Defining relations on the Hamiltonians of $XXX$ and $XXZ$ $R$-matrices and new integrable spin-orbital chains

P. N. Bibikov

Defining relations on the Hamiltonians of XXX and XXZ R-matrices and new integrable spin-orbital chains

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arxiv nlin/0703017 v1 pith:XE4KKJCZ submitted 2007-03-10 nlin.SI

Defining relations on the Hamiltonians of XXX and XXZ R-matrices and new integrable spin-orbital chains

P. N. Bibikov This is my paper

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Several complete systems of integrability conditions on a spin chain Hamiltonian density matrix are presented. The corresponding formulas for $R$-matrices are also given. The latter is expressed via the local Hamiltonian density in the form similar to spin one half $XXX$ and $XXZ$ models. The result is applied to the problem of integrability of $SU(2)\times SU(2)$- and $SU(2)\times U(1)$-invariant spin-orbital chains (the Kugel-Homskii-Inagaki model). The eight new integrable cases are found. One of them corresponds to the Temperley-Lieb algebra, the others three to the algebra associated with the $XXX$, $XXZ$ and graded $XXZ$ models. The last two $R$-matrices are also presented.

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