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arXiv:cond-mat/9502090 · Europhysics Letters (EPL)

Integrable models of coupled Heisenberg chains

Holger Frahm, Claus Rödenbeck

Published Jan 1, 1996• 29 citations
arXivHTMLPDF

Abstract

We show that the solutions of the Yang-Baxter equation invariant under the action of the Yangian Y ( sl 2 ) lead to inhomogeneous vertex models. Starting from a four-dimensional representation of Y ( sl 2 ), we obtain an integrable family of coupled Heisenberg spin-½ chains. Some thermodynamical properties of this model are studied by means of the algebraic Bethe ansatz.

DOI
10.1209/epl/i1996-00302-7

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We show that the solutions of the Yang-Baxter equation invariant under the action of the Yangian Y ( sl 2 ) lead to inhomogeneous vertex models. Starting from a four-dimensional representation of Y ( sl 2 ), we obtain an integrable family of coupled Heisenberg spin-½ chains. Some thermodynamical properties of this model are studied by means of the algebraic Bethe ansatz.
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A crash course on the Yang–Baxter equation and its applications in the statistical mechanics of lattice vertex models is presented. Using the simple example of a one-dimensional lattice gas, the basic terminology and standard mathematical procedures of statistical mechanics are illustrated. The coproduct notation is introduced via a discussion of the two-dimensional dimer model, and the transfer matrix formulation is elaborated further. The algebraic Bethe Ansatz (ABA) is introduced in the context of the ice model, leading to the Yang–Baxter equations. The motivation for introducing the ABA is developed, emphasizing the similarity to the quantum oscillator problem, in features such as the generation of “excited states”, and the determination of the eigenvalue spectrum. Finally, these methods from the theory of lattice models are applied to the case of a quantum many-body problem, the Heisenberg chain. From the condition that the monodromy matrices satisfy the Yang–Baxter equation, a complete set of mutually commuting operators is derived, and the Heisenberg chain problem is completely solved.PACS Nos.: 02.20.Uw, 24.10.Cn, 87.10.Hk, 05.20.–y
A new integrable model which is a variant of the one-dimensional Hubbard model is proposed. The integrability of the model is verified by presenting the associated quantum R-matrix which satisfies the Yang–Baxter equation. We argue that the new model possesses the SO(4) algebra symmetry, which contains a representation of the η-pairing SU(2) algebra and a spin SU(2) algebra. Additionally, the algebraic Bethe ansatz is studied by means of the quantum inverse scattering method. The spectrum of the Hamiltonian, eigenvectors, as well as the Bethe ansatz equations, are discussed.