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arXiv:hep-th/9910253 · Journal of Physics A Mathematical and General

Solution of the quantum inverse problem

Frank Göhmann, V. E. Korepin

Published Feb 9, 2000• 126 citations
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Abstract

We derive a formula that expresses the local spin and field operators of fundamental graded models in terms of the elements of the monodromy matrix. This formula is a quantum analogue of the classical inverse scattering transform. It applies to fundamental spin chains, such as the XYZ chain, and to a number of important exactly solvable models of strongly correlated electrons, such as the supersymmetric t - J model or the EKS model.

DOI
10.1088/0305-4470/33/6/308

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We derive a formula that expresses the local spin and field operators of fundamental graded models in terms of the elements of the monodromy matrix. This formula is a quantum analogue of the classical inverse scattering transform. It applies to fundamental spin chains, such as the XYZ chain, and to a number of important exactly solvable models of strongly correlated electrons, such as the supersymmetric t - J model or the EKS model.
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For the integrable spin- s XXZ chain we express explicitly any given spin- s form factor in terms of a sum over the scalar products of the spin-1/2 fundamental operators of the algebraic Bethe ansatz. Here, they are given by the operator-valued matrix elements of the monodromy matrix of the spin-1/2 XXZ spin chain. We derive the reduction formula of higher-spin form factors by the fusion method in detail. We call an arbitrary matrix element of a local operator between two Bethe eigenstates a form factor of the operator in this paper. We thus revise the derivation of the higher-spin XXZ form factors given in a previous paper. The revised method has several interesting applications in mathematical physics. For instance, we express the spin- s XXZ correlation function of an arbitrary entry at zero temperature in terms of a sum of multiple integrals.
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 study integrable models solvable by the nested algebraic Bethe ansatz and possessing the GL(3)-invariant R-matrix.We consider a composite model where the total monodromy matrix of the model is presented as a product of two partial monodromy matrices.Assuming that the last ones can be expanded into series with respect to the inverse spectral parameter we calculate matrix elements of the local operators in the basis of the transfer matrix eigenstates.We obtain determinant representations for these matrix elements.Thus, we solve the inverse scattering problem in a weak sense.