On quantum group symmetry and Bethe ansatz for the asymmetric twin spin chain with integrable boundary

Abstract

Motivated by a study of the crossing symmetry of the `gemini' representation of the affine Hecke algebra we give a construction for crossing tensor space representations of ordinary Hecke algebras. These representations build solutions to the Yang--Baxter equation satisfying the crossing condition (that is, integrable quantum spin chains). We show that every crossing representation of the Temperley--Lieb algebra appears in this construction, and in particular that this construction builds new representations. We extend these to new representations of the blob algebra, which build new solutions to the Boundary Yang--Baxter equation (i.e. open spin chains with integrable boundary conditions). We prove that the open spin chain Hamiltonian derived from Sklyanin's commuting transfer matrix using such a solution can always be expressed as the representation of an element of the blob algebra, and determine this element. We determine the representation theory (irreducible content) of the new representations and hence show that all such Hamiltonians have the same spectrum up to multiplicity, for any given value of the algebraic boundary parameter. (A corollary is that our models have the same spectrum as the open XXZ chain with nondiagonal boundary -- despite differing from this model in having reference states.) Using this multiplicity data, and other ideas, we investigate the underlying quantum group symmetry of the new Hamiltonians. We derive the form of the spectrum and the Bethe ansatz equations.

arXiv comment43 pages, multiple figures

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• Introduction to Quantum Integrability
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• Scalar products of the open XYZ chain with non-diagonal boundary terms
• Determinant representations of scalar products for the open XXZ chain with non-diagonal boundary terms
• Determinant formula for the partition function of the six-vertex model with a non-diagonal reflecting end
• On boundary fusion and functional relations in the Baxterized affine Hecke algebra
• Generalized T-Q relations and the open spin-s XXZ chain with nondiagonal boundary terms
• A representation basis for the quantum integrable spin chain associated with the su(3) algebra
• A Tensor Space Representation of the Symplectic Blob Algebra
• Drinfeld twist and symmetric Bethe vectors of the open XYZ chain with non-diagonal boundary terms
• Junction type representations of the Temperley-Lieb algebra and associated symmetries