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Progress of Theoretical Physics

Quantum Inverse Scattering Method and Yang-Baxter Relation for Integrable Spin Systems

Kiyoshi Sogo, Miki Wadati

Published Jul 1, 1982• 32 citations
Publisher

Abstract

The Lax pair for spin 1/2 Heisenberg XYZ model is presented. This gives a basis for the quantum inverse scattering method. A generalization of the Yang-Baxter relation to arbitrary spin is presented through the factorized S-matrix theory. An equivalent vertex model is solved exactly by the quantum inverse scattering method.
DOI
10.1143/ptp.68.85

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The quantum inverse scattering method relates the Bethe Ansatz to the theory of classical completely integrable differential equations. These are sometimes called soliton equations. The modern way to solve them is called the classical inverse scattering method. In a sense this is a nonlinear generalization of the Fourier transform. In this Part the quantum inverse scattering method is expounded. The main statements of the classical inverse scattering method necessary for quantization are given in Chapter V where the Lax representation is introduced. The Hamiltonian structure of integrable models is also discussed along with the infinite number of integrals of motion. The most convenient method of analyzing the Hamiltonian structure relies on the classical r -matrix. Some concrete models will be considered. Chapter VI is devoted in particular to the quantum inverse scattering method. The R-matrix, which is the main object of this method, is introduced. The Yang-Baxter equation for the R-matrix is discussed. The main statements of the method are given and a number of examples are presented. The algebraic formulation of the Bethe Ansatz, one of the main achievements of the quantum inverse scattering method, is presented in Chapter VII. The notion of the determinant of the transition matrix in the quantum case is introduced in this chapter. (This is closely related to the concept of the antipode in quantum groups.)
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.
In the case of two-component models, all possible solutions for factorization equations of S-matrix are obtained. These solutions give a classification of solvable vertex models. It is found that Yang-Baxter relation is more general than the factorization equation. The six vertex model by Yang, Yang and Sutherland is derived not from the factorization equation but from Yang-Baxter relation. Thus it is concluded that Yang-Baxter relation lies at the basis of all the known solvable two component models. Furthermore, spin Hamiltonians corresponding to our vertex models are presented in explicit forms.
Introduction to the quantum inverse scattering method The quantum inverse scattering method is the modern algebraic theory of exactly solvable quantum systems. It arose [404, 410, 411] as an attempt to carry over the concepts of the inverse scattering method for classical non-linear evolution equations [2, 134] into quantum mechanics. As a result, our understanding of both the theory of integrable partial differential equations and the theory of exactly solvable quantum systems changed, and the algebraic roots of the exact solvability became apparent. These roots originate from the Yang-Baxter equation and its classical counterpart. Before turning to our actual subject, which is the application of the quantum inverse scattering method to the Hubbard model, we give a brief general introduction. We shall limit our exposition basically to the material which is needed later for the understanding of the algebraic structure of the Hubbard model. The reader who is interested in the general scope of the method and in the history of its development is referred to the excellent books and review articles [131, 270, 276, 277, 407]. Integrability As a motivation for the definition of the Yang-Baxter algebra in the following subsection we shall first recall the concept of integrability in classical mechanics. Then, by considering the elementary example of the harmonic oscillator, we shall see that this concept does not directly apply to quantum mechanical systems and needs to be extended.