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Journal of Physics Conference Series

Integrable chains with Jordan - Schwinger representations

R. Kirschner

Published Jan 28, 2013• 7 citations
Publisher

Abstract

The restiction to the class of Jordan - Schwinger representations of sℓ ( n + 1) results in simple relations for the L matrices and in explicit expressions for the general Yang-Baxter operators as products of two parameter permutation operators. Limits are studied which are related to the finite dimensional representations and to degenerate Yangians. The analogy to the sℓ (2) case leads to analogous forms of global spin chain operators.

DOI
10.1088/1742-6596/411/1/012018

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The restiction to the class of Jordan - Schwinger representations of sℓ ( n + 1) results in simple relations for the L matrices and in explicit expressions for the general Yang-Baxter operators as products of two parameter permutation operators. Limits are studied which are related to the finite dimensional representations and to degenerate Yangians. The analogy to the sℓ (2) case leads to analogous forms of global spin chain operators.
In this thesis we theoretically investigate the close relation between spin chains and Yangians. First, we investigate the Haldane-Shastry model, which symmetry algebra is a representation of the Yangian. We show this in full detail for the spin-1/2 model. We also give an argument why a spin-1 chain with Yangian symmetry cannot exist. \nAfter that, we turn to integrable spin chains which are constructed using the quantum inverse scattering method. Using this method, an integrable spin chain can be constructed from each so called R-matrix, which is a solution of the Yang-Baxter equation. Our main result is the construction of an R-matrix that is invariant under the adjoint representation of SU(n), which is found using an intertwining operator for the corresponding Yangian representation. The integrable\nspin chain that corresponds to this R-matrix is non-Hermitian, making its physical interpretation unclear. These results have been published recently as a preprint under arXiv:1606.02516. \nFinally, integrable spin chains with defects are studied. We present the Hamiltonian for the Heisenberg model and the Babujian-Takhtajan model with defects, such that these models keep there integrability while staying Lie algebra invariant.
We develop an approach for constructing the Baxter -operators for generic sl ( N ) spin chains. The key element of our approach is the possibility of representing a solution of the Yang–Baxter equation in the factorized form. We prove that such a representation holds for a generic sl ( N ) invariant -operator and find the explicit expression for the factorizing operators. Taking trace of monodromy matrices constructed of the factorizing operators one defines a family of commuting (Baxter) operators on the quantum space of the model. We show that a generic transfer matrix factorizes into the product of N Baxter -operators and discuss an application of this representation for a derivation of functional relations for transfer matrices.