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arXiv:2407.11066

Quantum inverse scattering for the 20-vertex model up to Dynkin automorphism: 3D Poisson structure, triangular height functions, weak integrability

Pete Rigas

Published Jul 12, 2024• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2407.11066

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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 compute the action-angle variables for a Hamiltonian flow of the inhomogeneous six-vertex model, from a formulation introduced in a 2022 work due to Keating, Reshetikhin, and Sridhar, hence confirming a conjecture of the authors as to whether the Hamiltonian flow is integrable. To demonstrate that such an integrability property of the Hamiltonian holds from the action-angle variables, we make use of previous methods for studying Hamiltonian systems, implemented by Faddeev and Takhtajan, in which it was shown that integrability of a Hamiltonian system holds for the nonlinear Schrodinger's equation by computing action-angle variables from the Poisson bracket, which is connected to the analysis of entries of the monodromy and transfer matrices. For the inhomogeneous six-vertex model, an approach for computing the action-angle variables is possible through formulating several relations between entries of the quantum monodromy, and transfer, matrices, which can be not only be further examined from the structure of LLL operators, but also from computing several Poisson brackets parameterized from entries of the 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.
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.
Abstract The quantum inverse scattering method is a scheme for solving integrable models in 1 + 1 dimensions, building on an R-matrix that satisfies the Yang–Baxter equation (YBE) and in terms of which one constructs a commuting family of transfer matrices. In the standard formulation, this R-matrix acts on a tensor product of vector spaces. Here, we relax this tensorial property and develop a framework for describing and analysing integrable models based on planar algebras, allowing non-separable R-operators satisfying generalised YBEs. We also re-evaluate the notion of integrals of motion and characterise when an (algebraic) transfer operator is polynomial in a single integral of motion. We refer to such models as polynomially integrable. In an eight-vertex model, we demonstrate that the corresponding transfer operator is polynomial in the natural Hamiltonian. In the Temperley–Lieb loop model with loop fugacity β ∈ C , we likewise find that, for all but finitely many β-values, the transfer operator is polynomial in the usual Hamiltonian element of the Temperley–Lieb algebra T L n ( β ) , at least for n ⩽ 17 . Moreover, we find that this model admits a second canonical Hamiltonian, and that this Hamiltonian also acts as a polynomial integrability generator for small n and all but finitely many β-values.
Inspired by the integrable structures appearing in weakly coupled planar N=4 super Yang-Mills theory, we study Q-operators and Yangian invariants of rational integrable spin chains. We review the quantum inverse scattering method along with the Yang-Baxter equation which is the key relation in this systematic approach to study integrable models. Our main interest concerns rational integrable spin chains and lattice models. We recall the relation among them and how they can be solved using Bethe ansatz methods incorporating so-called Q-functions. In order to remind the reader how the Yangian emerges in this context, an overview of its so-called RTT-realization is provided. The main part is based on the author's original publications. Firstly, we construct Q-operators whose eigenvalues yield the Q-functions for rational homogeneous spin chains. The Q-operators are introduced as traces over certain monodromies of R-operators. Our construction allows us to derive the hierarchy of commuting Q-operators and the functional relations among them. We study how the nearest-neighbor Hamiltonian and in principle also higher local charges can be extracted from the Q-operators directly. Secondly, we formulate the Yangian invariance condition, also studied in relation to scattering amplitudes of N=4 super Yang-Mills theory, in the RTT-realization. We find that Yangian invariants can be interpreted as special eigenvectors of certain inhomogeneous spin chains. This allows us to apply the algebraic Bethe ansatz and derive the corresponding Bethe equations that are relevant to construct the invariants. We examine the connection between the Yangian invariant spin chain eigenstates whose components can be understood as partition functions of certain 2d lattice models and tree-level scattering amplitudes of the four-dimensional gauge theory. Finally, we conclude and discuss some future directions.
A general approach to the computation of exact correlation functions of lattice quantum integrable models proposed recently is reviewed.It is based on the resolution of the so-called quantum inverse scattering problem, namely, the reconstruction of local operators in terms of the (operator) entries of the quantum monodromy matrix obeying a Yang-Baxter algebra.The application of this method to the XXZ Heisenberg spin-1 2 chain in a magnetic field leads to multiple integral representations of its n-point correlation functions.For zero magnetic field, it gives a proof, in both the massless and massive (anti-ferromagnetic) regimes, of the formulas obtained from the q-deformed KZ equations (massless regime) and the representation theory of the quantum affine algebra U q ( ŝl 2 ) together with the corner transfer matrix approach (massive regime).