the concept of intertwiners associated with the quantum loop algebra U q ( 2 ) at q N = 1, a three-parameter family of auxiliary matrices is constructed. The elements of this family satisfy a functional relation with the transfer matrix allowing one to solve the eigenvalue problem of the model and to derive the Bethe ansatz equations. This functional relation is obtained from the decomposition of a tensor product of evaluation representations and involves auxiliary matrices with different parameters. Because of this dependence on additional parameters, the auxiliary matrices break in general the finite symmetries of the six-vertex model, such as spin-reversal or spin-conservation. More importantly, they also lift the extra degeneracies of the transfer matrix due to the loop symmetry present at rational coupling values. The extra parameters in the auxiliary matrices are shown to be directly related to the elements in the enlarged centre Z of the algebra U q ( 2 ) at q N = 1. This connection provides a geometric interpretation of the enhanced symmetry of the six-vertex model at rational coupling. The parameters labelling the auxiliary matrices can be interpreted as coordinates on a hypersurface Spec Z ⊂ ℂ 4 which remains invariant under the action of an infinite-dimensional group G of analytic transformations, called the quantum coadjoint action.
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.