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arXiv:0911.5592 · Theoretical and Mathematical Physics

Jordanian deformation of the open XXX spin chain

P. P. Kulish, N. Manojlović, Z. Nagy

Published May 1, 2010• 11 citations
arXivHTMLPDF
DOI
10.1007/s11232-010-0047-x

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Jordanian deformation of the non-compact and sl2 \mathfrak{s}{\mathfrak{l}}_2 sl2​-invariant XXX−1/2 spin-chain
A bstract Using a Drinfeld twist of Jordanian type, we construct a deformation of the non-compact and sl2 \mathfrak{s}{\mathfrak{l}}_2 sl2​ s l 2 -invariant XXX −1/2 spin-chain. Before the deformation, the seed model can be understood as a sector of the psu \mathfrak{psu} psu psu (2, 2|4)-invariant spin-chain encoding the spectral problem of N \mathcal{N} N N = 4 super Yang-Mills at one loop in the planar limit. The deformation gives rise to interesting features because, while being integrable, the Hamiltonian is non-hermitian and non-diagonalisable, so that it only admits a Jordan decomposition. Moreover, the eigenvalues of the deformed Hamiltonian coincide with those of the original undeformed spin-chain. We use explicit examples as well as the techniques of the coordinate and of the algebraic Bethe ansatz to discuss the construction of the (generalised) eigenvectors of the deformed model. We also show that the deformed spin-chain is equivalent to an undeformed one with twisted boundary conditions, and that it may be derived from a scaling limit of the non-compact U q ( sl2 \mathfrak{s}{\mathfrak{l}}_2 sl2​ s l 2 )-invariant XXZ −1/2 spin-chain.
s l 2 Gaudin model with jordanian twist is studied. This system can be obtained as the semiclassical limit of the XXX spin chain deformed by the jordanian twist. The appropriate creation operators that yield the Bethe states of the Gaudin model and consequently its spectrum are defined. Their commutation relations with the generators of the corresponding loop algebra as well as with the generating function of integrals of motion are given. The inner products and norms of Bethe states and the relation to the solutions of the Knizhnik-Zamolodchikov equations are discussed.