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Journal of Physics A Mathematical and General

Diagonalisation of GL(N) invariant transfer matrices and quantum N-wave system (Lee model)

P. P. Kulish, Nicolai Reshetikhin

Published Nov 11, 1983• 253 citations
Publisher

Abstract

The algebraic Bethe ansatz is constructed for the GL(N) invariant transfer matrices with arbitrary GL(N) spin. For this purpose the notion of vacuum subspace is introduced. It is shown that the GL(N) magnet can be interpreted as an integrable discrete approximation of vector or matrix nonlinear Schrodinger models or of the quantum N-wave system.

DOI
10.1088/0305-4470/16/16/001

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The algebraic Bethe ansatz is constructed for the GL(N) invariant transfer matrices with arbitrary GL(N) spin. For this purpose the notion of vacuum subspace is introduced. It is shown that the GL(N) magnet can be interpreted as an integrable discrete approximation of vector or matrix nonlinear Schrodinger models or of the quantum N-wave system.
We present an 'algebraic treatment' of the analytical Bethe ansatz. For this purpose, we introduce abstract monodromy and transfer matrices which provide an algebraic framework for the analytical Bethe ansatz. It allows us to deal with a generic gl ( )-spin chain possessing on each site an arbitrary gl ( )-representation. For open spin chains, we use the classification of the reflection matrices to treat all the diagonal boundary cases. As a result, we obtain the Bethe equations in their full generality for closed and open spin chains. The classifications of finite-dimensional irreducible representations for the Yangian (closed spin chains) and for the reflection algebras (open spin chains) are directly linked to the calculation of the transfer matrix eigenvalues. The local Hamiltonian associated with a given integrable spin chain needs to be computed case by case. A general formula is still lacking at that point. As examples, we recover the usual Hamiltonian and Bethe equations for closed and open spin chains; we treat the alternating spin chains and the closed spin chain with an impurity. We also compute the Hamiltonian and Bethe equations for the open spin chain model used in the context of large- N QCD.
The algebraic Bethe ansatz can be performed rather abstractly for whole classes of models sharing the same R -matrix, the only prerequisite being the existence of an appropriate pseudo vacuum state. Here we perform the algebraic Bethe ansatz for all models with 9 × 9, rational, gl(1|2) invariant R -matrix and all three possibilities of choosing the grading. Our Bethe ansatz solution applies, for instance, to the supersymmetric t – J model, the supersymmetric U model and a number of interesting impurity models. It may be extended to obtain the quantum transfer matrix spectrum for this class of models. The properties of a specific model enter the Bethe ansatz solution (i.e. the expression for the transfer matrix eigenvalue and the Bethe ansatz equations) through the three pseudo vacuum eigenvalues of the diagonal elements of the monodromy matrix which in this context are called the parameters of the model.
We have recently constructed a large class of open quantum spin chains which have quantum-algebra symmetry and which are integrable. We show here that these models can be exactly solved using a generalization of the analytical Bethe Ansatz (BA) method. In particular, we determine in this way the spectrum of the transfer matrices of the Uq[su(2)]-invariant spin chains associated with A1(1) and A2(2) in the fundamental representation. The quantum-algebra invariance of these models plays an essential role in obtaining these results. The BA equations for these open chains are “doubled” with respect to the BA equations for the corresponding closed chains.
This chapter addresses the long-standing challenge of constructing the Bethe wave function in large but finite volumes via the diagonalization of the transfer matrix within the spin representation. While the coordinate Bethe ansatz provides effective tools for specific models, it becomes increasingly cumbersome in systems with internal degrees of freedom, such as the Lieb–Liniger model. The algebraic Bethe ansatz offers a more general and systematic framework, particularly well suited for handling nested structures. Spin chains serve as a natural setting for this formalism, offering both mathematical richness and physical relevance as models of interacting quantum spins. Starting from the Heisenberg spin chain, the chapter introduces the algebraic structure underpinning the method, including the R -matrix formulation, quantum group symmetries, and the construction of the transfer matrix. To connect the algebraic formalism with thermodynamic behavior, the chapter explores the string hypothesis, which organizes solutions to the Bethe equations into regular complex patterns. This leads naturally to the Bethe–Takahashi equations, which govern the thermodynamic limit of integrable spin chains. These tools enable a tractable analysis of excited states and physical observables, establishing a foundation for applying the algebraic Bethe ansatz to a broader class of quantum integrable systems.
We present an "algebraic treatment" of the analytical Bethe Ansatz. For this purpose, we introduce abstract monodromy and transfer matrices which provide an algebraic framework for the analytical Bethe Ansatz. It allows us to deal with a generic gl(n)-spin chain possessing on each site an arbitrary gl(n)-representation. For open spin chains, we use the classification of the reflection matrices to treat all the diagonal boundary cases. As a result, we obtain the Bethe equations in their full generality for closed and open spin chains. The classifications of finite dimensional irreducible representations for the Yangian (closed spin chains) and for the reflection algebras (open spin chains) are directly linked to the calculation of the transfer matrix eigenvalues. As examples, we recover the usual closed and open spin chains, we treat the alternating spin chains and the closed spin chain with impurity.
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