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arXiv:solv-int/9903003 · Journal of Physics A Mathematical and General

New integrable models from fusion

Z. Maassarani

Published Jan 1, 1999• 1 citation
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Abstract

Integrable multistate or multiflavour/colour models were recently introduced. They are generalizations of models corresponding to the defining representations of the quantum algebras. Here I show that a similar generalization is possible for all higher-dimensional representations. The R -matrices and the Hamiltonians of these models are constructed by fusion. The sl (2) case is treated in some detail and the spin-0 and spin-1 matrices are obtained in explicit forms. This provides in particular a generalization of the Fateev-Zamolodchikov Hamiltonian.

DOI
10.1088/0305-4470/32/27/310

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Integrable multistate or multiflavour/colour models were recently introduced. They are generalizations of models corresponding to the defining representations of the quantum algebras. Here I show that a similar generalization is possible for all higher-dimensional representations. The R -matrices and the Hamiltonians of these models are constructed by fusion. The sl (2) case is treated in some detail and the spin-0 and spin-1 matrices are obtained in explicit forms. This provides in particular a generalization of the Fateev-Zamolodchikov Hamiltonian.
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We define multi-colour generalizations of braid-monoid algebras and present explicit matrix representations which are related to two-dimensional exactly solvable lattice models of statistical mechanics. In particular, we show that the two-colour braid-monoid algebra describes the Yang-Baxter algebra of the critical dilute A-D-E models which were recently introduced by Warnaar, Nienhuis and Seaton(1992) as well as by Roche(1992). These and other solvable models related to dense and dilute loop models are discussed in detail and it is shown that the solvability is a direct consequence of the algebraic structure. It is conjectured that the Baxterization of general multi-colour braid-monoid algebras will lead to the construction of further solvable lattice models.
A class of recently introduced su (n) "free-fermion" models has recently been used to construct generalized Hubbard models. I derive an algebra defining the "free-fermion" models and give new classes of solutions. I then introduce a conjugation matrix and give a new and simple proof of the corresponding decorated Yang–Baxter equation. This provides the algebraic tools required to couple in an integrable way two copies of free-fermion models. Complete integrability of the resulting Hubbard-like models is shown by exhibiting their L and R matrices. Local symmetries of the models are discussed. The diagonalization of the free-fermion models is carried out using the algebraic Bethe Ansatz.
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The authors have generalized Sklyanin's approach of constructing open integrable quantum spin chains to the case of PT-invariant R matrices. They formulate a fusion procedure for such chains. In particular, they show that the fused transfer matrix can be expressed in terms of products of the original transfer matrix and products of certain quantum determinants which can be explicitly evaluated. Applications of these results include constructing open integrable higher-spin chains, as well as obtaining functional equations for transfer-matrix eigenvalues, which may be solved by an analytical Bethe ansatz.
Models generalizing the su(2) XX spin-chain were recently introduced. These XXC models also have an underlying su(2) structure. Their construction method is shown to generalize to the chains based on the fundamental representations of the Am Lie algebras. Integrability of the new models is shown in the context of the quantum inverse scattering method. Their R-matrix is found and shown to yield a representation of the Hecke algebra. The diagonalization of the transfer matrices is carried out using the algebraic Bethe Ansatz. I comment on eventual generalizations and possible links to reaction-diffusion processes.